‰lie Cartan (1869-1951)

Translations of

MATH EMATICALMONO G "PHS

Volume 1.23

Elle Cartan.[1869-1951)

M. A. AkivisB, A. Rosenfeld

American Mathematical society'

Translations of Mathematical Monographs 123

Elie Cartan(1869-1951)

ELIE CARTANApril 9, 1869-May 6, 1951

Translations of

MATHEMATICALMONOGRAPHS

Volume 123

Elie Cartan(1869-1951)

M. A. AkivisB. A. Rosenfeld

American Mathematical SocietyProvidence, Rhode Island

IV

3JIH KAPTAH (1869-1951)

M. A. AICHBHcE. A. Po3eH$eni6

Translated by V. V. Goldberg from an original Russian manuscriptTranslation edited by Simeon Ivanov

1991 Mathematics Subject Classification. Primary 01A70;Secondary 01A60, 01A55.

ABSTRACT. The scientific biography of one of the greatest mathematicians of the 20th cen-tury, Elie Cartan (1869-1951), is presented, as well as the development of Cartan's ideas bymathematicians of the following generations.

Photo credits: p. iv-Centre National de la Recherche Scientifique; pp. 2, 3, 9, 10, 17, 19, 25,27, 28, 29-Henri Cartan; p. 31-Department of Geometry, Kazan University, Tatarstan, Russia

Library of Congress Cataloging-in-Publication Data

Akivis, M. A. (Maks Aizikovich)[Elie Kartan (1869-1951). English]Elie Cartan (1869-1951)/M. A. Akivis, B. A. Rosenfeld; [translated from the Russian by

V. V. Goldberg; translation edited by Simeon Ivanov].p. cm.-(Translations of mathematical monographs, ISSN 0065-92 82; v. 12 3)Includes bibliographical references.ISBN 0-8218-4587-X (acid-free)1. Cartan, Elie, 1869-1951. 2. Mathematicians-France-Biography. 3. Lie groups.

4. Geometry, Differential. I. Rozenfel'd, B. A. (Boris Abramovich) II. Title. III. SeriesQA29.C355A6613 1993 93-69325 16.3' 76' 092-dc20 CIP

Copyright Q 1993 by the American Mathematical Society. All rights reserved.The American Mathematical Society retains all rights

except those granted to the United States Government.Printed in the United States of America

The paper used in this book is acid-free and falls within the uide1inesestablished to ensure permanence and durability.

Information on Copying and Reprinting can be found at the back of this volume.This publication was typeset using AMS-TEX,

the American Mathematical Society's TEX macro system.

109876 5432 1 9796959493

Contents

Preface xi

Chapter 1.§ 1.1.

The Life and Work of E. CartanParents' home

§ 1.2. Student at a school and a lycee 2

§ 1.3. University student 4§ 1.4. Doctor of Science 6

§ 1.5. Professor 8

§ 1.6. Academician 17

§ 1.7. The Cartan family 24

§ 1.8. Cartan and the mathematicians of the world 27

Chapter 2. Lie Groups and Algebras 33

§2.1. Groups 33

§2.2. Lie groups and Lie algebras 37

§2.3. Killing's paper 42

§2.4. Cartan's thesis 45

§2.5. Roots of the classical simple Lie groups 46

§2.6. Isomorphisms of complex simple Lie groups 51

§2.7. Roots of exceptional complex simple Lie groups 51

§2.8. The Cartan matrices 53

§2.9. The Weyl groups 55

§2.10. The Weyl affine groups 60

§2.11. Associative and alternative algebras 63

§2.12. Cartan's works on algebras 67

§2.13. Linear representations of simple Lie groups 69

§2.14. Real simple Lie groups 73

§2.15. Isomorphisms of real simple Lie groups 78

§2.16. Reductive and quasireductive Lie groups 82

§2.17. Simple Chevalley groups 84

§2.18. Quasigroups and loops 85

vii

Viii CONTENTS

Chapter 3. Projective Spaces and Projective Metrics 87§3. 1. Real spaces 87§3.2. Complex spaces 93

§3.3. Quaternion spaces 95§3.4. Octave planes 96§3.5. Degenerate geometries 97§3.6. Equivalent geometries 101

§3.7. Multidimensional generalizations of the Hesse transferprinciple 1 07

§3.8. Fundamental elements 1 09

§3.9. The duality and triality principles 113§3.1 0. Spaces over algebras with zero divisors 116§3.11. Spaces over tensor products of algebras 118§3.12. Degenerate geometries over algebras 121

§3.13. Finite geometries 123

Chapter 4. Lie Pseudogroups and Pfaffian Equations 125§4.1 . Lie pseudogroups 125§4.2. The Kac-Moody algebras 127

§4.3. Pfaffian equations 129§4.4. Completely integrable Pfaffian systems 1 30

§4.5. Pfaffian systems in involution 1 32

§4.6. The algebra of exterior forms 134§4.7. Application of the theory of systems in involution 135§4.8. Multiple integrals, integral invariants, and integral

geometry 136§4.9. Differential forms and the Betti numbers 139

§4.10. New methods in the theory of partial differential equations 142

Chapter 5. The Method of Moving Frames and DifferentialG t 145eome ry

§5.1. Moving trihedra of Frenet and Darboux 145

§5.2. Moving tetrahedra and pentaspheres of Demoulin 147§5.3. Cartan's moving frames 148§5.4. The derivational formulas 150§5.5. The structure equations 152§5.6. Applications of the method of moving frames 153§5.7. Some geometric examples 1 54

§5.8. Multidimensional manifolds in Euclidean space 158§5.9. Minimal manifolds 160

§5.10. "Isotropic surfaces" 162

§5.11. Deformation and projective theory of multidimensionalmanifolds 1 66

CONTENTS ix

§5.12. Invariant normalization of manifolds 170§5.13. "Pseudo-conformal geometry of hypersurfaces" 174

Chapter 6. Riemannian Manifolds. Symmetric Spaces 177§6.1. Riemannian manifolds 177§6.2. Pseudo-Riemannian manifolds 181

§6.3. Parallel displacement of vectors 181

§6.4. Riemannian geometry in an orthogonal frame 183§6.5. The problem of embedding a Riemannian manifold into a

Euclidean space 184§6.6. Riemannian manifolds satisfying "the axiom of plane" 185§6.7. Symmetric Riemannian spaces 186§6.8. Hermitian spaces as symmetric spaces 191§6.9. Elements of symmetry 193

§6.10. The isotropy groups and orbits 196§6.11. Absolutes of symmetric spaces 198§6.12. Geometry of the Cartan subgroups 199§6.13. The Cartan submanifolds of symmetric spaces 200§6.14. Antipodal manifolds of symmetric spaces 201§6.15. Orthogonal systems of functions on symmetric spaces 202§6.16. Unitary representations of noncompact Lie groups 204§6.17. The topology of symmetric spaces 207§6.18. Homological algebra 209

Chapter 7. Generalized Spaces 211§ 7.1. "Affine connections" and Weyl's "metric manifolds" 211§7.2. Spaces with affine connection 212§7.3. Spaces with a Euclidean, isotropic, and metric connection 215§7.4. Affine connections in Lie groups and symmetric spaces

with an affine connection 216§7.5. Spaces with a projective connection 219§7.6. Spaces with a conformal connection 220§7.7. Spaces with a symplectic connection 221§7.8. The relativity theory and the unified field theory 222§7.9. Finsler spaces 223

§7.10. Metric spaces based on the notion of area 225§7.11. Generalized spaces over algebras 226§7.12. The equivalence problem and G-structures 228§7.13. Multidimensional webs 231

Conclusion 235

Dates of Cartan's Life and Activities 239

List of Publications of Elie Cartan 241

x CONTENTS

Appendix A. Rapport sur les Travaux de M. Cartan, by H. Poincare 263

Appendix B. Sur une degenerescence de la geometrie euclidienne,by E. Cartan 273

Appendix C. Allocution de M. Elie Cartan 275

Appendix D. The Influence of France in the Development ofMathematics 281

Bibliography 303

Preface

The year 1989 marked the 120th birthday of Elie Cartan (1869-19 51),one of the greatest mathematicians of the 20th century, and 1991 markedthe 40th anniversary of his death. The publication of this book is timed tothese two dates. The book is written by two geometers working in two dif-ferent branches of geometry whose foundations were created by Cartan. Themathematical heritage of Cartan is very wide, and there is no possibility ofdescribing all mathematical discoveries made by him, at least not in a bookof relatively modest size. Because of this, the authors pose for themselves amuch more modest problem to describe and evaluate only the most impor-tant of these discoveries. Of course, the authors are only able to describe indetail Cartan's results connected with those branches of geometry in whichthe authors are experts.

The book consists of seven chapters. In Chapter 1 the outline of E. Car-tan's life is given, and in Chapters 2-7 his main achievements are described,namely, in the theory of Lie groups and algebras; in applications of these the-ories to geometry; in the theory of Lie pseudogroups; in the theory of Pfaffiandifferential equations and its application to geometry by means of Cartan'smethod of moving frames; in the geometry of Riemannian manifolds; and, inparticular, in the theory of symmetric spaces created by Cartan; in the theoryof spaces of affine connection and other generalized spaces. In the same chap-ters the main routes of the development of Cartan's ideas by mathematiciansof the following generations are given. At the end of the book a chronology ofthe main events of E. Cartan's life and a list of his works are presented. Thereferences to Cartan's works are given by numerals without Cartan's name,and the other references by first letters of the names of the authors, with nu-merals added for multiple references. The appendices contain H. Poincare'sreference on Cartan's work (1912); Cartan's paper On a degeneracy of Eu-clidean geometry, which was omitted in his cEuvre Compl etes; his speech atthe meeting in the Sorbonne on the occasion of his 70th birthday (1939);and his lecture, The influence of France in the development of Mathematics(1940). Chapters 1-3 and 6 were written by B. A. Rosenfeld, Chapters 5 and7 were written by M. A. Akivis, and Chapter 4 was written by both authors.

Xi

xii PREFACE

The authors express their cordial gratitude to Henri Cartan, a son of E. Car-tan, who himself is one of the greatest mathematicians of this century, forproviding numerous facts for a biography of his father and for pictures fur-nished by him.

Moscow, Russia M. A. Akivis

University Park, PA, U.S.A. B. A. Rosenfeld

CHAPTER I

The Life and Work of E. Cartan

§11, Parents' home

Elie Joseph Cartan was born on April 9, 1869, in the village of Dolomieulocated between Lyons and Grenoble in the Departement Isere in the south-eastern part of France. The Isere river, after which the Departement wasnamed, has a very fast current, and several hydroelectric power plants arenow located along it. They supply the industrial district of Grenoble, thecenter of the Departement Isere, with electric power. The first hydroelectricpower plant on this river was built by Aristide Berges (1833-1904), the ownerof a paper mill in Lancey, in 1869, the year of Cartan's birth.

The Departement Isere is in the central part of the historic French provinceDauphine, which was a patrimonial estate of a dauphin, the eldest son of theking (the crown prince). Dauphine stretched from the Alps to the Rhone, theleft tributary of which is the Isere. Originally the capital of Dauphine wasthe town of Vienne, which is located on the Rhone just south of Lyon. Lateron, the capital was transferred to Grenoble.

During Cartan's childhood, Dolomieu had about 2,500 inhabitants. Pre-sently the population is about 1,600. Long ago the village was a center ofsilkworm breeding and silk spinning. Figure 1.1 (next page) shows SquareChamp-de-Mars (Martial Field) in Dolomieu (presently Place the Cartan)and the house where Cartan spent his childhood (he lived there from 1872 to1879). Cartan's family home is the second from the right. Dolomieu was alsothe home of the famous geologist Deodat Guy Silvain Gratet deDolomieu (1750-1801), one of the sons of Francois de Gratet, Marquis deDolomieu. Deodat Dolomieu was an academician and a participant in thefamous Egyptian campaign of Napoleon. He immortalized his own name andthe name of his home village through his discovery of the mineral dolomite.

Cartan's ancestors were peasants. His great-grandfather Benoit Cartan(1779-1854) was a farmer. Cartan's grandfather, whose first name was alsoBenoit (1801-1854), was a miller. Cartan's father Joseph (1837-1917) wasborn in the village of Saint Victor de Morestel, which is 13 kilometers fromDolomieu. After he married Anne Cottaz (1841-1927) the family settled inDolomieu, where Anne had lived. Joseph Cartan was the village blacksmith.Cartan recalled that his childhood had passed under "blows of the anvil,

i

2 1. THE LIFE AND WORK OF E. CARTAN

FIGURE 1.1

which started every morning from dawn", and that "his mother, during thoserare minutes when she was free from taking care of the children and thehouse, was working with a spinning-wheel" [189, p. 51].

Figure 1.2 shows a picture of Cartan's parents, taken approximately in1890. Cartan recollected later that his parents were "unpretentious peasantswho during their long lives demonstrated to their children an example ofjoyful accomplished work and courageous acceptance of burdens" [189, p.51].

the was the second oldest of the four Cartan children. His elder sisterJeanne-Marie (1867-1931) was a dressmaker, and his younger brother Leon(1872-1956) became a blacksmith, working in his father's smithy. Cartan'syounger sister Anna (1878-1923), not without the influence of her brother,graduated from L'Ecole Normale Superieure (the Superior Normal School)for girls and taught mathematics at different lycees (state secondary schools)for girls. She was the author of two textbooks for these lycees: Arithmetic andGeometry, for first-year students, and Geometry, for second-year students.Both textbooks were reprinted many times.

§ 1.2. Student at a school and a lycee

the Cartan began his education in an elementary school in Dolomieu.He later spoke very warmly of his teachers, M. Collomb and especially M.Dupuis, who gave one hundred boys a primary education, the importance ofwhich Cartan could appraise at its true worth only much later. the was the

§1.2. STUDENT AT A SCHOOL AND A LYCEE 3

FIGURE 1.2

best student in the school. M. Dupuis recollected: "Elie Cartan was a shystudent, but an unusual light of great intellect was shining in his eyes, and thiswas combined with an excellent memory. There was no question that couldbe a problem for him: he understood everything that was taught in class evenbefore the teacher finished his explanation." Cartan remembered that in theschool he "could, without a moment's 'hesitation, list all subprefectures ineach Departement" of France as well as the grammatical fine points "of therules of past participles" [189, p. 52].

the Cartan was of small stature and did not possess the physical strengthof his father and brother. That he became one of the most famous scientistsof France was due to the fact that the school where he was studying wasvisited by Antonin Dubost (1844-1921). Dubost was a remarkable person-ality in many respects. He was a republican journalist during the empire ofNapoleon III. After France became the Third Republic, he became a prefectof the Departement Orne, which is to the west of Paris. Later he moved tothe Departement Isere and was its representative in 1880-1897. During thisperiod he was the Minister of Justice in the cabinet of Grenobler CasimirPerier. In 1897 Dubost was elected to the French Senate and was the Pres-ident of the Senate from 1906 to 1920. Cartan described him as having "a


strong optimism, based on a strong belief in progress, in the power of in-tellect and in the hope of discovering truth and doing good". Later Cartannoted: "His visit changed my whole life" [189, p. 52]. Impressed by theunusual abilities of Cartan, Dubost recommended that he participate in acontest for a scholarship in a lycee. Cartan prepared for this contest underthe supervision of M. Dupuis.

At that time in France there were two kinds of secondary schools: colleges,belonging to local self-governments, and lycees, belonging to the Ministryof Public Education. (After restoration of royal power in France in 1615the lycees were renamed "royal colleges"; the name "lycee" was returned tothem only after the 1848 Revolution. The 1959 reform renamed the colleges"municipal lycees".) Young Cartan passed the contest exams in Grenoble,the main city of the Departement Isere. He remembered that he "passedthese competitions, which turned out to be not so difficult, without particularnervousness" [189, p. 52]. The brilliant success of Elie in this contest wasa source of special pride of M. Dupuis, who supervised his preparation forthe contest. Thanks to the continuing support of M. Dubost, who retaineda fatherly interest in Cartan's scientific career and achievements throughouthis life, Cartan received a full scholarship in the College of Vienne (Vienneis the ancient capital of the province Dauphine'). Elie was 10 years old atthat time.

Cartan spent the next ten years in colleges and lycees far from home. Hisfirst five years (1880-1885) were at the College of Vienne. After this hisscholarship was transferred to the Lycee of Grenoble, where he was a studentfrom 1885-1887. The teaching in colleges and lycees at that time to a consid-erable extent consisted of a medieval curricula of "trivial" and "quadrivial"sciences. The first group, the so-called trivium (three-path), were formed byGrammar, Rhetoric, and Philosophy, and the second group, the quadrivium(four-path), was formed by Mathematical Sciences. Originally they had beencomprised of Arithmetic, Geometry, Astronomy and Music. Cartan com-pleted the study of the trivium in the Grenoble college (after passing Rhetoricand Philosophy). To study mathematical sciences, in 1887 he moved to Paris,to the Janson-de-Sailly Lycee ("Grand Lycee"), where he was a student un-til 1888. With special warmth Cartan remembered two professors from thislcee: Salomon Bloch, who taught "elementary mathematics", and E. Lacour,who taught "special mathematics". One of his classmates in this lycee wasJean-Baptiste Perrin (18 70-1942 ), who later became one of the most famousphysicists in France. A close friendship between Cartan and Perrin, whichbegan during these years, continued throughout their lives.

§1.3. University student

After graduation from the Lycee Janson-de-Sailly, Cartan decided to be-come a mathematician. At that time in Paris there were three educationalinstitutions with mathematical majors: the Sorbonne (University of Paris),

§ 1.3. UNIVERSITY STUDENT 5

which was founded by Robert de Sorbon in 1253; l'Ecole Polytechnique (thePolytechnic School); and l'Ecole Normale Superieure (the Superior NormalSchool). The latter two had been founded during the French Revolution. ThePolytechnic School, where one would study for three years (later changed totwo years), gave a mathematical and general technical education, after whichone was supposed to study a specialization in practical higher technical insti-tutions. L'Ecole Normale Superieure, where, according to the Convent's de-cision, "the art of teaching, not science itself" should be taught, was a higherpedagogical educational institution in which one would study for three years.

Cartan chose l'Ecole Normale Superieure and enrolled in 1888. Of theprofessors whose lectures he attended in this school and the Sorbonne, Cartanthought most highly of "a mathematical giant, Henri Poincare, whose lectureswere flying over our heads" [ 189, p. 54). Poincare (1854-1912), about whomCartan wrote that "there was no branch of mathematics which was not underhis influence" [ 18 9, p. 54], was a mathematician, physicist, astronomer, andphilosopher who created in 1883 the theory of automorphic functions, whichis closely connected with group theory and hyperbolic geometry. He attractedCartan's attention to geometric applications of group theory.

Listing professors who influenced him, Cartan indicated Charles Hermite(1822-1901), a specialist in analysis, algebra, and number theory, who in-troduced "Hermitian forms" for problems in number theory forms whichplay an important role in geometry; Jules Tannery (1848-19 10), one of thefounders of French set theory; Gaston Darboux (1842-1917), one of thefounders of the method of moving frames, who is also known by his workin the theory of differential equations; Paul Appell (1855-1930), a special-ist in analysis and mechanics; Emile Picard (1856-1941), a specialist in thetheory of differential equations who widely used geometric and group theorymethods in his work; and Edouard Goursat (1858-1936), a specialist in thetheory of differential equations, who also was interested in transformationgroups. (In 1889 Goursat wrote a paper on finite groups of motions of afour-dimensional Euclidean space that are generated by reflections.)

L'Ecole Normale Superieure at that time was closely connected with theNorwegian mathematician Sophus Lie (1842-1899), who from 1886 to 1889was head of the Department of Geometry in Leipzig University. In 1888-1889, upon the recommendation of Tannery and Darboux, several Frenchmathematicians, including Ernest Vessiot (1865-1952) and Arthur Tresse(1868-1958 ), studied under Lie in Leipzig. Picard was also very much inter-ested in Lie's papers. After Vessiot returned to Paris, he and Picard publishedpapers on applications of continuous groups to the problem of integrability ofdifferential equations. These papers were a further development of Lie's re-search. In the investigations of Lie, Picard, and Vessiot, the so-called solvableor integrable Lie groups played a special role. This gave rise to the problemof listing all so-called simple Lie groups, since the presence of simple sub-groups in a group indicates that it is nonsolvable. Cartan's interest in these


problems, to a considerable extent, can be explained by the influence of hisclassmate Tresse.

After graduation from l'Ecole Normale Superieure in 1891, Cartan wasdrafted into the French army, where he served one year and attained therank of sergeant.

§ 1.4. Doctor of Science

While Elie Cartan served in the army, his friend Arthur Tresse was astudent of Sophus Lie at Leipzig University. When Tresse returned fromLeipzig, he informed Cartan that W. Killing's paper, The structure of the fi-nite continuous groups of transformations [Kil2], had been published in theLeipzig journal Mathematische Annal en, in 18 88-1890. In this paper impor-tant results on the classification of simple Lie groups were obtained.

Tresse also told Cartan that, after publication, this paper was found tocontain incorrect statements concerning nilpotent groups ("groups of zerorank"), and that the mathematician F. Engel from Leipzig, who was workingjointly with Klein and Lie, assigned the task of correcting Killing's inaccu-racies to his student Carl Arthur Umlauf (1866-?). Umlauf accomplishedthe mathematical objective assigned to him and defended his doctoral dis-sertation, on the structure of the finite continuous groups of transformations,especially groups of zero rank [Um] (1891). Tresse advised Cartan to inves-tigate whether the main part of Killing's paper also contained inaccuracies.From this came the subject of Cartan's thesis.

Cartan worked on this subject for two years (1892-1894) in Paris. Asan excellent student of l'Ecole Normale Superieure, he was a recipient ofthe grant ("bourse") of the Peccot Foundation, founded in 1885 to supporttalented young scientists of l'Ecole Normale Superieure. (The Peccot Foun-dation is still in existence.)

Following Tresse's advice, Cartan studied the Killing paper and becameconvinced that the principal parts of this work were correct and that thenew method, which was used by Killing and which was based on the studyof "roots" of simple Lie groups, is an exceptionally powerful method forstudying this kind of group. Simultaneously, Cartan discovered a number ofinaccuracies and incomplete statements. A rigorous classification of simpleLie groups constituted the main part of Cartan's doctoral dissertation.

In 1892, at the invitation of Darboux and Tannery, Lie came to Paris andspent six months there. However, the main purpose of Lie's visit to Paris wasto meet Cartan. (This information was given by his son Henri in a letter toone of the authors of this book.) Lie and Cartan had discussions on severaloccasions. Cartan recollected that Lie was interested "with a great good willin the research of young French mathematicians" [201, Engl. tr., p. 265]and that at that time Lie "could often be seen with them around the table atthe Cafe de la Source, on the Boulevard Saint-Michel; it was not unusual forthe white marble table top to be covered with formulas in pencil, which the

§ 1.4. DOCTOR OF SCIENCE 7

master had written to illustrate the exposition of his ideas" [201, Engl. tr.,p. 265]. In the same article Cartan gave his impression of Lie's personality:"Sophus Lie was of tall stature and had the classic Nordic appearance. Afull blond beard framed his face and his gray-blue eyes sparkled behind theeyeglasses. He gave the impression of unusual physical strength. One alwaysimmediately felt at ease with him, certain beforehand of his sincerity and hisloyalty." He also evaluated Lie's influence on mathematics: "Posterity willsee in him only the genius who created the theory of transformation groups,and we French shall never be able to forget the ties, which bind us to himand which make his memory dear to us." [201, Engl. tr., p. 267].

In 1893 Cartan published his first scientific papers two notes, The struc-ture of simple finite continuous groups [ 1 ] and The structure of finite con ti nu-ous groups [2] in Comptes Rendus des Seances de 1'Academie des Sciences(Paris). They were presented for publication by Picard. In these notes Car-tan's results on simple Lie groups were presented briefly. The details weregiven in the paper The structure of finite groups of transformations [3], pub-lished in German in Mitteilungen (Communications) of University of Leipzigand recommended for publication by Lie. These results comprised Cartan'sdoctoral dissertation, The structure of finite continuous groups of transforma-tions, which he defended in 1894 in the Faculty of Sciences in the Sorbonne,and which was published as a book [5].

From 1894 to 1896 Cartan published a few more papers on the theoryof simple Lie groups: the notes On reduction of the group structure to itscanonical form [4] (18 94) and On certain algebraic groups [8) (1895), andthe paper On reduction of the structure of a finite and continuous group oftransformations to its canonical form [9] (1896). In 1894 two papers byCartan [6], [7] were published in which he gave a new proof of Bertrand'stheorem concerning permutation theory. Cartan's proof was based on theproperties of complete permutation groups. In 1896 Cartan's first paper onintegral invariants, The principle of duality and certain multiple integrals intangential and line spaces [10], was published.

Also between 1894 and 1896 Cartan was a lecturer at the University ofMontpellier, one of the oldest scientific centers in France. Then, during theyears 1896 through 1903, he was a lecturer in the Faculty of Sciences at theUniversity of Lyons. At this time he continued his intense scientific work:in 1897 his two notes, On systems of complex numbers [I I ] and On realsystems of complex numbers [12], and, in 1898, his paper, Bilinear groupsand systems of complex numbers [ 13], were published. Following Frenchtradition, by systems of complex numbers Cartan meant associative algebras,also called systems of hypercomplex numbers. In these articles, which arealso connected to the direction of the Lie school, many notions arising in thetheory of Lie groups were generalized for associative algebras. In particular,Cartan gave a classification of both the complex and real simple associativealgebras.


Cartan's Teflections on differential forms, which he dealt with in his pa-pers on Lie groups and in his paper on integral invariants, brought him tothe so-called Pfaff problem the theory of integration of the Pfaffian equa-tions, which are equivalent to a system of partial differential equations. In1899 he published his first paper, On certain differential expressions and thePfaff problem [ 14], on this topic, which was followed by the papers On somequad natures, whose differential element contains arbitrary functions [ 151, Onthe integration of the system of exact equations [ 16], and On the integrationof certain Pfaffian systems of character two [171 (1901); two notes, On theintegration of completely in tegrabl e differential systems [181, [18a), and thenote On the equivalence of differential systems [ 19] (1902).

In 1903, while in Lyons, Cartan married Marie-Louise Bianconi (1880-1950), whose father Pierre-Louis Bianconi (1845-1929), a Corsican by birth,had been Professor of Chemistry in Chambery and was, at that time, "in-specteur d'Academie" in Lyons.

§1.5. Professor

In 1903 Cartan became a professor in the Faculty of Sciences at the Univer-sity of Nancy. Nancy is the capital of the Departement Meurthe-et-Mosellein the part of Lorraine that was not ceded to Germany after the 1870-1871war. He worked in Nancy until 1909. In Nancy, Cartan also taught at theInstitute of Electrical Engineering and Applied Mechanics. While in Nancy,Cartan's sons Henri (1904) and Jean (1906) were born. Figure 1.3 is a 1904portrait of Cartan.

After publishing the note On the structure of infinite groups [20] in 1902,Cartan published two long papers, On the structure of infinite groups of trans-formations [211 [22], in 1904-1905. They were followed by the note Sim-ple continuous infinite groups of transformations [23] (1907) and the pa-per The subgroups of continuous groups of transformations [26] (1908). Inthese articles Cartan studied the structure of infinite-dimensional analoguesof Lie groups. While for Lie groups Cartan used the name "finite continuousgroups", for their infinite-dimensional analogues he used the name "infinitecontinuous groups". Now they are called "Lie pseudogroups". While classi-cal Lie group theory was connected with the theory of systems of ordinarydifferential equations, the theory of Lie pseudogroups turned out to be con-nected with the theory of systems of partial differential equations and withthe theory of systems of Pfaffian equations, which are equivalent to the lat-ter. In these articles the foundations of the method of moving frames andof Cartan's method of exterior forms were laid. Later these methods playeda very important role in the development of differential geometry. In 1908,in the French edition of Encyclopaedia of Mathematical Sciences, Cartanpublished the article Complex numbers. This article was Cartan's extendedFrench translation of the paper The theory of usual and higher complex num-bers by Eduard Study (1862-1930) [Stu I ], from the German edition of this

§1.5. PROFESSOR 9

FIGURE 1.3

Encyclopaedia. Cartan's translation was four times as long as the originalStudy paper.

In 1907-1908 Cartan also published two geometric notes under the sametitle, On the definition of the area of a part of a curved surface [24], [25]. In1909 Cartan moved his family to Paris. In Paris he worked as a lecturer inthe Faculty of Sciences in the Sorbonne and in 1912 became Professor, basedon the reference he received from Poincare [Poi6]. Appendix A contains theEnglish translation of this reference, which was not included in Poincare'sEuvres [Poi]. In 1909 Cartan built a house in his home village Dolomieu

(Figure 1.4, next page), where he regularly spent his vacations. In DolomieuCartan continued his scientific research but sometimes went to the familysmithy and helped his father and brother to blow blacksmith bellows (Figure1.5 shows a 1932 picture of Cartan working in his garden).

In 1910, in the note On isotropic developable surfaces and the method ofmoving frames [29] and in the paper The structure of continuous groups oftransformations and the method of a moving trihedron [31 ], Cartan for the firsttime connected the theory of Lie groups and the theory of Pfaffian equationswith the method of moving frames. This method later became the basicmethod in the geometric work of Cartan.


FIGURE 1.4

FIGURE 1.5

§ 1. 5. PROFESSOR II

In the same year, Cartan's paper The Bfafj?an systems with five variablesand partial differential equations of second order [30] was published. In 1911his papers Variational calculus and certain families of curves [32] and onsystems of partial differential equations of second order with one unknownfunction and three independent variables in involution [331 appeared, and in1912 he published two notes, On the characteristics of certain systems of par-tial differential equations [34] and on groups of contact transformations andnew kinematics [35).

In 1913 Cartan returned to the theory of simple Lie groups and publishedan important paper titled Projective groups, under which no plane manifoldis invariant [37]. In the same year, his Notes on the addition of forces [36]appeared.

In 1914, in the paper Real simple finite continuous groups [38], Cartansolved the problem of classification of real simple Lie groups. This problemis similar to that which he solved in his thesis for complex Lie groups. Inthe paper Real continuous projective groups, under which no plane manifold isinvariant [39] (1914), he constructed the theory of linear representations ofthese groups. Also in 1914 his notes On the integration of certain systems ofdifferential equations [40] and on certain natural families of curves [411, thepaper On the absolute equivalence of certain systems of differential equationsand on certain families of curves [42], and the popular paper Theory of groups[43] appeared.

In 1915, when Cartan was 46 years old, he was drafted into the Army andserved at the rank of sergeant (the rank that he attained in 1892) in a hospitalset up in the building of l'Ecole Normale Superieure. While he served in thisposition, until the end of World War I, Cartan continued his mathematicalstudies.

In 1915 Cartan's papers, on the integration of certain indefinite systems ofdifferential equations [44] and on Backlund transformations [45], appeared.In the same year, Cartan wrote an extended French translation, Theory ofcontinuous groups and geometry [46], of Gino Fano's (1871-1952) paperContinuous geometric groups. Group theory as a geometric principle of classi-fication [Fan] for the French edition of Encyclopaedia of Mathematical Sci-ences (from the German edition of this Encyclopaedia). However, after thebeginning of World War I, the French edition of this Encyclopaedia, whichwas in the process of publication in Leipzig and Paris, was discontinued, andin 1914 only 21 pages of the paper [46] were published. The complete textof this paper, taken from proofs of 1915, was published only after Cartan'sdeath in his cEuvres Completes [207].

From 1916 to 1918 Cartan studied the theory of deformation of hypersur-faces. At that time he published the papers The deformation of hypersurfacesin the real Euclidean space of n dimensions [47] (1916), The deformation ofhypersurfaces in the real conformal space of n > 5 dimensions [48] (1917),and on certain hypersurfaces in the real conformal space of five dimensions

12 1. THE LIFE AND WORK OF t. CARTAN

[49] (1918). In 1918 Cartan published four notes on three-dimensional man-ifolds of n-dimensional Euclidean space: in the note [50] the general theoryof three-dimensional surfaces of this space is constructed; in the note [50a]Cartan gives the theory of developable surfaces, i.e., surfaces with vanishingcurvature; in [SOb] he develops the theory of surfaces of constant negativecurvature, which he calls "Beltrami manifolds"; and in [SOc] Cartan presentsthe theory of surfaces of constant curvature, which he calls there "Riemannmanifolds". In 1919-1920 Cartan published the paper on the manifolds ofconstant curvature of a Euclidean and non-Euclidean space [51], [521, wherethe results of notes [50]-[SOc] are generalized to p-dimensional surfaces ofboth Euclidean and non-Euclidean spaces. Also appearing in 1920 were thenotes, on the projective deformation of surfaces [53] and On the projectiveapplicability of surfaces [53a], the paper on the projective deformation of sur-faces [541, and Cartan's lecture on the general problem of deformation [55)given at the International Congress of Mathematicians in Strasbourg.

In 1922 Cartan's articles on Einstein's gravitation theory appeared: thepaper on the equations of gravitation of Einstein [56] and the note on ageometric definition of Einstein's energy tensor [57 ]. The research in generalrelativity theory and the attempts to create a unitary* field theory broughtCartan to his theory of " generalized spaces"; in the same year he publishedthe notes on a generalization of the notion of Riemannian curvature [58], ongeneralized spaces and relativity theory [ 59 ], On generalized conformal spacesand the optical universe [60], on the structure equations of generalized spacesand the analytic expression of the Einstein tensor [611, and on a fundamentaltheorem of Weyl in the theory of metric spaces [62]. In 1922 Cartan publishedthe paper On small oscillations of a fluid mass [63] and the book Lectures onintegral invariants [64], in which he summarized his research on integralinvariants and gave applications of this theory to mechanics.

The results of the note [62] were given in detail in the paper on a fun-damental theorem of Weyl [65] (1923). Departing from the ideas of Weyl,who constructed a generalization of Riemannian geometry to create one ofthe first "unitary field theories", Cartan came up with the notions of spaceswith Euclidean connection, metric connection, affine connection, and, later,with conformal and projective connection. The space with Euclidean con-nection differs from the Riemannian manifold in the way that the space withmetric connection differs from the Weyl space namely, by the presence oftorsion. The geometry of spaces with affine, Euclidean, or metric connectionwas presented in the paper On manifolds with an affine connection and gen-eral relativity theory, which appeared in three parts: [66] (1923), [69] (1924),and [80] (1925). In the same paper Cartan considered a space with degen-erate Euclidean geometry, which at present is called "the isotropic space";the generalized isotropic space is called today "the space with an isotropic

*Editor's note. Or unified.

§ 1.5. PROFESSOR 13

connection". The geometry of spaces with conformal connection was pre-sented in the paper [68] (1923), and the geometry of spaces with projectiveconnection was presented in the paper [70] (1924).

In 1923 Cartan also published the paper Non-analytic functions and singu-lar solutions of first order differential equations [67]. In 1924 he published thepaper Recent generalizations of the notion of a space [711 delivered the talksRelativity theory and generalized spaces [72] at the International Philosoph-ical Congress in Naples and Group theory and recent research in differentialgeometry [731 at the International Congress of Mathematicians in Toronto;and published the notes On differential forms in geometry [74], On the affineconnection on surfaces [76], and On the projective connection on surfaces [77].

From 1924 to 1940 Cartan held the University chair in the Department ofHigher Geometry at the Faculty of Sciences at the Sorbonne. From 1917 to1936 Cartan lived with his family in the village Le Chesnay near Versailles,and in 1936 he rented an apartment in a multistory house at 95 BoulevardJourdan in the southern part of Paris, near the square Porte d'Orleans. Cartanlived in this apartment until his death. At present his son Henri and his wifeNicole Weiss live in this apartment.

In Figure 1.6 (next page) a letter by Cartan is reproduced; it is his letter to ayoung mathematician, Olga Taussky (b. 1906), written in Le Chesnay on thestationary of the Department of Higher Geometry at the Faculty of Sciencesof the Sorbonne. In 1936 Taussky, who worked at that time in Austria andwho later moved to England and eventually to the U.S.A., wrote a letter toCartan in which she explained her results on the theory of division algebras.In his response Cartan wrote: "Mademoiselle, thank you for your letter. Yourproof, which relates to the systems of hypercomplex numbers without zerodivisors, is very simple and elegant. It would be a great pleasure for me tomeet you personally in Oslo in July."

In 1925, Cartan published his book Geometry of Riemannian manifolds[84] and gave the talk Holonomy groups of generalized spaces and topology[77] at a session of the Association for the Development of Science in Greno-ble. He also published the following papers: Irreducible tensors and simpleand semisimple linear groups [81 ], on the theory of linear representationsof simple and semisimple Lie groups; The duality principle and the theoryof simple and semisimple groups [82], where he considered analogues of theduality principle in projective geometry based on the bilateral symmetry ofthe systems of simple roots of some simple Lie groups, and also the "trialityprinciple", based on the trilateral symmetry of one of those systems; On themotions depending on two parameters [83]; and also Note on generation offorced oscillations [78], written jointly with his son Henri.

In 1926 Cartan delivered the talk Applications of Riemannian manifoldsand Topology [85] at a session of the Association for the Development ofScience in Lyons, and published the notes On certain differential systems, inwhich the unknowns are Pfaffian forms [86], and On Riemannian manifolds,


ME"

PACULTE DES SCIENCES 1936

GlEOMETRIE SUPIERIEURE_Xc

/7. NU-V--107 a

I

Courtesy of Olga Taussky Todd

FIGURE 1.6

in which parallel translation preserves the curvature [87], where for the firsttime Cartan considered an important class of Riemannian manifolds that helater named "symmetric Riemannian spaces". He also published the papersHolonomy groups of generalized spaces [88], On spheres of three-dimensionalRiemannian manifolds [89], and The axiom of plane and metric differentialgeometry [90] (the latter in the collection of articles "In Memoriam N. I.Lobachevsky" in Kazan), and two notes written jointly with J. A. Schouten:On the geometry of the group-manifold of simple and semi-simple groups [91 ],and On Riemannian geometries admitting an absolute parallelism [92] (inEnglish and Dutch, published in Proceedings of the Amsterdam Academy ofScience). In 1926 and 1927 his papers On a remarkable class of Riemannianmanifolds [93], [94] appeared. In this two-paper series Cartan gave a detailedtreatment of the geometry of symmetric Riemannian spaces.

In 1927, in the paper The geometry of transformation groups [ 101 ], Cartanconstructed the theory of symmetric spaces with affine connection. In the

§ 1. 5. PROFESSOR 15

same year, he published the notes on geodesic lines of spaces of simple groups[96], On the topology of real simple continuous groups [97], On the geodesicdeviation and some related problems [98], On certain remarkable Riemannianforms of geometries with a simple fundamental group [99], and on Rieman-nian forms of geometries with a simple fundamental group [1001 - the papersThe geometry of simple groups [10 3), and Group theory and geometry [ 105];and the important paper on certain remarkable Riemannian forms of geome-tri es with a simple fundamental group [ 107 ], with the same title as the note[99]. These works were devoted to various aspects of the geometry of sym-metric Riemannian spaces. In the same year, the note on curves with zerotorsion and developable surfaces in Riemannian manifolds [95] and the pa-pers On certain arithmetic cycles [ 102] and on the possibility of imbedding aRiemannian manifold into an Euclidean space [104] appeared. In the Bul-letin of Kazan Physics Mathematics Society Cartan published his report onthe Schouten memoir The Erlangen program and the theory of parallel trans-lation. New point of view on foundations of geometry [ 106], which was de-voted to the geometry of "generalized spaces". In Mathematichesky Sbornik,Moscow, he published the paper on a problem of the calculus of variations inplane projective geometry [108). From 1926 to 1927 at the Sorbonne, Cartandelivered a series of lectures, the notes of which were published in 1960 inRussian translation (translated by S. P. Finikov) under the title Riemanniangeometry in an orthogonal frame [108a).

In 1928 Cartan's Lectures on the geometry of Riemannian manifolds [ 114]appeared, in addition to his notes On complete orthogonal systems of func-tions in certain closed Riemannian manifolds [109] (Cartan's term "closed"means "compact"); On closed Riemannian manifolds admitting a transitivecontinuous transformation group [I 10 1, On the Betti numbers of spaces ofclosed groups [ 111 ] (where the algebraic topology of compact Lie groups wasreduced to the algebraic theory of Lie algebras); and the complement [ 113] tothe memoir The geometry of simple groups [ 103]. This complement was de-voted to finite groups of the Euclidean space generated by reflections. In thesame year, Cartan's lecture at the International Congress of Mathematiciansin Toronto, entitled On the ordinary stability of Jacobi ellipsoids [112], waspublished. This lecture was devoted to the development of the well-knownPoincare research on the stable forms of a rotating fluid mass. In 1928 healso delivered the talk On imaginary orthogonal substitutions [115] at a ses-sion of the Association for the Development of Science in La Rochelle. Inthe same year, Cartan gave the talks On a geometric representation of non-holonomic material systems [ 119] and On closed spaces admitting a transitivefinite continuous group [ 120] (i.e., on compact spaces admitting a transitiveLie transformation group) at the International Congress of Mathematiciansin Bologna.

In 1929 Cartan published the papers On the determination of a completeorthogonal system of functions on a closed symmetric Riemannian space [117]


and On the integral invariants of certain closed homogeneous spaces and topo-logical properties of these spaces [ 118]. He also published the paper Closed andopen simple groups and Riemannian geometry [ 116], in which he presented aclassification of noncompact simple Lie groups using his theory of symmetricRiemannian spaces. (This method is much simpler than the method he usedin [38].)

In 1930 the book The theory of finite continuous groups and Analysis situs[128] (analysis situs is the old name for topology) appeared, in addition tohis notes Linear representations of the group of rotations of the sphere [ 1211;The linear representations of closed simple and semisimple groups [122]; twonotes entitled The third fundamental Lie theorem [ 123], [123a]; the noteA historic note on the notion of absolute parallelism [124] (devoted to theapplication of this notion to general relativity theory); and the papers Onlinear representations of closed groups [ 125] and On an equivalence problemand the theory of generalized metric spaces [ 126]. In the same year, Cartantook part in the First Congress of Mathematicians of the U.S.S.R. in Kharkovand gave the talk Projective geometry and Riemannian geometry [ 127). On hisway from Kharkov, Cartan made a stop in Moscow and delivered the courseof lectures The method of moving frames, the theory offinite continuous groupsand generalized spaces [ 144] at the Institute of Mathematics and Mechanicsin Moscow University. This course was published in Russian in 1933 and1962 (translated by S. P. Finikov) and in French in 1935.

In 19 31 the book Lectures on complex projective geometry [ 134] appeared.In this work a detailed investigation is given of symmetric spaces, whosefundamental groups are the group of projective transformations of three-dimensional complex projective space or its subgroups. In the same year,Cartan published the paper Absolute parallelism and unitary field theory [ 130],devoted to results he had obtained in 1920 and rediscovered by Einstein in1928. Also in 1931, the expository paper Euclidean geometry and Rieman-nian geometry [1291 and the papers On the theory of systems in involutionand its application to relativity theory [ 131 ] and On the evolvents of a ruledsurface [132] appeared, and Cartan gave the talk The fundamental group ofthe geometry of oriented spheres [ 133].

In 1931 Cartan also published a survey of his mathematical works (187],which was later republished with supplements in a collection of his articles[204] and in the complete collections of his papers [207], [209].

During the twenty years after Cartan defended his doctoral dissertation,his ideas were not developed further by other mathematicians. The situationchanged in the beginning of the 1920s when Hermann Weyl (1885-1955)became interested in Cartan's works. In 1924-1925 Weyl obtained impor-tant results in the theory of simple Lie groups. These results were developedfurther in 1933 by Bart el Leendert van der Waerden (b. 1903). On the otherhand, Cartan's papers on the geometry of "generalized spaces" were closelyconnected with the papers of Weyl and Jan Arnoldus Schouten (1883-1971)

§1.6. ACADEMICIAN 17

FIGURE 1.7

on the geometry of spaces with affine connection, which appeared respec-tively in 1918 and 1921. That Cartan was isolated during the two decadesafter receiving his doctoral degree is due to his extreme modesty and to thefact that in this period the center of attention of French mathematicians wasin set theory and function theory. In the 1930s the mathematical commu-nity in different countries recognized the scientific importance of the direc-tions of Cartan's research. Cartan was elected a Foreign Member of severalAcademies of Sciences: Polish Academy in Cracow (1921), Norway Academyin Oslo (1926), and the famous National Academy dei Lincei ("of lynxes")in Rome (1927). Finally in 1931 Cartan was elected a Member of the ParisAcademy of Sciences. In Figure 1.7 a portrait of Cartan taken in 1931 isreproduced.

§ 1.6. Academician

After being elected as a Member of the Paris Academy of Sciences, Cartanremained a modest man. He continued his intensive research.

In 1932 Cartan published the papers On the group of the hypersphericalgeometry [ 135] and On the pseudo-conformal geometry of hypersurfaces of the


space of two complex variables (which appeared in two parts [136], [136a]);and gave two lectures, On the pseudo-conformal equivalence of two hypersur-faces of the space of two complex variables [ 139] and Symmetric Riemannianspaces [ 13 8 ], at the International Congress of Mathematicians in Zurich. Thefirst four of these were devoted to the geometry of real hypersurfaces of thetwo-dimensional complex space with respect to analytic transformations ofthis space, which form a Lie pseudogroup. In the same year, in the mathe-matical journal of the University of Belgrade, Yugoslavia, Cartan publishedthe paper On the topological properties of complex quadrics [ 13 7 ], in which hestudied globally one of the most important symmetric Riemannian spaces.

In 1933 the book Metric spaces based on the notion of area [ 140] appeared,and in Moscow a translation of his course of lectures [ 144] was published.In the same year, there appeared the notes Newton's kinematics and spaceswith Euclidean connection [ 140a] and Finsler spaces [141 ], and two noteson Finsler spaces: the letter to the Indian geometer Damodar DharmanandKosambi (1907-1966) [141 a] and the note on the paper of the Polish geome-ter Stanislaw Go14 (1902-?) [1 40b).

On October 22, 1933, in Nimes, Cartan gave a speech in memory of one ofhis teachers, G. Darboux [ 188], during festivities accompanying the unveilingof a bust of the scientist.

In 1934 the book Finsler Spaces [ 142], two notes [ 142a), [ l 42b] concerningA. Weil's communications, and the note Tensor calculus in projective geom-etry [1431 appeared. In the same year, Cartan wrote the manuscript Theunitary (field) theory of Einstein-Mayer [ l 43a], which was published only inthe (Euvres Completes of his works [207], and gave three talks at the Inter-national Conference on Tensor Differential Geometry in Moscow, U.S.S.R:Finsler spaces [ 1521, Spaces with projective connection [ 15 3 ], and The topol-ogy of closed (i.e., compact) spaces [ 154]. Figure 1.8 shows Cartan's arrivalin Moscow (Cartan is on the left in the first row; on the right in the samerow is the Chairman of the Conference, V. F. Kagan).

I n 1 93 6 the French text of Cartan's lectures [ 144] appeared, in addition tohis paper Homogeneous bounded domains of the space of n complex variables[14515 the notes on the communications of L. S. Pontryagin on the Betti num-bers of compact simple groups [145a] and of G. Bouligand (b. 1889) [146];and the paper Projective tensor calculus [147] in Matematichesky Sbornik(Moscow). Cartan also gave the talk On a degeneracy of Euclidean geom-etry [147a] at a session of the Association for the Development of Sciencein Nantes, in which he considered the two-dimensional isotropic geometry.The text of this talk, which was not included in the Euvres Completes ([207,209]), is reproduced in Appendix B of this book.

In 1936 the papers The topology of spaces representing Lie groups [ 150]and The geometry of the integral f F(x, y , y' , y") dx [148], and the noteOn the fields of uniform acceleration in restricted relativity (theory) [ 149] ap-peared. In the same year, Cartan delivered the lecture The role ofgroup theory

§ 1.6. ACADEMICIAN 19

FIGURE 1.8

in the evolution of modern geometry [ 151 ] at the International Congress ofMathematicians in Oslo.

In 1937 the following works were published: the books Lectures on thetheory of spaces with a projective connection [ 155] and The theory of finite con-tinuous groups and differential geometry considered by the method of movingframes [157]; the talks [152]-[154] at the International Conference on Ten-sor Differential Geometry in Moscow, U.S.S.R., in Proceedings of the Vectorand Tensor Analysis Seminar at Moscow University (in French and Rus-sian); the papers Extension of tensor calculus to non-affine geometries [ 156];the talk The role of analytic geometry in the evolution of geometry [158] atthe International Philosophical Congress; the papers Groups [ 159], Geometryand groups [ 160 ], and Riemannian geometry and its generalizations [ 161 ] inFrench Encyclopaedia; and the talks The problems of equivalence [ 16 la] andThe structure of infinite groups [ 161 b] in Proceedings of French MathematicalSeminar.

In 1938 the book Lectures on the theory of spinors [164] was published.This book was devoted to the linear representations of the group of orthog-onal matrices, which were discovered by Cartan as far back as 1913. In the1930's they were named the spinor representations because of their appli-cations in physics, which are connected with the spin of an electron. In thesame year the papers Linear representations of Lie groups [ 162], Galois theoryand its generalizations [165], and Families of isoparametric surfaces in spacesof constant curvature [ 166], and his note Generalized spaces and integration


of certain classes of differential equations [ 16 31 were published.In 1939 the papers On remarkable families of isoparametric surfaces in the

spherical spaces [ 167] and The absolute differential calculus in light of recentproblems in Riemannian geometry [169] appeared. He also gave the talkOn certain remarkable families of hypersurfaces [168] at the MathematicalCongress in Liege, Belgium.

On May 18, 1939, at the Sorbonne, a celebration in honor of Cartan's 70thbirthday was held. The chairman of the meeting was the well-known physi-cian and biologist, Academician Gustave Roussy (1874-1948 ), the rector ofthe Sorbonne, who opened the meeting. One of Cartan's teachers, Emile Pi-card, who was at that time the permanent secretary of the French Academyof Sciences, gave a short description of Cartan's works in the theory of Liegroups and the theory of differential equations, Riemannian geometry, andthe theory of "generalized spaces". Picard stressed that Cartan was not only"a pure mathematician, an artist and a poet in the world of numbers andforms", but also that he was dealing with problems of physics, connectedwith relativity theory, and had written a book on spinors.

In his greetings, the Dean of the Faculty of Sciences of the Sorbonne, thefamous geodesist Charles Maurain (1871-1967) recollected all the universi-ties of the world where Cartan had worked or delivered talks or courses. Oneof the founders of the method of moving frames, the Belgian AcademicianAlphonse Demoulin (1869-1947), greeted Cartan on behalf of the scientistsof the entire world. Without mentioning his own name, Demoulin told thatin 1904 "one of Darboux's students" generalized the Darboux method of amoving trihedron for non-Euclidean spaces and noted the further stages ofits development, which brought Cartan in 1910 to the general formulation ofthe method of moving frames.

Arthur Tresse, one of Cartan's former schoolmates in l'Ecole NormaleSuperieure and the honorary general inspector of secondary schools, greetedCartan on behalf of his schoolmates and told how the student Cartan deliv-ered lectures to his schoolmates in l'Ecole Normale Superieure. Tresse alsogreeted the successors of the scientific "dynasty" of E. Cartan, the mathemati-cians Henri and Helene Cartan, and the physicist Louis Cartan. He spokewarmly about the composer Jean Cartan, E. Cartan's deceased son.

The famous physicist and director of the School of Physics and Chemistry,Academician Paul Langevin (1872-1946), described the works of Cartan re-lated to physics. Georges Bruhat (1887-1944), a physicist and the deputydirector of l'Ecole Normale Superieure, noted the many connections Cartanhad with l'Ecole Normale Superieure. Professor of the Sorbonne, mathemati-cian, and Academician Gaston Julia (1893-1965) recalled how he listened toCartan's lectures in l'Ecole Normale Superieure and how he again met Cartanin a hospital, which was set up at the same school during the war. Julia, ayoung officer, was seriously wounded in the face and was undergoing a reha-bilitation in this hospital after a series of successive plastic surgeries in the


hospital Val-de-Grace, where his nose was reconstructed.The President of the French Mathematical Society, Antoine Joseph Henri

Vergne (18 79-1943 ), greeted Cartan as an active member of the Society.Professor of Mathematics at the University of Nancy, Jean Dieudonne (b.1906), saluted Cartan on behalf of young mathematicians. Cartan himselfgave a speech at this meeting. In his speech he gave his recollections of hisentry into science, from which we have previously quoted. He also replied toeach of the speakers who had greeted him. The speeches at this celebrationmeeting were published in a book [Ju], and Cartan's speech can be foundin Appendix C to this book. At the conclusion of this celebration meeting,the orchestra, under conductor Charles Munch (1891-1968), performed thecomposition To the memory of Dante, written by Jean Cartan.

On the date of this celebration the collection of selected Cartan papers,Selecta [204], was published. It includes his works [37], [70], [118], [150],[ 161 a], and [ 162], as well as Cartan's survey of his own works [ 187] and thelist of his mathematical works.

Cartan retired as Professor of the Sorbonne in 1940, after 30 years ofservice in this university. While working in the Sorbonne, Cartan also wasProfessor of Mathematics at the School of Industrial Physics and Chemistryin Paris.

In 1940 the papers On a theorem of J. A. Schouten and W. van der Kulk[ 170], on the linear quaternion groups [ 171 ], and on families of isoparamet-ri c hypersuffaces in the spherical spaces of five and nine dimensions [ 172 ]appeared. In the same year in Moscow, U.S.S.R., in the collection of articlesdevoted to the memory of the Soviet Academician D. A. Grave (1863-1939),Cartan's paper On a class of suffaces similar to the suffaces R and the sur-faces of Jonas [ 180] was published. (This paper was published in France in19444)

In 1940, in the Yugoslavian journal Saturn, the Serbian translation of Car-tan's lecture The influence of France in the development of Mathematics [ 191 ],delivered during his visit to Belgrade in February of 1940, appeared. In 1941this translation was published as a separate booklet. The English translationof this lecture from Serbian, compared with its French text, is given in Ap-pendix D. The introduction to this book was written by the famous Serbianmathematician Mihailo Petrovic (1868-1943), who was Cartan's schoolmatein l'Ecole Normale Superieure. Cartan started this lecture from the worksof F. Viete (1540-1603) and finished the section on Viete with the followingwords: "I should tell you that for quite some time Viete was in contact withone of your (i.e., Yugoslavian) first mathematicians, Marino Ghetaldi (MarinGetaldic) (1556-1626), who was born in Dubrovnik and who, in Paris, inthe year 1600, published one of Viete's last works." [ 191, p. 6]. Later Cartanconsidered works of R. Descartes, B. Pascal, P. Fermat, A. C. Clairaut, J. B.D'Alembert, J. L. Lagrange, P. S. Laplace, A. M. Legendre, G. Monge, J. B.Fourier, A. L. Cauchy, J. V. Poncelet, E. Galois, Ch. Hermite, G. Darboux,


and 11 Poincare, and in passing he mentioned the names of other famousFrench mathematicians.

The last of those mentioned by Cartan was Jacques Herbrand (1908-193 1),who defended his thesis in 1930 and was tragically killed the following sum-mer in an accident in the mountains. His thesis was related to proof theory.He also worked in the theory of fields of classes. His last paper, publishedin 1931, was written jointly with young Claude Chevalley (1909-1984). Car-tan said of Herbrand's works that "his works, mercilessly interrupted by hisearly death, were announcing of a great mathematician, perhaps similar toEvariste Galois". We present here the last paragraph of the lecture, whereCartan expressed his general view on mathematics: "More than any other sci-ence, mathematics develops through a sequence of successive abstractions. Adesire to avoid mistakes forces mathematicians to find and isolate the essenceof the problems and entities considered. Carried to an extreme, this proce-dure justifies the well-known joke according to which a mathematician is ascientist who neither knows what he is talking about or whether whateverhe is indeed talking about exists or not. French mathematicians, however,never enjoyed distancing themselves from reality; they do know that, althoughneeded, logic is by no means crucial. In mathematical activity, like in anyother type of human activity, one should find a balance of values: there is nodoubt that it is important to think correctly, but it is even more important toformulate the right problems. In that respect, one can freely say that Frenchmathematicians not only always knew what they were talking about, but alsohad the right intuition to select the most fundamental problems, those whosesolutions produced the strongest influence on the overall development of sci-ence. ,,

In 1942, the paper On pairs of applicable suf faces with preservation of pf i n-ciple curvatures [ 1761 appeared. In this year Cartan also wrote the paper Theisotropic suffaces of a quadric in a seven-dimensional space [ 1771, which isstill unpublished. H. Cartan sent us the manuscript of this paper. We willconsider this paper in Chapters 3 and 5. In the same year, Cartan wrotethe obituary of the Italian geometer Tullio Levi-Civita (187 3-194 1) and thepaper A centenary: Sophus Lie [201 ], on the occasion of the 100th birthdayof Lie. In the latter he recalled his meetings with the founder of the theoryof Lie groups during Lie's visit to Paris. This paper was only published in1948.

In 1943 the papers on a class of Weyl spaces [ 1781 and Surfaces admittinga given second fundamental form [ 179] and the obituary of the mathematicianGeorges Giraud (1889-1943) [193] were published. In 1944, the paper [ 1801,published in the U.S.S.R. in 1940, was published in France.

In 1945 Cartan published Exterior differential systems and their geomet-ric applications [ 181 ] and the paper On a problem of projective differentialgeometry [ 1821. In the same year, in Moscow, U.S.S.R., he participated incelebrations on the occasion of the 220th anniversary of the founding of the


Academy of Sciences of the U.S.S.R.In 1946 a new edition of Lectures on the geometry of Riemannian man-

ifolds [183] was published. Cartan included in this edition the topics thathe originally intended to include in the second volume of this book. Inparticular, the study of Riemannian manifolds by means of moving frames(published in Russian translation in [ 108a]) was included. In the same year,Cartan published the paper Some remarks on the 28 double tangents of aplane quartic and the 27 straight lines of a cubic surface [ 18 41.

In the first half of 1946, when the President of the Paris Academy ofSciences was sick, Cartan replaced him and chaired the weekly meetings ofthe Academy. During these meetings Cartan informed the audience aboutFrench and foreign members of the Academy who had passed away. Thesecommunications by Cartan were brief but detailed obituaries of eminent sci-entists. During this time Cartan delivered obituaries of the following FrenchAcademicians: the head of French geodesic service General Georges Per-rier (1872-1946) [194]; the metallurgist Leon Alexandre Guillet (1873-1946)[ 196]; the bacteriologist Louis Martin (1864-1946) [198]; the famous physi-cist Langevin [ 199]; and the two foreign members of the Academy: the Amer-ican biologist Thomas Hunt Morgan (1866-1945) (195], the founder of thestudy of genes as carriers of heredity and their localization in chromosomes,and the American pathology anatomist and biologist Simon Flexner (1863-1946) [197]. In the same year, Cartan wrote an article on the occasion of the80th birthday of his old friend E. Vessiot [200] and a note on the occasionof the 200th birthday of Gaspard Monge (1746-1818) (198a]. These publi-cations show that Cartan was very familiar with the status of many sciences,including some that are rather far from mathematics.

In 1947 Cartan published the paper A real anallagmatic space of n dimen-sions [1851 on the geometry of an n-dimensional conformal space, which,following the old French tradition, he named "anallagmatic space", and theshort book The group theory [ 185a].

In 1948, in the collection of articles Great currents of mathematical thought,which was prepared for publication by Francois Le Lionnais (1901-1984)during World War II, the paper [201 ] was published. In the same year, Car-tan published the 30-page book [202] under the same title Gaspard Monge:his life and work as his earlier note [198a]. In this book Cartan publishedfor the first time a series of Monge's letters. This was the reason why thehistorian of science Rene Taton (b. 1915) referred many times to this bookin his research Scientific works of Gaspard Monge [Ta].

In 1949 Cartan published his last two papers: Two theorems of real anallag-matic space of n dimensions [186], relating to the n-dimensional conformalgeometry, and The life and works of Georges Perrier [204]; [194] is a shortobituary of Perrier.

After Cartan retired in 1940, he spent the last years of his life teachingmathematics at the Ecole Normale Superieure for girls.


Elie Cartan died in Paris on May 6, 1951, after a long illness.Immediately after Cartan's death, in the years 1952-1955, a facsimile edi-

tion of his papers [207] was published. It consisted of three parts, and eachpart appeared in two volumes. In Part I, a list of publications by Cartan, hissurvey [ 187] of his own scientific works, and his papers on the theory of Liegroups and the theory of symmetric spaces were reproduced. In Part II, Car-tan's papers on algebra, theory of Lie pseudogroups, and theory of systemsof differential equations were included. Part III contains Cartan's papers ondifferential geometry and some other areas. In particular, in Part III, for thefirst time, a complete text [46] of his extended translation of the paper [Fa]of Fano and the paper [143a] on a unitary field theory of Einstein-Mayerwere published. In 1984 a new edition [209] of Cartan's papers was released.In this edition Parts I and II of the 1952-1955 edition were each placedin one volume; at the end of the second volume of Part III, the papers ofShiing-shen Chern and Claude Chevalley [ChC] and of J. H. C. Whitehead[Wh] were added, in which analyses of Cartan's mathematical work werepresented.

Cartan's best-known students are the French mathematicians Andre Lich-nerowicz (b. 1915) and Charles Ehresmann (1905-1979). Andre Weil (b.1906) was also greatly influenced by Cartan. He dedicated his book Integra-tion on topological groups and its applications [Wel] to Cartan.

In addition to the papers [ChC] and [Wh] on the life and research ofCartan, the articles of Dieudonne [Die], Hodge [Hod], and Saltykov [Sal],and the articles in the memorial collection [ECR] published by Roumanianmathematicians on the occasion of Cartan's 100th birthday, and the researchof Hawkins [Hawl ]-[Haw3], are also worthy of note.

§ 1.7. The Cartan family

Elie Cartan and his wife Marie-Louise had four children: the mathemati-cian Henri, the composer Jean, the physicist Louis, and daughter Helene,who, like her father and eldest brother, became a mathematician. Figure 1.9shows a 1928 picture of the Cartan family: in the first row from left to rightare Louis, Helene, and Jean, and in the second row from left to right are ElieCartan, Henri Cartan, and Marie-Louise Bianconi-Cartan.

Henri Cartan (b. 1904), the eldest son of E. Cartan, became one of themost prominent contemporary mathematicians. He graduated in 1926 froml'Ecole Normale Superieure, the same school from which his father grad-uated. From 1928 to 1929 he taught in the Lycee Malherbe in Caen, thecenter of the Departement Calvados in Normandie. From 1929 to 1931 hewas a lecturer in the Faculty of Sciences at the University of Lille. From1931 to 1935 he was a lecturer and from 1936 to 1940 a professor in theFaculty of Sciences at the University of Strasbourg. From 1940 to 1949 he

§1.7. THE CARTAN FAMILY 25

FIGURE 1.9

was a lecturer in the Faculty of Sciences at the Sorbonne, except for the pe-riod 1945-1947 when he again worked in Strasbourg. From 1949 to 1969 heworked as a professor in the Faculty of Sciences at the Sorbonne. Between1940 and 1965 he also taught in l'Ecole Normale Superieure, and from 1969to 1975 he was a professor of the Faculty of Sciences at the University ofOrsay, a southern suburb of Paris (this university was later renamed the Uni-versity of Paris-Sud). Since 1975, H. Cartan has been a professor emeritusof this university. In 1935 H. Cartan, with Chevalley, Jean Frederic Delsarte(1903-1968), Jean Dieudonne, and Andre Weil (b. 1906) organized a groupwhich wrote the mathematical encyclopaedia Elements of Mathematics un-der the pseudonym Nicolas Bourbaki [Bou]. H. Cartan worked in this groupuntil 1954, when he was 50 years old. This collective work exceptionallyinfluenced the development of mathematics throughout the entire world. In1965 H. Cartan was elected as a corresponding member of the Paris Academyof Sciences, and in 1974 he became a member of this Academy. From1967-1970 H. Cartan was the President of the International MathematicalUnion. In 1980 he and Andrei N. Kolmogorov (1903-1987) were the recip-ients of a very prestigious Wolf Prize in Mathematics. H. Cartan is a foreign


member of many Academies of Sciences, including the London Royal Soci-ety and the National Academy of Sciences, U.S.A. He is also a honoris causaDoctor of Sciences of many universities. He is the author of several well-known books: The elementary theory of analytic functions [CaH2], Homolog-ical algebra (jointly with S. Eilenberg) [CaE], and Differential calculus anddifferential forms [CaH3]. These books have been translated into many lan-guages. He also is the author of numerous papers in the theory of ana-lytic functions, algebraic topology, homological algebra, and potential theory[CaH 1 ]. He has five children: Jean (b. 1936) is an engineer, Frangoise (b.1939) is a teacher of English, Etienne (b. 1941) is a teacher of mathemat-ics, Mireille (b. 1946) is an expert in ecology, and Suzanne (b. 1951) is amanagement expert.

Jean Cartan (1906-1932) was a student of Paul Dukas (1865-1935) in theParis Conservatory from 192 5-1931. After graduation from the Conserva-tory, J. Cartan was a composer: he is the author of two string quartets, asonatina for flute and clarinet, a composition for choir and orchestra includ-ing words from the Lord's Prayer, and a composition for orchestra, To thememory of Dante, mentioned earlier. J. Cartan died of tuberculosis at theage of 25.

Louis Cartan (1909-1943) was a talented physicist who specialized inatomic energy. He was a student of Maurice de Broglie (1875-1960). Heworked in the X-ray physics laboratory in Paris, and after that became a pro-fessor of the Faculty of Sciences at the University of Poitiers. He authored thebook Mass spectrography. Isotopes and their masses [CaL], and, jointly withJean Thibaud and Paul Comparat, the book Some actual technical questionsin nuclear physics. Method of trochoid: positive electrons. Mass spectrography:isotopes. Counters of particles with linear acceleration. Geiger's and Muller'scounters [TCC]. During World War II L. Cartan was an active participantin the Resistance in Poitiers. In 1942 he was arrested by the police of theVichy government and was handed over to the German occupation forces.In February 1943 he was taken to Germany, and in December 1943 he wasdecapitated. The poor parents learned of Louis's death only in May 1945. Atpresent three of Louis's children are alive: Annette (b. 1936) is a teacher ofEnglish, Isabelle (b. 1938) is a teacher of mathematics, and Pierre (b. 1940)is a financier.

The youngest Cartan child, daughter Helene (1917-1952 ), was a math-ematician. She graduated from the Ecole Normale Superieure, as had herfather and brother. She taught in several lycees and authored several mathe-matical papers.

Figure 1.10 shows the grave of E. Cartan, his wife, and their two childrenin a cemetery in Dolomieu. On the vertical tombstone there is the inscription

The CARTAN FAMILY

The inscription on the left half of the horizontal tombstone reads:

§1.8. CARTAN AND THE MATHEMATICIANS OF THE WORLD 27

FIGURE 1.10

Jean CARTAN, December 1, 1906 - March 26, 1932

Marie-Louise BIANCONI, the spouse of the CARTAN

February 18, 1880 - May 21, 1950

Elie CARTAN, April 9, 1869 - May 6, 1951

The right half of the same horizontal tombstone readsHelen CAR TAN, October 12, 1917 - June 7, 1952

§1.8. Cartan and the mathematicians of the world

the Cartan visited many countries and was connected by friendship withmany mathematicians. In 1920, 1924, 1928, 1932, and 1936 he partici-pated in the International Congresses of Mathematicians held in Strasbourg,Toronto, Bologna, Zurich, and Oslo. In 1939 he participated in the Mathe-matical Congress in Liege. In 1940, in Belgrade, he delivered the lecture onthe role of French mathematicians in the history of mathematics.

Cartan greatly influenced mathematicians of many countries. Among Ger-man mathematicians, Ernst August Weiss (1900-1942), a student of Eduard

28 1. THE LIFE AND WORK OF E CARTAN

FIGURE 1.11

Study, was most influenced by Cartan; Weiss spent two semesters with Car-tan and developed further Cartan's idea on the "triality principle". Manypapers of Shiing-shen Chern (b. 1911), a student of Wilhelm Blaschke, alsoreflected Cartan's influence.

In April and May of 1931 Cartan made a trip to Romania and Poland.In Romania he delivered a series of lectures in Cluj, Bucharest, Ia§i (Yassy),and Cernau]i (Chernovcy, now in the U.S.S.R.). In the same year, Cartanwas elected an honorary member of the Romanian Academy of Sciencesin Bucharest. In 1934 Cartan was made a corresponding member of theRoyal Society of Sciences in Liege, Belgium; in 1937 he was elected a foreignmember of the Amsterdam Academy of Sciences (Netherlands). In 1949 hebecame a foreign member of the National Academy of Sciences of the U.S.A.and a member of the National Academy of Forty in Rome. Cartan was alsoelected an honoris causa Doctor of Sciences at Harvard University (1936) andthe Universities of Liege (1934), Brussel and Louvain (1947), and Bucharestand Pisa (1948).

Cartan corresponded with many scientists. However, although many ofhis letters have been preserved, only his correspondences with A. Einstein[210] and the Romanian geometers Gheorghe Titeica (1873-1939), Alexan-dru Pantazi (1873-1939), and Gheorghe Vranceanu (1900-1979) [211] havebeen published.

Figure 1.11 shows a group of participants at the Congress in Zurich. Fromleft to right in this picture are Ferdinand Gonseth (1890-?), the Cartan,


FIGURE 1.12

FIGURE 1.13

Gustave Juvet (1896-1936), Gaston Julia, Mrs. Julia, and Mrs. Gonseth.Figure 1.12 shows a group of participants at the Congress in Oslo. From leftto right in this picture are George David Birkhoff (1884-1944), Elie Cartan,and Constantine Caratheodory (1873-1950). Figure 1.13 is a picture of agroup of mathematicians in Paris at the beginning of 1935. In the first rowfrom left to right in this picture are: Emil Artin (1892-1962), Gaston Julia,Francesco Severi (1879-1961), and the Car-tan.


Cartan had a close friendship with many Soviet geometers. Being in Parisin 1926-1927, Serge P. Finikov (1883-1964) attended the course of lecturesdelivered by Cartan. Later Finikov founded a Soviet differential-geometricschool that dealt with applications of the method of exterior forms and themethod of moving frames. From 1927 to 1928 in the Sorbonne, GeorgiN. Nikoladze (1888-1931), under Cartan's supervision, prepared and de-fended his doctoral dissertation On continuous families of geometric figures.Before 1917 Nikoladze worked as an engineer-metallurgist in the factoriesof Donbass. From 1919 he taught mathematics at the University of Tbilisi.After his return to Tbilisi, Nikoladze became a professor at the Universityof Tbilisi and founded the Georgian geometric school. Cartan also was onfriendly terms with Veniamin F. Kagan (1869-1953), the founder of the So-viet tensor differential-geometric school.

We have already mentioned Cartan's publications in Moscow and Kazan.In 1937, in the VIII International Lobachevsky competition, the Lobachev-skian prize was awarded to Cartan for his work in geometry. Cartan visitedthe U.S.S.R. three times: in 1930 he participated in the First All-UnionMathematical Congress in Kharkov and later delivered a series of lectures atMoscow University; in 1934 he participated in the International Conferenceon Tensor Differential Geometry in Moscow; and in 1945 he was presentduring the celebration of the 220th anniversary of the Academy of Sciencesof the U.S.S.R.

Ten books and collections of papers by Cartan appeared in Russian transla-tions in the U.S.S.R. In 1933 the translation of the course of lectures [ 144] de-livered by Cartan in 1930 appeared in Moscow (translated by S. P. Finikov).In 1936 in Moscow the translation of the book [114] under the title Geometryof Riemannian manifolds was published (translated by G. N. Berman; editedby A. Lopshits). In 1937 Cartan's lectures [152]-[1541 at the InternationalConference on Tensor Differential Geometry were published in Proceedingsof the Vector and Tensor Analysis Seminar. In 1939 a collection [205] ofCartan's papers [88], [105], and [140] was published in Kazan (translatedby P. A. Shirokov and B. L. Laptev). In 1940 these translations were re-published in a collection, The VIII International Lobachevsky Competition.In the same year, the Russian translation, titled The integral invariants, ofthe book [64] was published in Moscow (translated by G. N. Berman; editedby V. V. Stepanov). In 1947 the Russian translation, titled The theory ofspinors, of the book [ 164] was published in Moscow (translated by P. A.Shirokov). In 1949 a collection of Cartan's papers [93], [94], (101], [103],[ 116], and [ 128], titled Geometry of Lie groups and symmetric spaces [206],was published in Moscow (translated by B. A. Rosenfeld; edited by P. K.Rashevsky). In 1960, 1962, and 1963 the Russian translations of Cartan'sbooks [ 108a], [ 144], [157], and [ 181 ] were published in Moscow (translatedby S. P. Finikov). In 1962 a collection of Cartan's papers [66], [68]-[70], and[80], titled Spaces with afj?ne, projective and conformal connection [208], was


Courtesy of Department of Geometry, Kazan University, Tatarstan, Russia

FIGURE 1.14

published in Kazan (translated by P. A. Shirokov, V. G. Kopp, B. L. Laptev,and others; edited by P. A. Shirokov).

Figure 1.14 shows a meeting of Cartan (left) with the mathematicians fromKazan: Petr A. Shirokov (1895-1944) (center) and Nikolai G. Chebotarev(1894-1947) during one of Cartan's visits to Moscow.

The method of exterior differential forms was developed by Finikov inthe book The Cartan method of exterior differential forms in differential ge-ometry [Fin]. This method was applied to solutions of a very large numberof problems in differential geometry by Finikov and his numerous studentsand followers in Moscow, Kiev, Vilnius, Tomsk, and other cities of theU.S.S.R. Also, further development in theory of Riemannian manifolds andspaces with affine connection, particularly symmetric spaces, was achievedin papers of Kagan, Shirokov, and other geometers from Moscow, Kazan,Saratov, Penza, and other cities.

During Cartan's first two visits to Moscow, the authors of this book werehigh school students. During his third visit to Moscow in May of 1945, theauthors were serving in the Soviet Army. At that time B. A. Rosenfeld'smilitary unit was located near Moscow, and he had the good fortune to seeCartan and discuss with him his own results and plans.

The scientific activities of M. A. Akivis in the field of differential geometry,which started a few years after World War II ended, also were very closelyconnected with the development of Cartan's ideas.

CHAPTER 2

Lie Groups and Algebras

§2.1. Groups

The 1870s, when the Cartan was a lad taking his first steps in his father'sblacksmith shop and in the elementary school of Dolomieu, were criticalyears in the history of France as well as in world history and in the history ofmathematics. In 1870, after its defeat in the Franco-Prussian war, the SecondEmpire of France fell, and France again became a republic. In the 1870s anew period of world history began-the Industrial Revolution. At that time anew period in the history of mathematics also began. Two great discoveries,made in the first half of the 19th century, were understood: the discovery ofgroup theory by Evariste Galois and the discovery of non-Euclidean geometryby Nikolai I. Lobachevsky.

The mathematical implications of these apparently unrelated discoveries,which were arrived at independently, were very closely related. Before Galoisit was believed that only one arithmetic of real and complex numbers wasconceivable. Galois showed that there are many different arithmetics definedby different groups and fields. Before Lobachevsky, it was believed that onlyone geometry, namely Euclidean geometry, was conceivable. Lobachevskydiscovered a new geometry, which was as much noncontradictory as Eu-clidean geometry but quite different from it. The discoveries of Galois andLobachevsky were the principal manifestations of creations of new "alge-bras" and "geometries" in the 19th century. Along with Galois groups andfields, a series of new numerical systems was discovered at that time. Later,this series was named "hypercomplex numbers" and "algebras". Along withLobachevskian geometry, during the 19th century, other geometries, differ-ent from classical Euclidean geometry, were also discovered: affine, projec-tive, multidimensional geometries, and finally the Riemannian geometries-geometries of curved spaces. Group and algebra theories as well as non-Euclidean and other geometries discovered at that time played an importantrole in Cartan's mathematical research. In the mid-1870s another importantdiscovery was made: the set theory of Georg Cantor (1845-1918). This the-ory and the theory of functions of a real variable, which is closely connectedwith set theory, became the main areas of research of French mathemati-cians at the end of the 19th century and the beginning of the 20th century.

33

34 2. LIE GROUPS AND ALGEBRAS

Originally these two theories were not reflected in Cartan's work.Group theory was created by the young Evariste Galois (1811-1832), who

was killed in a duel. However, in his short lifetime he published a few works,and, on the night before the fatal duel, he wrote a summary of his maindiscoveries. This was later published by a friend. Galois was a student at thesame Ecole Normale Superieure where Cartan later studied. Galois made hisdiscovery while trying to determine the solvability by radicals of algebraicequations. If one is given an algebraic equation

(2.1)a0xn+aixn-1+...+an_1x+an=0

with rational coefficients, real or complex, then the values of x in this equa-tion which make it an identity are called the roots of the equation. In the caseof quadratic equations (n = 2), the roots xI and x2 are expressed in termsof the coefficients ao , al , and a2 by commonly known formulas, found atthe beginning of the 9th century by Muhammed al-Khwarizmi (circa 783-850). These formulas involve quadratic radicals. In the 16th century NiccoloTartaglia (circa 1500-1557) and Girolamo Cardano (1501-1576) found the"Cardano formula", through which the roots of a cubic equation (n = 3) areexpressed in terms of the coefficients ao, al , a2 , and a3 ; the "Cardano for-mula" involves cubic radicals. Cardano's student, Luigi Ferrari (1522-1565),solved a similar problem for n = 4. For a few centuries mathematicians triedto find a formula expressing the roots of equation (2.1) for n > 5, in termsof the coefficients of this equation. However, this problem was solved onlyfor the simplest particular cases of this equation, for example, for "binomialequations" xn = a (one root of this equation is expressed by the radicalx = a and others are the products of this radical and powers of the com-plex number e = cos ? + i sin 2n) . In 1829 Niels Henrik Abel(1802-1829), in his Demonstration of the impossibility of the algebraic reso-lution of general equations surpassing fourth degree [Ab], distinguished a classof equations solvable by radicals, and this class was wider than the binomialequations. In the paper Memoir on conditions of solvability of equations byradicals [Gal], written before his duel, Galois gave a complete solution tothe problem. The Galois solution is based on the notion of groups which heintroduced and which was implicitly contained in the paper Reflections on so-lution of equations [Lag 1 ] by Joseph Louis Lagrange (1736-1813) and in thepaper Arithmetic investigations [Gaul by Carl Friedrich Gauss (1777-1855).In many branches of mathematics one can find such operations on objects,which assigns to each pair of objects of a set an object from the same set.Examples of such operations are: the addition of numbers, vectors, or matri-ces; the multiplication of numbers or matrices; and the successive realizationof transformations. At the very beginning of human civilization, the conceptof the natural number, which includes the sets of different objects consistingof the same number of objects, and later the arithmetic of integers and thealgebra of rational, real, and complex numbers, were introduced. In a similar

§2.1. GROUPS 35

way, the theory, including very diverse arithmetic, algebraic, and geometricoperations, appeared next. The term "group" was introduced by Galois, who,in his Memoir on the conditions of solvability of equations by radicals, wroteabout substitutions: "If in such a group there are substitutions S and T,then there is the certainty of there being the substitution ST" (see [Gal, p.47] or [Ro8, p. 328]). Note that Galois used the term "group" in a widersense than we do. In the famous letter to his friend written on the eve ofhis fatal duel, Galois wrote: "When a group G contains another group H,the group G can be decomposed into groups" (see [Gal, p. 173] or [Ro8, p.329]), where these "groups" are right cosets of G with respect to H . (In theEnglish translation of this letter in [Sm, p. 279] the word "sets" was usedinstead of "cosets".) At present a group consisting of elements a, b, c, .. .is defined as a set of elements such that

1 ° . To each two elements a and b there corresponds an element c =aob.

2°. (aob)oc=ao(boc) forany a, b, and c.3°. There exists a "neutral element" e such that e o a = a o e = a for

every a .4°. For each element a there exists a "complementary element" a such

that aoa =aoa =e.

If within a group the following property holds:

5°. a o b = b o a for every two elements a and b (the group operationis commutative),

the group is called commutative or abelian.In the case of integers, rational, real, and complex numbers, and the opera-

tion of addition, the "neutral element" is 0 and the "complementary element"for a number a is the number -a . For the last three classes of numberswithout 0 and the operation of multiplication, the "neutral element" is 1 andthe "complementary element" for a number a is its reciprocal a-1 . In bothcases the property 5° is satisfied.

The addition of numbers: a + b = c, vectors: a + b = c, and matrices:A + B = C and the multiplication of numbers: ab = c are commutative.Numbers, vectors, and matrices with these operations form commutativegroups. The simplest example of a noncommutative group is the group ofpermutations

a1 a2 an

bl b2 bn

of n elements, i.e., substitutions of each element ai of the upper row by thecorresponding element bl of the lower row, where the elements b1 , b2 , ... ,

bn of the lower row are the same elements a 1 , a2 , ... , an of the upper row


but arranged in another order. Here the group operation has the form

a1 a2 ... an x b1 b2 bn = a1 a2 anb1 b2 ... bn (Cl c2 ... cn (Cl C2 Cn

the role of the neutral element is played by the identity permutation

a1 a2 . ana 1 a2 ... an '

and the permutation inverse to a permutation

al a2 ... anis ('s' b2 bn

b2

... n a1 a2 an

The multiplication of nonsingular matrices: AB = C is also noncommuta-tive. In the group of nonsingular matrices, where the operation is the matrixmultiplication, the neutral element is the identity matrix I and the comple-mentary element for a matrix A is its inverse matrix A-1

A subset H of a group G is said to be a subgroup if H itself is a groupwith respect to multiplication in G. If H is a subgroup of a group G, thenthe products aH and Ha of the elements of this subgroup and an arbitraryelement a of G from the left and the right are called a left and right cosetof the subgroup H. If every right coset of a subgroup H is also a left coset,then the subgroup H is said to be invariant or normal (or a normal divisor).In this case multiplication of cosets can be defined, and the cosets with thismultiplication form a group. This group is called a quotient group (or factorgroup) of the group G by its invariant subgroup H and is denoted by G/H.

Simple groups play a special role in group theory. A group G is simpleif it does not have invariant subgroups except the group G itself and thesubgroup consisting of the neutral element of G only. In the case where inG there is a sequence of subgroups G = Go) G1, G2 , ... , Gk = e such thateach subgroup G1+1 is an invariant subgroup of Gi and each quotient groupGi+1 /Gi is abelian, the group G is called solvable.

Galois introduced the notion of the group which is now called the Galoisgroup of an algebraic equation. This group is the group of automorphismsof a field which is such an extension of the field F (to which belong thecoefficients of the equation defined by the roots of this equation) that leavesits subfield F invariant. This group is a finite group which in general canbe represented as a permutation group of roots of this equation. The Galoiscriterion of solvability of the algebraic equation (2.1) by radicals is that theGalois group of this equation is solvable. In the case of the binomial equationxn = a this group is cyclic. In the case of equations that were considered byAbel, this group is the general commutative (abelian) group. (This explainsthe origin of the name "abelian".) These two groups are examples of solvablegroups.

§2.2. LIE GROUPS AND LIE ALGEBRAS 37

Besides the notion of a group, Galois introduced the concept of a field.A field is a commutative additive group, and its elements, excluding 0, forma multiplicative group, where multiplication is distributive with respect toaddition. If the multiplicative group of a field is commutative, the field iscalled commutative.

Examples of commutative fields are: the field Q of rational numbers, thefield R of real numbers, the field C of complex numbers, and the field FP ofremainders modulo a prime integer p (i.e., the numbers 0, 1, 2 , ... , p - 1 ,where the sum or the product is the remainder resulting from the division ofthe sum or the product of the corresponding numbers by p). The field FPconsists of p elements. Galois also constructed more general finite fields:Galois fields Fq where q is a positive integer power pk of a prime numberp. In the same way as the field C consists of elements a + bi, where aand b are elements of R and i is the "imaginary unit", i.e., a root ofthe equation x2 + 1 = 0 , the field Fq consists of elements of the forma! + >la., , where i1 , i2 ... , ik_1 are "Galois imaginaries"-roots of anirreducible polynomial of degree k with coefficients from F . A similarextension of fields determined by algebraic equations plays an important rolein Galois theory.

The Galois memoir on solvability of algebraic equations by radicals, whichwas originally published by his friend in an obscure publication, was repub-lished in 1846 by Joseph Liouville (1809-1882) in the Journal de Mathema-tiques Pures et Appliquees, of which Liouville was the editor. Galois's ideaswere recognized only after Camile Jordan (1838-1922) in 1865 and 1869published his comments on Galois's memoir and in 1870 released a funda-mental Treatise on permutations and algebraic equations [Jo 11. in which hepresented the theory of the permutation group, the Galois theory, and itsapplication to the problem of solvability of algebraic equations by radicals.

§2.2. Lie groups and Lie algebras

In 1870, not long before the Franco-Prussian war, two friends, SophusLie and the young German mathematician Felix Klein (1849-1925), came toFrance. In Paris the friends attended the lectures of Darboux, had discus-sions with Jordan, and carefully studied his recently published book [Jo 11. Al-though Jordan's book was mainly devoted to discrete and even finite groups,Lie and Klein, whose first papers were in geometry, were interested in con-tinuous groups and their importance for geometry.

Examples of continuous groups are the following group transformations ofgeometric spaces: the groups of motions of a Euclidean and a non-Euclideanspace, the groups of rotations about a point in these spaces, the group oftranslations and the group of similarities of a Euclidean space, the group ofaffine transformations, and the group of collineations (i.e., projective trans-formations). It is well known that the groups of rotations and translations


are subgroups of the group of motions of a Euclidean space (moreover, thegroup of translations is an invariant subgroup), the group of motions is asubgroup of the group of similarities, the group of similarities is a subgroupof the group of affine transformations, and the latter is a subgroup of thegroup of collineations. In 1871, while constructing his famous interpreta-tion of Lobachevskian geometry, Klein proved that the group of motions ofLobachevskian space is also a subgroup of the group of collineations. In 1872he arrived at his "Erlangen Program" [Kle]. According to this program, ev-ery geometry is defined by a group of transformations, and the goal of everygeometry is to study invariants of this group.

Sophus Lie chose another way. As early as in his geometric paper On com-plexes, in particular, on complexes of straight lines and spheres (1872) [Lie 1],which was written in Paris and was very highly regarded by Cartan (in hispaper on Lie, Cartan wrote: "It was in Paris that Sophus Lie made one of hismost beautiful discoveries, the famous transformation which bears his nameand which establishes an unforeseen relation between lines and spheres inspace on the one hand and between asymptotic lines and lines of curvature ofsurfaces on the other" [201, Engl. tr., p. 263]), Lie connected geometric trans-formations with differential equations ("Lie transformations", which Cartanmentioned in the above quotation, are imaginary transformations sendingstraight lines into spheres and sending asymptotic lines of surfaces into theircurvature lines). Cartan wrote further: "But the theory of transformationgroups itself, its technique, has not been created and nothing indicated thepath to be followed for that creation. Sophus Lie devoted himself to this workfrom 1873 on and by intense labor rapidly managed to construct the funda-mental theorems from which he quickly deduced very many consequences.In 1882, upon reading a paper of the French mathematician Halphen, So-phus Lie realized that his earlier research enabled him to see in perspectivethe problem considered by Halphen" [201, Engl. tr., pp. 264-265]. Thepaper by Georges Halphen (1844-1889) mentioned by Cartan is the memoirReduction of a linear differential equation to integrable forms (1884) [Hall; itwas written earlier and in 1881 received an award from the Paris Academy ofSciences. The problem considered by Halphen is the problem of integrabilityof differential equations by quadratures, i.e., the expression of the solutionsof these equations in terms of integrals of known functions. In the paperClassification and integration of ordinary differential equations admitting agroup of transformations [Lie2] (1883-1884), Lie considered the problem ofintegrability by quadratures of differential equations as an analogue of theproblem of solvability by radicals of algebraic equations and tried to solvethis problem by the Galois method. And, in fact, the Lie criterion of solvabil-ity of differential equations by quadratures proved to be similar to the Galoiscriterion: with each differential equation, a continuous group "admitted bythis equation" is connected, and the Lie criterion is that this group must besolvable. In this connection Lie, according to Cartan, "felt the necessity of

§2.2. LIE GROUPS AND LIE ALGEBRAS 39

expounding in one great didactic work the results of his earlier researches,particularly those dealing with group theory. Thanks to the devoted collabo-ration of a young German mathematician; Friedrich Engel (1861-1941), theprojected work was written and published after nine years' labor; it appearedsuccessively in three volumes between 1888 and 1893" [201, Engl. tr., p.265]. In particular, and as in the case of finite groups, with which Galoisdealt, Lie had to study properties of simple and solvable continuous groups.

First of all, in his Theory of transformations groups (1888-1893) [LiE], Lieconsidered a wide class of continuous groups whose elements depend on afinite number of real or complex parameters. Lie himself called such groups"finite continuous groups" or, since he always presented these groups in theform of transformation groups, "finite transformation groups". At present,these groups are called Lie groups.

Lie considered transformations of the form

I i t 1 n 1 r(2.2) x = f (x ,... x ;a ,... a ),

where the x` and 'x` are coordinates of a transformable point and a trans-formed point and the as are parameters of the group. Our notation differsfrom the notation used by Lie and Cartan: in their time all indices werewritten as subscripts, but we write them as superscripts to be able to usetensor notation. In addition, the parameters as , ba , and ca defining twotransformations and their product (the result of their successive realization)are connected by the relations

a a 1 r 1 r(2.3) c =q (a ,... a ;b ,... ,b).

The parameters a' are chosen such that the values a' = 0 correspond tothe identity transformation 'x` = x` , i.e., x` = f'(x', ... , xn ; 0 5... , 0).Next, Lie considered the "infinitesimal transformations", i.e., transforma-tions infinitesimally close to the identity transformation. They can be writtenin the form 'x ` = x ` + (from here on, we shall adhere to thesummation convention: whenever the same index symbol appears in a termof an algebraic equation both as a subscript and a superscript, the expressionshould be summed up over the range of that index). If we denote daa = ead tand aft /aaa = ,a , then the transformation (2.2) can be written in the form

(2.4) x =x`+ea adt+...

If F (x 1, x2 , ... , xn) is an arbitrary differentiable function, then d F =But, by (2.4), dx` = eaad t . Therefore,

dF a i aF(2.5) dt

=eax


The last expression is a linear combination of the expressions X F =c,,',OF/axi . Thus to an infinitesimal transformation of a Lie group therecorresponds an operator X = e"X = ea a /ax` , which is a linear combi-nation of the basis operators X = a / a x i .

In particular, for the group 7" of translations

(2.6) x` = x` +aI

of the Euclidean space R" we have r = n and s = aft/aaa = o, (i.e., 1

for i = a and 0 for i a) and Xa = J'a/ax` , i.e., Xi = aax` .

For the group 0n of rotations

(2.7) x` = u`xJ

of the Euclidean space R" , where U = (ut) are orthogonal matrices of ordern, we have r = n(n - 1)/2. If, in a neighborhood of the identity element ofthe group Q" , we represent the matrix (ut) in the form bjJ+ aJt. + , where

at are infinitesimals of the first order and the dots denote higher degreeterms, and substitute these expressions into the condition of orthogonalityIk u i uk = aij 3, then we obtain

E(jk k +...)( k k

k

It follows from the last equation that the matrix aj' is skew-symmetric: a _-ai . The elements of the matrix (aJt) with i < j can be taken as parametersof the group 0,. Since

/x` = u`xJ =xI +a IXj +...J J

the infinitesimal operators Xa of the group 0(n) can be written in the formXi = xia/axl - xla/axi .

For the group of motions

(2.s) xj=uxJ+a'

of the space R" we have r = n(n + 1)/2, and the operators Xa of this groupare Xi = a/ax` and XiJ . = x'8/8x - x'a/ax` . The groups Tn and Onare subgroups of this group; moreover, 7,t is an invariant subgroup.

For the operators X of a transformation group, an operation of transitionfrom operators Xa and X to their "Poisson bracket" is defined as

(2.9) [XaX0 = Xa(X f) - X# (X 0

X2.2. LIE GROUPS AND LIE ALGEBRAS 41

This new operation is anticommutative:

(2.10) [Xa Xf ] = - [Xf x a]

and satisfies the "Jacobi identity":

(2.11) [ X[X,8Xy]] + [Xp[XYX ]] + [Xy[X Xp]] = O.

:In addition, the bracket [X, X,6 J is a linear combination of the operators X.

(2.12) [XX8 1 = c XyQ

where the numbers cY, are constants, which are called the structure constantsof the group. Lie called the set of operators X = e' X with the operation(2.9) the "infinitesimal group" and proved that if an "infinitesimal group"is given, then it completely defines the group of transformations (2.2) in aneighborhood of the identity transformation 'x` = x` , and the functions(2.2) are solutions of a certain system of differential equations.

At present, Lie groups are considered independently from their "represen-tations" in the form of a group of transformations of a certain space. Theyare considered as manifolds with a group structure in a neighborhood of theidentity element in which coordinates a' are introduced. In this case, insteadof the operators X = e'X4 the tangent vectors to this manifold with coordi-nates ea = d as/dt are considered. To lines a(t) and b(t) emanating fromthe group identity element (one-parameter subgroups a (t1 + t2) = a (t 1) a (t2 )are usually taken), there correspond tangent vectors e = {ea} and f = {fQ}

and to their product a(t)b(t) there corresponds the sum e + f of the vectors.To the product a(t)b(t)a-1(t)b-1(t) there corresponds the commutator [ef] ,

which is anticommutative:

(2.13) [ef] _ -[fe]

and satisfies the Jacobi identity

(2.14) [e[fgj] + [f[geJJ + [g[ef]J = 0.

These properties are similar to properties (2.10) and (2.11) of the opera-tors a . In particular, instead of considering the group 7'n as the group oftranslations (2.6), one considers it as a group of vectors a with respect toaddition. Similarly instead of considering the group O, as the group of rota-tions (2.7), one considers it as a group of orthogonal matrices, and the groupof motions (2.8) is considered as a group consisting of orthogonal matrices Uand vectors a with multiplication defined by (U, a) (V, b) _ (U V, a + Ub) .

The "infinitesimal group" is a vector space with the operation [ef] , whichmay be considered as vector multiplication. At present, a vector space withmultiplication is called an "algebra". Because of this, Hermann Weyl (1885-1955) in his paper The structure and representations of continuous groups


(1 935) [Wey4] suggested replacing the term "infinitesimal group" with theterm "Lie algebra" which is universally accepted nowadays. In the case inwhich a Lie group is a multiplicative matrix group, the corresponding Liealgebra consists of the matrices A = (d U/d t)o , where the derivative of thefunction U(t) is taken at the identity element of the group, which corre-sponds to the value t = 0 of the parameter. The commutator [AB] of twomatrices A and B is connected with their usual product by the relation

When a Lie group is a group of vectors a with respect to addition, thecorresponding Lie algebra consists of the vectors e = (da/dt)0, where thederivative of the function a is taken at the identity element of the group,and the commutator [efl of any two vectors e and f is equal to 0. In thecase of the group 0, of orthogonal matrices, the corresponding Lie groupconsists of skew-symmetric matrices A = (a), aj = -a,. .

§2.3. Killing's paper

The paper by W. Killing, which determined the subject of Cartan's thesis,was published in the journal Mathematische Annal en under the title Continu-ous finite transformation groups [Ki12]. Wilhelm Killing (1847-1923), a stu-dent of Karl Weierstral3 (1815-1 897), was very familiar with the Weierstraf3theory of elementary divisors and normal form of matrices. In his doctoraldissertation, which was defended in Berlin in 1872, he successfully appliedthese theories to the investigation of mutual disposition of two quadrics (sur-faces of second order) in a projective space. (This problem is equivalent tothe problem of classification of quadrics in a non-Euclidean space.) Based onthe recommendation of Weierstral3, Killing became Professor of Mathematicsin the Catholic Lyceum Hosianum in the city of Braunsberg in Eastern Prussia(now this city, which is located in Olsztyn wojewodstwo in Poland, has re-turned to its original name Braniewo). The lyceum was a college for trainingRoman Catholic clergy, founded in 1565 by Polish bishop Stanislaus Hosius(1 504-1579). When Killing became a professor of this college, he joined theholy order of tertiaries (the biography of Killing written by P. Oellers [Oel]has the subtitle "The university professor in tertiary cloth").

In Braunsberg, Killing continued his mathematical research. FollowingWeierstral3's advice, he studied the problem of space forms building on thework in the well-known papers of Klein and William Kington Clifford (18 34-1879). This problem brought Killing to consider infinitesimally small mo-tions. In 1884, Killing published in Braunsberg the program titled Extensionof the notion of space [Kill ], in which he, independently of Lie, arrived at thenotions of Lie group and Lie algebra and posed the problem of classificationof real simple Lie groups. Killing sent this program to Klein. In turn, Klein

§2.3. KILLING'S PAPER 43

informed Killing that his friend Sophus Lie was studying similar problems inChristiania (at the beginning of the 20th century its ancient name Oslo wasreturned to this city) and gave Lie's address to Killing. Upon Killing's re-quest, Lie sent preprints of his own papers to Killing. When Killing found outthat Lie had not studied the problem of classification of simple Lie groups,he returned these reprints to Lie. Lie was offended by this rather fast returnof reprints, and his relations with Killing were spoiled for good.

Killing started to correspond with Engel, a colleague of Klein and Lie.Engel helped him publish his paper [Ki12] in Mathematische Annalen. Aftera wrong statement on the "groups of zero rank" was discovered in this pa-per, Engel advised Killing to assign the correction of this mistake to one ofKilling's students. After Killing replied that in Braunsberg he did not haveany students in mathematics, Engel assigned this task to his own student,Umlauf, who in 1891 successfully defended his dissertation on this subject(there is no further information on Umlauts subsequent life and works).

In the paper [Ki12] Killing did not solve the problem of classification of realsimple Lie groups but solved the simpler problem of classification of complexgroups of this type. In this paper he applied the theory of eigenvalues ofmatrices with which he was very familiar and showed that in addition to thefour infinite series of groups which were discovered by Lie, there are five more"exceptional" simple groups of dimensions 14, 52, 78, 133, and 248. Hispaper [Ki12] is a very important event in the development of mathematics.Albert John Coleman (b. 1918) [Co12] even considers it as "the greatestmathematical paper of all time".

The four infinite series of complex simple Lie groups discovered by Lieare: the group of collineations of a complex projective space CPn ; the groupof motions of a complex non-Euclidean space CSn , i.e., a subgroup of theprevious group which leaves fixed a quadric alJ .x'xJ = 0(a1J . =

a a the groupof of the space CP2n -1 , which leaves fixed a linear complex ofstraight lines, i.e., a set of straight lines whose Pliicker coordinates p ii =x`y' - xjy' satisfy the equation a1J.p'J = 0 (a1J. = -aJ.1). Killing named thegroups of the first series "the system A", the groups of the second series foreven n "the system B" and for odd n "the system D", and the groups of thethird series "the system C". The groups of "system A" are locally isomorphicto the groups CSLn+I of complex unimodular matrices of order n + 1 . Thegroups of "system B" are locally isomorphic to the groups CO2n+1 of complexorthogonal matrices of order 2n + 1 . The groups of "system C" are locallyisomorphic to the groups CSp2n of complex symplectic matrices of order2n . The groups of "system D" are locally isomorphic to the groups C02n ofcomplex orthogonal matrices of order 2n .

The term "symplectic" was introduced by Weyl in his book [WeyS]. Inhis lectures [Wey4] he translated the term "Komplex-Gruppe" as "complex


group". However, these words also denote any Lie group with complex pa-rameters. Because of this, he suggested the groups of the third series be called"symplectic groups". The word "symplectic" originates from the Greek word"symplektikos", which has the same meaning as the Latin word "complexus"-complex. The complex dimensions of the groups CSLn+1 , Co2n+1, CSp2nand Co2n are equal to n(n + 2) , n(2n + 1) , n(2n + 1), and n(2n - 1) ,

respectively.Killing considered the problem on eigenvalues of a linear operator gener-

ated in the "infinitesimal group" by its fixed element eaX . Its action on anarbitrary element X = AaX has the form

(2.15) [eCX, X] = e')![X X ] = eaca ,!X ,a J3 y

and the eigenvectors are defined by the equation

[eaX, X]=coX,

which can be written in coordinate form as

(2.16) eacY 2,3 = c0AY.

The eigenvalues of this operator are roots of the equation

(2.17) A(a)) = det(eaca - wok) = 0 ,

which is called the characteristic equation of a Lie group. This equation canbe written

r r-1 r-2 1 r(2.18) co - W1 co + v2cc - .. + (_)r-

r-ice + 1) Vr = 0,

where the i/a are homogeneous functions of the parameters ea :

- a 1(cy 6 Y 6W1 Ca/Je V2 ayCflg - CQJ ar)e e

Ca cc cc26(2.1 9) 1ae

a of 1 vc e e e , ... .V/3 = 3 det c#a

c,0,&c p

ca cC ccvC Y Y

Killing defined as the group rank the number of functionally indepen-dent coefficients y/a of the characteristic equation. For the groups CSLn+

1

CO2n+1 , CSp2n and CO2, the rank is equal to n. Killing showed that fora simple group the same operator 2flX

Qis an eigenvector for all matrices

of linear transformations (2.15) corresponding to the operators eaX fromthe subgroup of zero rank which contains the infinitesimal transformation ofgeneral type. However, Killing's proof of this fact was invalid. From thisstatement, which was proved later, Killing showed that for simple groups the

X2.4. CARTAN'S THESIS 45

characteristic equation can be written in the form

Co'" 11(a) - a(h)) = 0,a

where the roots a(h) are linear functions of infinitesimal transformations hfrom the subgroup of zero rank. Next, considering different possible combi-nations of these roots, Killing gave the classification of complex simple Liegroups.

Killing denoted complex simple Lie groups by Roman numerals equal tothe group rank and by one of the capital letters A, B, C, D, E, and F. Hefound the isomorphisms of the simple groups IA, IB, and IC of dimen-sion three as well as of the simple groups IIB and IIC and of the simplegroups IIIA and IIID of dimension 15 and proved that the group IID isnot simple and consists of two groups IA. For the exceptional groups thathe discovered he used the notation IC (since the group of series C of rank2 is isomorphic to the group IIB ), IVE, VIE, VIIF, VIIIE, and IVF andproved that the dimensions of these groups are equal to 14, 52, 78, 133, 248,and 52 respectively. (Thus, Killing assumed that there are two nonisomor-phic complex simple groups of rank four and dimension 52.) Killing calledgroups which are composed of a few simple groups semisimple groups.

§2.4. Cartan's thesis

As early as in his note The structure of simple finite continuous groups [ 1 ](1893), Cartan, noting "exceptionally important results" of the Killing paper,indicated: "Unfortunately, in the considerations which led Mr. Killing tothese results, the rigor is missing. Therefore, it is desirable to perform thisresearch again, indicating which of Killing's theorems are inaccurate andproving those of his theorems that are correct" [1, p. 784-785]. This workwas performed by Cartan in his thesis [5].

Cartan's research was concerned with those Lie groups which, followingKilling, he called "semisimple groups". However, he defined these groups asthe groups not possessing a solvable invariant subgroup. The groups satisfyingthis definition are semisimple in the sense of Killing's definition. Note thatall noncommutative simple Lie groups are semisimple and that commutativesimple Lie groups, namely, the one-parameter group of translations and thegroup 02 (which can be considered as the group ID), are not semisimple.Cartan showed that, when the coefficient w (e) = 0 , the form 2 y/2(e) hasthe form

-2Vr2(e)=cac(3Yaeae4c.ay

When the form w, (e) is not zero, the expression on the right-hand side ofthis equation can be written in the form


(2.20) co(e) _ 2 - 2y/= ca cy eaep1 2 ay ,i

The condition that a Lie group be semisimple is the nondegeneracy of theform ap(e) , and the condition that a Lie group be solvable is the vanishing ofthis form (the vanishing of this form for commutative groups is obvioussince in this case cY,o = 0). Since the forms (2.19) were introduced byKilling and the value of the form (e) in deciding whether a Lie groupis semisimple or solvable was discovered by Cartan, this form is called the"Killing-Cartan form". Since for semisimple groups the form V(e) is anondegenerate quadratic form (the metric in the Lie algebra in which thesquare of the length of the vector e is equal to the value of this form forthis vector), in the Lie algebra of a complex semisimple Lie group, this formdefines the metric of a complex Euclidean space CRr . Moreover, since a Liealgebra can be considered as the tangent space to a Lie group at its identityelement, this form defines the metric of the complex Riemannian manifoldC Vr in the complex semisimple group itself. At present this Riemannianmetric in semisimple Lie groups is called the Cartan metric.

Cartan gave a rigorous proof of the fact that the "subgroup of zero rank"of a semisimple Lie group is commutative and can be considered as a set ofgroup elements that commute with a general element ("regular element") ofthe group. Because of this fact, at present this subgroup is called the Cartansubgroup of a semisimple Lie group, and the subalgebra of the Lie algebracorresponding to this subgroup is called the Cartan subalgebra.

Cartan slightly changed Killing's notations of simple Lie groups: he sug-gested that groups in the classes A, B, C, and D of rank n be denoted byAn) Bn , Cn, and Dn, respectively, and the groups VIE, VIIE, and VIIIEby E6, E7 , and E8 . In addition, he proved that the group IVE is isomor-phic to the group IVF and suggested that these two groups be denoted byF4 and the group IIC by G2 .

Thomas Hawkins (b. 1938) in [Haw3] made a thorough comparison ofthe Cartan thesis with the Killing paper [Ki12]. He noted all instances whereCartan corrected errors or omissions of Killing. In particular, he noted thatwhile considering the group E8 , Cartan, who was a skilled and intrepid calcu-lator, checked the Jacobi identities for all (238) = 2 , 51 1, 496 combinationsof the basis elements of the Lie algebra of this group taken three at a time,and Killing did not accomplish this.

§2.5. Roots of the classical simple Lie groups

We see that Killing gave to the word "root", already heavily used in math-ematics, one more very important meaning. The word "root" first appearedin the works of medieval Arab mathematicians. They translated the San-skrit word "pada", whose meaning is the base of a wall or the root of a tree,

§2.5. ROOTS OF THE CLASSICAL SIMPLE LIE GROUPS 47

as "jidhr", whose meaning is the root. The Indians used the word pada asa translation of the Greek word "basis", which was used by Pythagoreansfor bases "square numbers" (they represented these numbers in the formof squares). The Arabs began to use the word "jidhr" not only for nota-tion of roots of numbers, i.e., roots of the equations xn = a , but also fornotation of roots of any algebraic equation and notation of unknown quan-tities. European mathematicians who wrote in Latin began to use the Latintranslation of this word, "radix", and mathematicians who wrote in Ger-man, French, and English used the words "Wurzel", "racine", and "root",respectively, with the same meaning. This application of botanical termsin mathematics inspired one of the founders of projective geometry, GerardDesargues (1593-1662), to use such terms as "trunk", "branch", "shoot","tree", "stump", and "involution" the twisted form of young leaves. Onlythe latter term was widely used later and in significantly wider meaning thanin Desargues's works. Recently in mathematics the term "tree" has been usedin the sense of a connected graph without cycles and the term "forest" in thesense of a disconnected graph without cycles, i.e., a set of "trees".

The botanical term "root" of Killing was added to the system of termssimilar to the Desargues system by Hans Freudenthal (1905-1990) and H. deVries (b. 1932) in the book Linear Lie groups [FdV] (1969). In this book,Freudenthal and de Vries used the word "trunk" for the Cartan subgroup ofa Lie group and the Cartan subalgebra of a Lie algebra, the word "branches"for eigenvectors corresponding to the roots, and the word "nodes" for thecommutators of branches corresponding to opposite roots.

For the group CSLn+ 1 , the Cartan subgroup consists of those diagonalmatrices for which rL ea = 1 , and the corresponding Cartan subal-gebra consists of those diagonal matrices (h5), for which a h' = 0. Theeigenvectors of the linear transformation x -' [h, x] are the matrices E,,,having I in the intersection of the ath row and the 8th column and zerosin all other places. If we denote h = EY hYEYY, then

[Ii, E ] = hYE E - Eyy = (ha - h9)Ea E yy a EhYEa aY Y

Thus, the eigenvalue corresponding to the eigenvector E. of the linear trans-formation x -' [h, x] is a linear form on the Cartan subalgebra, whose valueon a vector h of this subalgebra is ha - h6 . Because of this property, wewill denote this linear form by co' - co

In the case of the group C02,,, which consists of the matrices preservingthe quadratic form i xAx2n-A-1 (here and further A=O, 1, ... , n- 1),the Cartan subgroup consists of the diagonal matrices (e5), for whiche2n-A-1 = (eAyl, and the Cartan subalgebra consists of the diagonal ma-trices (ha 5a) , for which h 2n -A- I = -h). As in the case of the groups


CSLn+I , it follows that the eigenvalues of the linear transformation x -+1h, x] corresponding to the eigenvectors EAI1 , EA112n-p-1 ,E2n-A-1 and

E2n-A-1 2n - -1 are a - c o" , a + -(O'1 and -CO11 + (0# , respec-tively.

For the group C02n+1

)which consists of the matrices preserving the qua-dratic form EA xAx2n-'1 + (x")2, the Cartan subgroup consists of the diagonal

matrices (e'J) j or which e2n -' = (eA)_l, e n = 1 , and the Cartan subalge-

bra consists of the diagonal matrices (h'J), for which hen-' = -hA , hn =0. It follows from this that the eigenvalues of the linear transformation x -*th , x] corresponding to the eigenvectors EA., EA, 2n-I1 , E2n-A,µ , E2n-A, 2n-µ ,E2 n , 2E n- n n t , and En n _ are loll - o/ , oil + UI1 ,

-W'- Co#,2

-Uv1 + W# , LEA , -WA , - USA , and w'1, respectively.In the case of the group CSp2n , which consists of the matrices preserving

the bilinear form EA (x 't y2n-'-1 - yAx2n -,I-1) , the Cartan subgroup con-sists of the diagonal matrices (e'6'), for which e2n-A-1

= (e) A-1 , and theCartan subalgebra consists of the diagonal matrices (h5), for whichhen-A-1 - -h'1 . It follows that the eigenvalues of the linear transformationx -+ [h, x] corresponding to the eigenvectors EA# , EA, 2n-1-1 , E2n-A-1,E2_2_1 2n- -1 (A 9) , EA 2n-A-1 , and E2.-A- l A are (J1 - co# , Ui1 +

W" , - 0JA - oI , -d + wu , 2d . and -2d, respectively.Thus, in the case of simple groups .An , Bn , Cn , and Dn , the roots of their

characteristic equations can be written as:

(2.21)

An : W1 - (I1 (,1 W'1 = 0)Bn : ±Lc)A ± Lt) , ±WA ,

Cn: ±WA+W,u,+2WA

Dn: ±WA+WI1.

Killing noticed that all these roots are linear combinations with integercoefficients of a certain number of forms composing a basis, that these coef-ficients can take only the values + l , +2, and ±3, and that the number offorms in this basis is equal to the group rank. We can also find a basis formedby the roots of a semisimple group. For the simple groups An , Bn , C., andDn , the roots composing the latter basis are:

(2.22)10 t

toAn : al = -Bn al = 1 - 12,

Cn al =w1-12,Dn : a I = W 1 - w2 ,

an = Can-1

+ cc)n.

an = n-1 - Lt)n ;_ n-1 n nan-1=(,c1 - ,an=( ;

an-1 OJn-1 Conacn2cvn

an-1 = 0n - 03

§2.5. ROOTS OF THE CLASSICAL SIMPLE LIE GROUPS 49

A2

(01-OJ2

a)

WQ_w2

-w'

B2

b)

C2

2w2

D2

w1fW2

2& (82-w1

-w1-w2

FIGURE 2.1

d)

The Cartan-Killing theory was significantly simplified by Weyl in the paperTheory of representations of semisimple continuous groups by linear transfor-mations (1925) [Wey3]. Developing the results of Weyl's paper, van derWaerden in his paper The classification of simple Lie groups [Wae] (1933)introduced a very visual representation of the roots of a simple Lie group byvectors of the Euclidean space R" . The possibility of such a representationfollows from the fact that the Cartan metric in a complex simple Lie groupdetermines the metric of the complex Euclidean space Rr in the Lie algebraof this group, and the Cartan subalgebra is an n-dimensional plane in thisspace; i.e., this algebra is the space CR" . Since the roots of a simple Liegroup are linear forms in the Cartan subalgebra, they can be represented bythe vectors of the space CR" , and since all the roots are linear combina-tions with integer coefficients of n linearly independent forms, these rootscan also be represented by vectors of the real Euclidean space R" . In thisrepresentation, a root a2 coy is represented by a vector a with coordinatesa2 . Figure 2.1 shows such systems for the groups A2 , B2 , C2 , and D2 .

A further simplification of the classification of complex simple Lie groupswas made by Eugene B. Dynkin (b. in 1924) in his paper [Dyl] of 1946under the same title as the van der Waerden paper [Wae] (see also [Dyn2]).The paper [Dyn I] was written in 1944 when the author was 19 years old;Dynkin followed the advice of Gel'fand, whose seminar he participated atthat time.

Dynkin introduced the notion of "simple roots" of semisimple Lie groups.If we write the roots as linear combinations a2coA , A = 1, 2, ... , n orA = 0 , 1, 2, ... , n , with integer or rational coefficients a2 , we will say

that a2coA > 0 if the first nonzero coefficient a2 is positive and that a roota = a2coA is greater than a root b = bA if the difference a - b is positive.A root is called simple if it is positive and cannot be represented as the sumof other positive roots. Any positive root can be represented as the sum ofsimple positive roots with positive coefficients. The Cartan-Killing basis rootsconsidered above are simple roots in Dynkin's sense.

50

al

a) A,, 0(X1

b) B,, 0

2. LIE GROUPS AND ALGEBRAS

a2

a2

a3

a3

a4 a,:-1 a,:

a4

Oa,1 a,,

a 1 a2 a3 a4 a,,-1(X,:

c) C,, --- -C=>=::)

d)

FIGURE 2.2

Dynkin introduced a very simple representation of systems of simple rootsin the form of graphs in which simple roots are represented by the graph dots.These dots are not joined if the corresponding vectors are orthogonal, theyare joined by a line if the angle between vectors is 120° , and they are joinedby a double line if the angle between vectors is 13 50 . In his papers [Dyn 1 ]and [Dyn2], Dynkin indicated the lengths of vectors representing the rootsby special marks next to the corresponding dots. Later, in the 1950s, herepresented the longer vectors by black dots and the shorter vectors by whitedots. Lev S. Pontryagin (1908-1988) in his book Topological groups [Pon2],used the Dynkin graphs*, but he did not show the lengths of vectors. JacquesTits (b. 1930), in the paper On certain classes of homogeneous spaces ofLie groups [Ti 1 ] (1955), suggested, in the case when the vectors representingthe roots have different lengths, putting the sign > in the direction of thedot representing the vector of smaller length. At present, the majority ofmathematicians use the Dynkin graphs in the form suggested by Tits, andthe white and black dots are used in the modification of the Dynkin graphssuggested by Ichiro Satake (b. 1927) for another purpose. Nevertheless,Joseph A. Wolf (b. 1936), who in the first editions of his book Spaces ofconstant curvature [Wo2] used the the Dynkin graphs in the Tits form, inthe last edition of this book, returned to the form of these graphs used byDynkin in the 1950s. Dynkin himself called his graphs "schemes of angles".Tits in the paper [Til] called them the "Schlafli figures". Wolf, in the book[Wo2], used the term the "Schlafli-Dynkin diagram". Because of a similarityof the Dynkin graphs with the Coxeter diagrams for groups generated byreflections (we will discuss these groups later), these graphs are sometimescalled "Coxeter-Dynkin graphs". Figure 2.2 shows the Dynkin graphs in theTits form for the complex simple groups An , Bn, C , and Dn D.

*Editor's note. Or diagrams. Same for Coxeter and Satare graphs.

§2.7. ROOTS OF EXCEPTIONAL COMPLEX SIMPLE LIE GROUPS 51

§2.6. Isomorphisms of complex simple Lie groups

Killing noted the isomorphisms between some complex simple Lie groups:the isomorphism of the groups Al , B1 and C1 , the isomorphism of thegroups B2 and C2, and the isomorphism of the groups A3 and D3 inaddtion to the fact that the group D2 is not simple and is isomorphic to thedirect product of two groups A 1 . Since Killing actually considered not Liegroups but their Lie algebras, the isomorphism of groups stated by him is inreality a local isomorphism.

In the case when two simple groups are isomorphic or locally isomorphic,the vector systems representing their roots or the Dynkin graphs of thesegroups are similar. For the groups Al , B1 , and C1 , the vector systemsconsist of two opposite vectors a and -a and the Dynkin graphs consist ofone point alone (Figure 2.3a); for the groups B2 and C2, the vector systemshave the form shown in Figures 2.1 b and 2.1 c, and the Dynkin graphs havethe form shown in Figure 2.3c; and for the groups A3 and D3, the Dynkingraphs have the form shown in Figure 2.3d.

In the case of a semisimple group, which is a direct product of a few simplegroups, the vector systems of root systems consist of a few systems of vectorsfor simple groups located in mutually orthogonal subspaces, and the Dynkingraphs consist of a few Dynkin graphs for simple groups. An example of thelatter group is the group D2 , which is isomorphic to the direct product oftwo groups Al = B1 = C1 (Figures 2.1d and 2.3b).

Al=Bi=C1

Oa1 D2

al

0 B2

Oat C2

a) b)

FIGURE 2.3

A3al a2 a3

D3

d)

§2.7. Roots of exceptional complex simple Lie groups

The complete systems of roots of characteristic equations of the simpleLie groups in the five "exceptional classes" G2, F4, E6, E7, and E8 can bewritten as:

(2.23)

G2 : (.w ` - cw3 ,

F4 : acv`

E6 : cvr - to ,

±>l(d-3cv'), i, j =0, 1, 2;*(0 - f() , C.t14)

± v"2-

i , j = 1 , 2, 3, 4;(4wO+w1_w1_wk -(V!)

i,j,k,1=1,2,... 2 6;


2*-,2-WO--Wl

FIGURE 2.4

E7 : ±(Z> .w1-oh-o)+wk-o1),h, i, j, k, l = 0, 1, ... 97;

E8 : ±(0i CvJ , ±(-L >i Col - w ) , ± (2 > i Co 1 - Ct - Cok - Cv!

i,j,k,I=1,2,...,8.The systems of simple roots for these groups have the form:

(2.24)

G2: a1 =W1 -(02, a2 =W0+w1 -2(02;F 4 : a 1 =a)2 - W3 , a2 = C O - C04 , a3 = Coo

a4=

E6: a1 =cot -(02, a2 =Cv2-Cv3,a3 = Cv - C0 , a4 = CO -CO5

,

a5=Cv5-Cv6, a6= 2 w0-2(CV!+w2-+w -Cv4-Cv5-w );E7: a1 =Cv1 -Co2, a2=(02-(0 3,a3=Cv3-CO4, a4-Coo-to 5

a5 =CVS-CO6, a6 = -CV1 -CV2-c3+CV4+CV5+Cc)6+w 7 ),E8: a 1 = Cv2 _(0 3 , a2 = C03 - (04 , a3 = (04 - Cv5 , a4 = C05 - C06 ,

a5 =Ct)6-Co7

a6 = (07 -Cv8 a7 = 2(Cc)1 -Cv2 _(0 3 - Cv4+Cv5 +Co6+Cv7+Co8ag=(0 +CVg

For these groups there are also the vector systems of root systems and theDynkin graphs: Figure 2.4 represents the vector systems of root systems forthe group G2, and Figure 2.5 the Dynkin graphs for all five exceptionalsimple Lie groups.

In the case of the group G2 , the angle between vectors representing simpleroots is 150° : in this case the corresponding dots of a diagram are joinedby a triple line. Van der Waerden in his paper [Wae] showed that the vec-tors representing a root system of a simple Lie group can form only theangles 90°, 60°, 45°, 30°, 120°, 135° , 150° , and 180° ; the lengths of non-orthogonal vectors are in no way related to each other, the lengths of vectorsforming the angles 45° and 13 5° are related by b2 = 2a2 , and the lengthsof vectors forming the angles 30° and 1500 are related by b = 3a . These2 2

§2.8. THE CARTAN MATRICES 53

a) G2 b) F4 C) E6a1 a2 a1 a2 a3 a4 a1 0'2 a3 a4 a5

0 a6

d) E7a 1 a2 a3 a4 (X55 a6

0

e) E8

a1 a2 a3 a4 a5 a6 a7

0 ag0c7

FIGURE 2.5

results give a rather simple method of classification of simple Lie groups.The method of classification, used by Killing and Cartan and based on com-putation of determinants, is much more complicated.

Since in Cartan's thesis Lie groups were considered as transformationgroups, in this work he also gave a representation of the exceptional sim-ple Lie groups in the form of transformation groups: these groups are rep-resented there as certain subgroups of projective transformations. In whatfollows, we will present simpler geometric realizations of these groups basedon Cartan's later results.

§2.8. The Cartan matrices

In the calculations of Killing and Cartan, the integers a,j appeared often.These integers can be defined with the help of the inner products of thevectors representing simple roots ai as follows:

(2.25) a1 =2ai , ai

i (a,, a!)

Cartan called these numbers the "fundamental integers". Possibly this isthe reason why at present these matrices are called "Cartan matrices". For-mula (2.25) shows that all diagonal entries of the Cartan matrices are equalto two, and all nondiagonal entries of these matrices are nonpositive, and, ingeneral, these matrices are not symmetric. For the groups An , Bn , C , andDn these matrices have the form:

(2.26)

2 -1 0 0 0 2 -1 0 0 0

-1 2 -1 0 0 -1 2 -1 0 00 -1 2 0

0) ...

.........................o 0 0 2 -1 0 0 0 2 -20 0 0 -1 2 0 0 0 -1 2

0 0 0 - # - 2 -1) 0 0 0 -1 2 00 0 0 -1 2 00 0 0 -1 20 0 0 -1 0 2

.........................................

-1 2 -1 0 0 -1 2 0 0 0 U

f2-1O..-OO\ (2_i.-.0000\I

7

U U11 21- - -


and for the groups G2 2F4 , E6 , E7 and E8 , the Cartan matrices are:

(2.27)2 0 -i 0 0 0

2 -i 0 0 0 2 0 -1 0 02 - 1 ) -1 2 -2 0 -i 0 z -i 0 0

0 0 -i 2 0 0 0 -i 2 -10 0 0 0 -i 2

2 0 -1 0 0 0 0 02 0 -1 0 0 0 0

0 2 0 -1 0 0 0 00 2 0 -1 0 0 0 -1 0 2 -1 0 0 0 0-1 0 2 -1 0 0 0

0 -1 -1 2 -1 0 0 00 -1 -1 2 -1 0 0

0 0 0 -1 2 -1 0 00 0 0 -1 2 -1 0

0 0 0 0 -1 2 -1 00 0 0 0 -1 2 -1

0 0 0 0 0 -1 2 -10 0 0 0 0 -1 2

0 0 0 0 0 0 -1 2

Note that for the groups An , the determinants of the Cartan matrices areequal to n + 1 , for the groups Bn and Cn they are equal to 2, for the groupsDn they are equal to 4, for the groups G2 , F4 and E8 they are equal to 1, forthe groups E6 they are equal to 3, and for the groups E7 they are equal to 2.These numbers, which are called the connection indices, determine importantalgebraic and topological properties of these groups.

In the theory of simple Lie groups, the inverse matrices of the Cartanmatrices play an important role. For the groups An) Bn , Cn and Dn , thesematrices have the following form:

(2.2s)n n-1 n-2 3 2 1

n+1 n+1 n+1 n+ l n l n1 ln-1 2n-2 2n-2 6 4 2n+1 n+1 n+1 n} l n{ 1 n1 ln-2 2n-4 2n-6 9 6 3n+1 n+1 n+1 n+1 n+1 n+1...........................................................

2 4 6 2n-4 2n-2 n-1n+l n}1 n}1 n+1 n+1 n+1

1

nil niln-1 n-1 n

n+1 n+1 n+1 n+1 n+1 n+1

/1 1 1 ... 1 1

1 2 2 ... 2 2

1 2 3 ... 3 3...................1 2 3 ...

l 1 3

2 2 ... 2 2

1 1 1 ... 1 2

1 2 2 ... 2 1

1 2 3 ... 3 2.....................................

2 3 ... n- 1 n-2

1 2 3 ... n- 1 2

1 1 1 ... 1 2 21 2 2 ... 2 1 1

1 2 3 ... 3 3 32 2..............................................

1 2 3 ... n- 2 n-2 n-21 1 3 n-2 n n2-22

1

2n?2 n42

4

2 1 2 2 4 4

§2.9. THE WEYL GROUPS 55

and for the groups G2 , F4 , E6 , E7 and E8 , these matrices have the form:

(2.29)

2 3 4 2

C2 11 3 6 8 43 2

2

4

6

3 4

4 63

4 6

41 3 22 3

46

1 2

23

43

2

3

4 23 2 1

5 3

7 10 86 4 2

10 15 129 6 3

206 4

210

23 2

If we denote by 7ri the vectors 2a,/(a,, ai) and by it' the vectors of thebasis dual to the basis (7ri) (i.e. the inner products (7ni , 7t-) are equal toof ), then the Cartan integers are equal to the coordinates of the vectors aiin the basis (it'), i.e.

(2.30)iai = ail7r

and the coordinates of the vectors 7i' in the basis (a,) are equal to theentries A'J of the matrix A-' = (A') which is the inverse matrix of theCartan matrix, i.e.,

(2.31) 7i' = A'J aJ..

The integer multiples of the vectors ai define the root lattice of a simpleLie group, and the integer multiples of the vectors 7t' define the weight latticeof this group. These lattices are discrete additive groups of vectors; the firstof these groups is a subgroup of the second one, and the order of the quotientgroup of the second group by the first one is equal to the connection indexof the Lie group.

§2.9. The Weyl groups

Killing in the paper [Ki12] and Cartan in his thesis considered transforma-tions of the root systems of complex simple Lie groups. Since these roots areroots of the characteristic equation (2.18) of the group, these transformationscan be considered as elements of the Galois group of this equation. This waswhy Cartan called the group of these transformations the "Galois group of


the Lie group". Both Killing and Cartan connected an involutive substitutionSa of the system of roots with every root a, considered products of thesesubstitutions, and wrote these substitutions and their products in the formof linear transformations where the linear transformations corresponding tothe substitutions S' have the form of reflections.a

Weyl in his paper [Wey3J showed that the transformations correspondingto the substitutions S. can be written in the form

(2.32) a) a,

and, in the metric of the Euclidean space R" in the Cartan subalgebra, thesetransformations are reflections in hyperplanes of R" orthogonal to the vec-tors a. Moreover, the linear transformations corresponding to the productsof the substitutions Sa are rotations of the space R" , i.e. the matrices ofthese transformations are orthogonal matrices of the group o,, . The prod-ucts of n reflections S. corresponding to simple roots are especially im-portant. At present, these transformations are called the Coxeter transforma-tions. For any order of the factors Sa , the eigenvalues of the matrices of thesetransformations have the form eMi , where the numbers Mi have the form2ira1/h , and the numbers ai are integers called the exponents of a simpleLie group, and the number h is the Coxeter integer of this group. The latternumber is connected with the rank n of the group and its dimension r bythe relation

(2.33) hr - n

n

In the paper [Wey3), Weyl also considered the group of rotations of thespace Rn generated by these reflections. He used for this group the name"group (S)". At present, this group is called the Weyl group.

If at the common initial point of the vectors a . (simple roots of a Liegroup), we construct the hyperplanes H1 orthogonal to these vectors, thenconnected sets of points of the space R" not belonging to the hyperplanesH1 are called open Weyl chambers, and their closures are called closed Weylchambers. The Weyl chambers have the form of cones with vertices at thecommon initial point of the vectors a . and with faces that are faces of ann-faced angle. The Weyl chambers are the fundamental domains of the Weylgroup.

The exponents ai of simple Lie groups are also called the exponents aiof its Weyl group. These integers belong to the interval 1 < ai < h where his the Coxeter integer (2.33). All integers of this interval that are relativelyprime with h are integers al = h - ah_ i+ I . These integers are equal to:

§2.9. THE WEYL GROUPS 57

(2.34)for the groups An : 1 , 2) 3) ... , n - 1 , n (h = n + 1) ;for the groups Bn and Cn : 1, 3) 5) ... , 2n - 1 (h = 2n);for the groups Dn (n is even) : 1,3,5,... , n - 2 , n - 1, ... , 2n-3;for the groups Dn(n is odd) : 1 , 3, 5, ... , n - 3, n - 1, n - 1,

n+1, ... , 2n-3 (h=2n-2);

for the group G2 : 1, 5 (h = 6) ;for the group F4 : 1, 5, 7, 11 (h = 12) ;

(2.3 5) for the group E6 : 1, 41 51 71 81 11 (h = 12) ;for the group E7 : 1 , 5 , 7 , 9 , 11, 13 , 17 (h = 18) ;for the group E8: 1, 7, 11, 13, 17, 19,23,29 (h = 30).

As we will see later, the exponents of simple Lie groups play an importantrole in the most unexpected questions of the theory of simple Lie groupssuch as the topology of real simple Lie groups and the theory of finite groupswhich are the analogues of simple Lie groups.

Influenced by the Weyl paper [Wey3], in 1925 Cartan returned to the the-ory of roots of simple Lie groups and showed in the paper The duality princi-ple and the theory of si m pl e and semi si mpl e groups [82] that for all simple Liegroups, except the groups An , Dn and E6 , their Weyl groups coincide withthe Galois groups, and the Weyl groups of the excluded groups are invariantsubgroups of their Galois groups. Moreover, he proved that, for the groupsAn) Dn (n 4), and E6 , the quotient group of the Galois group by the Weylgroup is isomorphic to the multiplicative group { 1, -1 I , and, for the groupD4 , it is isomorphic to the general group of permutations of three elements.This is connected with the fact that the Dynkin graphs of the groups An) Dn(n 0 4), and E6 possess bilateral symmetry and also with the "duality prin-ciple" of the spaces where the groups An , Dn , and E6 act, and with the factthat the Dynkin graph of the group D4 (Figure 2.6) possesses the trilateralsymmetry and with the "triality principle" in the space where the group D4acts (the latter principle was introduced by Cartan in the paper [82] ).

The finite groups of reflections of the space Rn generated by reflections inhyperplanes of this space were studied by Harold Scott MacDonald Coxeter(b. 1907) in the paper Discrete groups generated by reflections [Cox 1 ] (1934)where he gave a complete classification of these groups and characterizedthem by means of graphs whose structure is very close to the Dynkin graphswhich appeared ten years later. The vertices of the Coxeter graphs represent


[3,: - 1 ] ...

n

[3,5]5

...[311-2,41

[3':_3, 1, 11 ...O----K-

[3,4,3]4

[3, 3, 5]5

[32' 2, 11

[33.2.1] 0

[34.2,1] O

0

FIGURE 2.7

hyperplanes of the space Rn , reflections in which generate the group. Thevertices are not joined if corresponding hyperplanes are orthogonal. Theyare joined by a line without a numerical mark if the angle between thesehyperplanes is 60° and with the mark n if this angle is 1800/n. Figure2.7 shows the Coxeter graphs of finite groups generated by reflections. If theCoxeter graphs of these groups consist of lines without marks, the groups aredenoted by [3n] if the graph consists of n lines and does not have branchesand by [31, m, n] if the graph consists of three branches having I, m , and nlines, respectively. If the Coxeter graph consists of a few lines with marksk, 1, and m (the absence of mark is counted as the mark 3) and does nothave branches, the group is denoted by [k, 1, m]. If the graph consists ofgraphs of different types, the group notation consists of the notation of thecorresponding groups.

The group [3n-1] is the group of symmetries of the regular n-dimensional

simplex of the space Rn . The group [3 n- 2 , 4] is the group of symmetries ofthe n-dimensional cube of the same space. The group [3n-3' 1,1 ] is the groupof symmetries of the n-dimensional "semicube", i.e., the convex polytopeobtained from the n-dimensional cube by selection of one vertex on eachedge and rejection of the other vertex of this edge. The group [n] is the

0

0---0

0

X2.9. THE WEYL GROUPS 59

group of symmetries of a regular n-angle. The group [3, 5] is the groupof symmetries of an icosahedron and a dodecahedron. The group [3, 4, 3]is the group of symmetries of a regular polytope of the space R4 with 24faces whose vertices are 16 vertices of the four-dimensional cube and thereflections of its center of symmetry in its eight faces. The group [3, 3, 5]is the group of symmetries of regular polytopes of the same space R4 with600 and 120 faces. The group [32 ,2' 1 ] is the group of symmetries of a cubicsurface with 27 rectilinear generators in the projective space B3

. The group[33' 2.1] is the group of symmetries of a quartic (a curve of the fourth order)With 28 double tangents in the plane B2 . The group [34. 2 ' 1 ] is a subgroupof the group of permutations of a set of 120 elements which was called byJordan "the first hypoabelian group".

The Weyl group of the complex simple group An is isomorphic to thegroup [3n-1 ] of symmetries of the regular n-dimensional simplex. TheWeyl groups of the complex simple groups Bn and Cn are isomorphic tothe group [3n-2 , 4] of symmetries of the n-dimensional cube. The Weylgroup of the complex simple group Dn is isomorphic to the group [3n-3' 1,1 ]

of symmetries of the n-dimensional "semicube". The Weyl groups of thecomplex exceptional simple groups G2 , F4 , E6 , E7 , and E. are isomorphicto the groups [6], [3, 4, 3], [32.2)1] , [33.2.1] and [34.2.1] , respectively.Isomorphism of the Galois groups of the characteristic equations of the lastthree Lie groups and three last finite groups was shown by Cartan as far backas 1894 in his note on reduction of the group structure to its canonical form[4] and was proved in the paper on reduction of the structure of a finite andcontinuous group to its canonical form [9] (1896) (for the group E,, Cartanmade this result more precise in the paper [82]). Commenting on the solutionof the characteristic equation for the "groups of type E" of rank 1, Cartanwrote in [9]: "For the latter ones, if I = 6, it is reduced to an equation of thesame nature as the equation defined by 27 generators of the cubic surface;if l = 7, it is reduced to an equation of the same nature as the equationdefined by 28 double tangents to the curve of the fourth order; and finally, ifl = 8, it is reduced to the equation of 120th degree whose group is the firsthypoabelian group of 120 letters" [9, p. 57].

Cartan returned to these groups in one of his last works, Some remarkson 28 double tangents of a plane quartic and 2 7 lines of a cubic surface [ 184](1946), where he, using the term "Galois group of a configuration" for thegroup of transformations (for the two cases which he considered, these trans-formations are collineations) keeping this configuration fixed, formulated thefollowing theorem: "The Galois group of the characteristic equation of thesimple Lie group of rank 7 and order 133 is isomorphic to the Galois groupof the configuration of 28 double tangents of a plane quartic without dou-ble points. The Galois group of the characteristic equation of the simpleLie group of rank 6 and order 78 is isomorphic to the Galois group of the


configuration of 27 lines of a cubic surface without double points" [ 184, pp.1-2].

Comparison of Figures 2.3 and 2.5 with Figure 2.7 shows that the Dynkingraphs of complex simple Lie groups differ from the Coxeter graphs of theWeyl groups of these groups only by presence of inequality signs. The Coxetergraphs were first applied to the theory of simple Lie groups by Coxeter himselfin the paper [Cox2] with the same title as [Coxl]. This paper [Cox2] waspublished as Appendix to the Weyl paper [Wey4] (1935). Later, in the paperGroups of reflections and enumeration of semisimple Lie rings [Wit] (1941),Ernest Witt (b. 1914) applied the Coxeter graphs to classification of simpleLie algebras.

§2.10. The Weyl affine groups

In the paper [Coxl] Coxeter found also all infinite discrete groups of mo-tions of the space R" generated by reflections in hyperplanes of this space.These groups are also described by graphs similar to the graphs of the finitegroups of this type. Figure 2.8 shows the notation and the Coxeter graphs

13':

[4' 3n-2, 4]- - -

4 4

3n-3,1,1] 0[4' V

4

[31, 1, n - 5, 1, 1]

[00]

[3, 6]

[3, 3, 4, 3]

132,2, 2]

[333 , 1]

[35,2,1]

0

00

0

6

Q

0

0

0 0

0

FIGURE 2.8

§2.10. THE WEYL AFFINE GROUPS 61

of these groups. The group [cxl is the group of motions of the Euclideanline R1 generated by reflections of this line in two of its points or the groupof motions of the space Rn generated by reflections of this space in twoof its parallel hyperplanes. The groups 3n are the groups of symmetriesof polyhedral angles. The group [4, 3n-1 , 4] is the group of symmetriesof honeycombs of the space Rn formed by tessellation of this space withn-dimensional cubes. The group [3, 6) is the group of symmetries of honey-combs of the plane R2 formed by tessellation of this plane with equilateraltriangles or regular hexagons. The group [3, 3, 4, 3] is the group of sym-metries of honeycombs of the space R4 formed by tessellation of this spacewith regular polytopes.

The affine Weyl group of a complex simple Lie group is defined as theinfinite discrete group of motions of the space Rn determined by the vec-tors of the root system in the Cartan subalgebra of the Lie algebra of thisgroup, and the metric of this space is induced by the Cartan metric in theLie group. To find the affine Weyl group, Coxeter supplemented the reflec-tions in hyperplanes generating the Weyl group by the reflection in one morehyperplane passing through the common initial point of the vectors of theroot system, the terminal point of one of these vectors and orthogonal to thisvector. As this vector, Coxeter took the vector representing the minimal rootin the order of roots which was later defined by Dynkin. The Dynkin graphs,supplemented by one more dot representing the root which is opposite to themaximal root, are called the augmented Dynkin graphs. It turned out thatthese graphs are very useful in solving many problems related to simple Liegroups. Figure 2.9 (next page) shows the extended Dynkin graphs for simpleLie groups. If the minimal root has the form u = >j, mia! , where ai aresimple roots, then on these graphs, the dots representing the roots ai aremarked by the number mi and the new dot is marked by the number 1.(Note that the Dynkin graph itself can be obtained from the extended oneby deleting any dot marked 1.) Actually,. the extended Dynkin graphs wereconsidered by Dynkin himself in the paper [Dyn2] as "impossible graphs"(see also the book by Pontryagin [Pon21).

Comparison of Figures 2.8 and 2.9 shows that the extended Dynkin graphsof complex simple Lie groups differ from the Coxeter graphs of the affine Weylgroups of these groups only by the presence of the inequality signs, i.e., injust the same way as the Dynkin graphs of complex simple Lie groups differfrom the Coxeter graphs of the Weyl groups of these groups.

Note that the connection index of simple Lie groups is exactly equal to1, 2, 3, 4, and n + 1 in the cases when the extended Dynkin graphs ofthese groups do not possess the symmetry or possess the bilateral, trilateral,quadrilateral symmetry, or the symmetry of order n + 1 , respectively. Theconnection index of simple Lie groups is equal to the numbers of marks 1 inits extended Dynkin graph.

62

a) Ant)

b)Bn 1 )

C) Cn (1)

d) D(1)

e)

0

G(1)2

F{1}4

2. LIE GROUPS AND ALGEBRAS

ON.

g) E(1) 0-6

h) E(1) 00

0

0

b

0

FIGURE 2.9

As for the usual Weyl groups, for the affine Weyl groups of simple Liegroups, it is also possible to define the fundamental domains which are thesimplices whose n faces coincide with the faces of the Weyl chamber. Some-times, these domains are called the Weyl alcoves. The fundamental domainsof the affine Weyl groups of simple Lie groups were first considered by Weylhimself in his work [Wey3]. However, their relations were not indicated.Weyl used this notion to prove that the connection group (the Poincare group)of compact real semisimple groups is finite (for simple Lie groups withoutcenter, the order of this finite group is equal to the connection index of the Liegroup, i.e., the determinant of its Cartan matrix). Weyl used the finiteness ofthese groups for proving the complete reducibility of linear representationsof complex semisimple Lie groups.

Shortly after the publication of the Weyl paper [Wey3], in the paper Thegeometry of simple groups [103] (1927), Cartan described the fundamentaldomains of usual and affine Weyl groups, and, in the addendum [ 113] (1928)to this paper, he proved that any irreducible finite group generated by reflec-tions of the space Rn in its hyperplanes possesses a fundamental domain

§2.11. ASSOCIATIVE AND ALTERNATIVE ALGEBRAS 63

which cuts a spherical complex on a hypersphere with center at the pointof intersection of the hyperplanes. In the same paper, Cartan proved theuniqueness of the maximal and minimal roots relative to an arbitrary systemof roots.

§2.11. Associative and alternative algebras

As we have already noted, in the 19th century, along with the groups andfields defined by Galois, a number of new numerical systems were introduced.These numerical systems, which are generalizations of the field of complexnumbers, originally were also called systems of complex numbers. Lateron, in order to distinguish them from the usual complex numbers, mathe-maticians began to call them systems of hypercomplex numbers or associativealgebras.

Along with associative algebras, i.e., vector spaces where an associativemultiplication of vectors is defined which is distributive with respect to theiraddition and commutes with multiplication of vectors by numbers, moregeneral algebras were considered. These new algebras differ from associativealgebras by the fact that multiplication of their elements is not associative.The Lie algebras which we discuss in this chapter are nonassociative algebras.

If in a vector space a basis {e1} is given (for algebras, the elements ejare often called the "units" of an algebra), multiplication of elements of analgebra is defined by the formula

(2.36) e.e = ck.ek.

Formulas (2.12) are a particular case of formulas (2.36). They differ in thatthe operators Xa in formulas (2.12) play the role of the vectors e! , and thecommutators [X X91 play the role of the products e! ej .

For an arbitrary algebra, the numbers ck are also called its "structureconstants". For Lie algebras, multiplication is neither commutative nor as-sociative. For these algebras, these properties are replaced by the property ofanticommutativity (2.13) and the Jacobi identity (2.14). While multiplica-tion of elements of Lie algebras is written in the form c = [ab) , multiplicationin associative algebras is written in the form c = ab. Multiplication of ele-ments of nonassociative algebras with the "alternativity" property (any twoelements of an alternative algebra generate an associative algebra) is writtenin the same form as in associative algebras.

The appearance of algebras was closely connected to the appearance ofvectors. The simplest algebra is the field C of complex numbers with theunits 1 and i, i2 = -1 . In the works of Leonhard Euler (1707-1783), JeanLe Rond D'Alembert (1717-1783), Gauss, and Augustin Louis Cauchy(17 89-1857 ), the geometric interpretation of complex numbers was estab-lished in the "plane of a complex variable" with addition according to the


parallelogram rule and multiplication according to which the moduli of com-plex numbers are multiplied and their arguments are added. Following this,in the first half of the 19th century, attempts were made to "generalize thecomplex numbers for the space", i.e., to construct a number system with threeunits. These algebras were constructed by Augustus de Morgan (1806-1871)and Charles Graves (1810-1860). However, in all such algebras there were"divisors of zero", i.e., elements a and b which are themselves different fromzero but whose product is zero. The best known among these algebras is thealgebra of "triplets" which is isomorphic to the direct sum of the fields R andC, i.e., to a set of pairs (a, a) of numbers where a is a real number and ais a complex number and where addition and multiplication are defined bythe following formulas: (a, a) + (b, /3) = (a + b, a + /3) , (a, a)(b(ab, a/3) .

In 1844 William Rowan Hamilton (1805-1865) discovered the algebraof quaternions, which is the algebra with four units 1, i, J, k) i2 = j2 =

1, i j = -j i = k . This algebra was of significantly greater importanceboth for algebra and geometry. Hamilton called expressions of the formxi + Y j + zk "vectors" and viewed quaternions of general type as sums ofscalars (real numbers) and vectors. The algebra of quaternions, which isdenoted by H after Hamilton, is a noncommutative field (skew field). Asin the field C, in the field H, a transition to the conjugate element a - a(which is multiplication of the quaternion units i, j and k by -1) is definedsatisfying the property:

(2.37) P =Rte.

The product as , which is equal to the sum of squares of coordinates of thequaternion, is called the square of the modulus lal of the quaternion. Themodulus lal in the fields H and C possesses the property

(2.38) lafll = lallfll.

Two algebras important for both algebra and geometry are connected withthe name of Arthur Cayley (1821-1895 ). The first of these algebras is thealgebra 0 of octaves with eight units 1, i , j , k , l , p , q. r , i 2 = j 2, l2 =

- 1 , ij= ji =k, it=-li=p, kp=-pk=q, jp=-pj=r. Octavesare often called "Cayley numbers" or "Graves-Cayley numbers" since almostat the same time as Cayley, they were discovered by John Thomas Graves(1806-187 0), brother of Charles Graves. The algebra 0, like the algebra H,is a skew field, and its multiplicaion is not associative but alternative in thesense indicated above. As in the fields C and H, in the field 0, the transi-tion to the conjugate element a -' a (which is multiplication of the octaveunits i , j , k , l , p , q , and r by -1) is defined satisfying property (2.37) inaddition to the modulus lal whose square 1a12 = as is equal to the sum ofsquares of coordinates of the octave and which satisfies property (2.38). We

§2.11. ASSOCIATIVE AND ALTERNATIVE ALGEBRAS 65

will see below that the field 0 is closely connected with exceptional simpleLie groups.

The second algebra discovered by Cayley is the algebra of matrices; itappeared in his Memoir on the theory of matrices [Cay 11 (1858). The al-gebra Rn of real matrices of order n consists of square arrays A = (air )which are added and multiplied according to the rules: A + B = (ail + bit )and AB = (Ej ai b .k) . The algebra Rn has n2 units Ei . discussed ear-lier. The algebras Cl , and Hn of complex and quaternion matrices can bedefined in the same manner. The algebras Rl , C,r , and H,r possess divi-sors of zero. In the algebras Rn and Cn , such elements are matrices withzero determinant. In the algebra Hn , such elements are matrices with zero"semideterminant" a real number equal to the determinant of a real or com-plex matrix representing the given quaternion matrix in the algebras C2n andRan containing a subalgebra isomorphic to Hn . If we introduce the notionof tensor product A ® B of algebras A and B with bases {e1} and {f} asan algebra with the basis {e1fj (ejfa = fe1), then the algebras Cn and Hncan be defined as the tensor products Rn ® C and Rn ® H.

The founder of multidimensional algebra and geometry Hermann Grass-mann (1809-1877) in 1844 in the work The science oflinear extension [Gral],having defined the n-dimensional linear space, also introduced the "exte-rior product" of vectors of this space. Later, in his geometrical works,Cartan often used this notion. At present, the exterior product of vec-tors x1 , x2 , ... , xk is written in the form x1 A x2 A A xk . It is un-changed by an even substitution of the vectors x1, x2 , ... , xk , multipliedby -1 for an odd substitution of them, and equal to zero when the vec-tors x1, x2 , ... , xk are linearly dependent (in particular, x A x = 0). Thevectors x1 , x2 , ... , xn , along with all their possible exterior products xi A

A - A xi , i 1 < i2 < < ik , form a basis of an algebra with 2" units.xi2k

A modification of the Grassmann algebra is the algebra with 2" unitsconstructed by Clifford in the paper Applications of Grassmann's extensivealgebra [C12] (1878). He wrote the units of his algebra in the form 1, el , e2 ,

A A A en and e, 112., , ik = el e2 ek where e2 _ -1 and the products e,1

i2

. , , ik

arenot changed if their indices undergo an even substitution and are multipliedby -1 if the indices undergo an odd substitution (i.e., products of distinctfactors behave in the same way as exterior Grassmann products).

If we denote the Clifford algebra with n units ei by the symbol Kn+1 ,

then the algebra K1 coincides with the field R of real numbers, the algebraK2 with the field C of complex numbers, and the algebra K3 with the fieldH of quaternions. Other algebras Kn+1 are generalizations of the field Hof quaternions but in a direction other than that of the field 0 of octaves.Namely, all algebras Kn+J are associative, and the algebra K4 is isomorphicto the direct sum H (D H of two fields H, the algebra K5 is isomorphicto the algebra H2 of quaternion matrices of second order, the algebra K6


is isomorphic to the algebra C4 of complex matrices of fourth order, andthe algebra K7 is isomorphic to the algebra R8 of real matrices of eighthorder. Clifford showed that the algebras Kn+1 are isomorphic to the followingalgebras: K8m+ 1 = R24 , K8m+2 = 0241 , K8m+3 = H24,,, , K8m+4 =H24 , K8m+5 = H24 +l , K8m+6 = C24 i+2 , K8m+7 = R24, ,+3 , Kg(m+l) = R2 4»i+3

R24m+3 .

In 1872, in the paper A preliminary sketch of biquaternions [C11), Cliffordintroduced two modifications of the algebra C now known, respectively, asthe algebra of split complex numbers and the algebra of dual numbers. Thesealgebras are denoted by 'C and °C , respectively. The algebra 'C has theunits 1 , e, e2 = 1 and the algebra °C has the units 1, e, e2 = 0 . If inthe algebra 'C we take the basis e+ = (1 + e)12, e_ = (1 - e)/2 for whiche2 = e , ei = e_ , e+e_ = 0, we see that this algebra is isomorphic to thedirect sum R ® R of two fields R. Next, Clifford extended the notion ofbiquaternions (complex quaternions) introduced by Hamilton, to split com-plex and dual quaternions. He called the Hamilton biquaternions hyperbolic,and he called split complex and dual quaternions elliptic and parabolic bi-quaternions, respectively. The algebras of hyperbolic, elliptic, and parabolicbiquaternions are the tensor products H ® C , H ®'C , and H ® °C .

The general notion of the associative algebra whose particular cases arethe algebras Rn , Cn , and Hn and similarly defined algebras 1 Cn and °Cnas well as the algebras Kn , was introduced by Benjamin Peirce (1809-1880)in his posthumously published paper Linear associative algebras [Pe] (1881).Peirce introduced the notion of nilpotent element one of the powers of whichis equal to zero (the "dual unit" e of the algebra °C is an example of suchan element) and the notion of "idempotent element" for which a2 = a (thesplit complex numbers (1 ± e)/2 are examples of such elements). He usedthese notions for classification of algebras of small dimensions.

In 1883-1885 several papers by outstanding mathematicians on the the-ory of algebras appeared. In 1883 WeierstraB wrote a letter to H. Schwartz,a fragment of which was published in 1884 in the form of the note To thetheory of complex quantities formed by n principal units [Wei]. In the sameyear, Poincare's note on complex numbers [Poi2) appeared, and in 1885 thepaper [Ded] by Richard Dedekind (1831-1916), with the same title as Weier-straQ's note [Wei], was published. Poincare studied the relation between al-gebras (which Poincare called "systems of complex numbers", as WeierstraBand Dedekind had done earlier) and continuous groups, namely, "bilineargroups", i.e., groups of transformations (2.2) for which the functions f arelinear in both the variables xk and the variables as . WeierstraB showed thatany commutative associative algebra is isomorphic to the direct sum of a fewfields R and C . He called the elements of associative algebras "complexquantities" and the basis elements (units) of these algebras "principal units".

Dedekind's paper was devoted to finite algebraic extensions of the field

§2.12. CARTAN'S WORKS ON ALGEBRAS 67

Q of rational numbers. These extensions are similar to the Galois fieldsFq discussed earlier which are finite algebraic extensions of the fields E. ofresidue classes. Both extensions can be considered as algebras over the fieldsQ and R, respectively.

In 1892 Theodor Molien (18 61-1941) defended in Dorpat (now Tartu)his doctoral dissertation on systems of higher complex numbers. In 1893 thisdissertation was published in Leipzig (see [Mol]) in the same journal Math-ematische Annalen in which the Killing paper [Ki12J had appeared. Molienwas in close contact with German algebraists. In his dissertation, he gener-alized the notions of simplicity and semisimplicity used by Killing for Liealgebras to associative algebras. He also found a criterion of semisimplicityof a complex algebra in the form of nondegeneracy of the quadratic form

(2.39)Ic,, ch aia'

similar to form (2.20), i.e., reducibility of this form to the sum of squares ofall coordinates a` . Molien's dissertation dealt with complex algebras, and itsmain result is that any simple complex algebra is isomorphic to the algebraC. and any semisimple complex algebra is isomorphic to the direct sum ofsuch algebras.

§2.12. Cartan's works on algebras

Cartan's works on classification of simple and semisimple associative alge-bras were a natural development of his works on classification of simple andsemisimple Lie groups and algebras. Two of his notes on systems of complexnumbers [11] and on real systems of complex numbers [ 12] (1897) and theextensive paper Bilinear groups and systems of complex numbers [13] (1898)were devoted to this problem.

Following Poincare, in the latter paper Cartan used the term "systems ofcomplex numbers" for algebras and considered "bilinear groups" of transfor-mations connected with algebras. In addition to solving problems of clas-sification of simple and semisimple algebras, Cartan revised the notions ofsimplicity and semisimplicity of algebras and introduced the notion of an"invariant subsystem of the system of complex numbers", which is similarto the notion of an invariant subgroup of a group and is the most impor-tant particular case of an invariant subsystem "pseudonull invariant sub-system". An "invariant system" is a subalgebra remaining invariant undermultiplication by an arbitrary element of the algebra from the right or fromthe left. At present, such subalgebras are called ideals of an algebra. Theterm "ideal" was originated from the term "ideal prime factors" introducedby Ernst Kummer (1810-1893) in his theory of algebraic integers. In thering Z of regular integers an ideal (defined for a ring in the same manner asfor an algebra since an associative algebra is a ring with respect to additionand multiplication) consists of numbers that are multiples of an integer. This


was the reason, when Kummer encountered ideals in the rings of algebraicintegers, he considered them as sets of numbers that are multiples of "idealfactors". The Cartan term "pseudonull invariant subsystem" arose from theword "pseudonull" which Cartan used for nilpotent elements of algebras. Atpresent, such subalgebras are called radicals. An algebra is called simple if itdoes not contain ideals different from the algebra itself and zero. An algebrais called semisimple if it does not contain a radical.

In his paper, Cartan proved that any complex or real algebra is a directsum of a semisimple algebra and a "pseudonull subsystem", i.e., a radical, andthat any semisimple algebra is isomorphic to a direct sum of simple algebrasand a "pseudonull subsystem". Moreover, we saw earlier that in 1892 Molienproved that any complex simple algebra is isomorphic to the algebra Cn ofcomplex matrices. For the algebra C2 , which is isomorphic to the algebra ofcomplex quaternions, Cartan used the term the "algebra of quaternions"; forthe algebra C3 , following James Joseph Sylvester (1814-1897), he used theterm the "algebra of nonions"; and for the algebra Cn he used the term the"algebra of n -ions".2

Next, Cartan proved that any real simple algebra is isomorphic to eitherthe algebra Rn of real matrices or the algebra Cn of complex matrices orthe algebra Hn of quaternion matrices. In the first note mentioned above,Cartan announced results related to complex matrices and in the second oneto real matrices. In particular, it follows from Cartan's results that all Cliffordalgebras are simple or semisimple.

In 1898, in the German Encyclopaedia of Mathematical Sciences, the sur-vey paper The theory of usual and higher complex numbers [Stul] by Studyappeared where the development of the theory of algebras in the 19th centurywas summarized. Following Molien, Study used for algebras the name "sys-tems of higher complex numbers". In this paper, by analogy with split com-plex and dual numbers, Study defined the algebra 'H of split quaternions withthe units 1 , i, e, f having the properties i2 = -1, e2 = I , ie = ei = f.This algebra is isomorphic to the algebra R2 of real matrices of second order.In this paper, Study also defined the algebra °H of semiquaternions with theunits 1 , i, e , q having the properties i2 = -1 "C2 = 0, it = -El = 1. Semi-quaternions are often called "Study's quaternions". As in the fields C andH, in the algebras I C) °C, 'H , and °H , then the transition to the conjugateelement satisfying property (2.30) and the modulus lal satisfying the prop-erty (2.31) can be defined. The difference is that while in 'C and 'H theproducts ad are algebraic sums of squares of all coordinates, in °C and °Hthey are sums of squares of coordinates in 1 and i only. In the same manner,if in the definition of an alternative skew field 0 of octaves, one replacesthe unit 1 by the units e and e with the same properties as in the algebras'C , °C, 'H and °H, then the algebra 'G of split octaves and the algebra °Gof semi-octaves will be obtained where properties (2.37) and (2.38) also hold.

§2.13. LINEAR REPRESENTATIONS OF SIMPLE LIE GROUPS 69

Study's paper was translated into French and significantly revised by Car-tan. This revised translation was published under the title Complex numbers[27] in the French edition of Encyclopaedia of Mathematical Sciences (1908).While Study's original paper was 34 pages long, Cartan's extended translationwas 140 pages long.

After presenting the theory of Clifford algebras Kn along Study's lines,Cartan added that "it is possible to consider more general systems" in whichsome squares of ei are equal to -1 and some are equal to + 1 . At present,the algebras which differ from the algebras Kn , by the fact that for l of itsunits e2 = +I and for the remaining n - I - 1 units e? _ -1 , are denotedby Kn . Cartan noted that "all these systems are simple or semisimple" andindicated the structure of the algebras Kn and Kn in the following way.After introducing the number h = 1 - J:i e? , he indicated that Kn = R2if h = 1 (mod 8), KI l = C2 if h = 2 (mod 8), Kn = H2 if h =_ 3{mod 8) , K1 = R2n R2n_ 1 if h - 0 (mod 8), and K,,, = H21 -1 EDif h - 4 (mod 8). Cartan denoted the algebras R2 1 , C2n-1 , and Hen - I

by Sm , CSm , and QSm , respectively, and the direct sums R2n ED 1

and H2 - , by 2Sm and 2 QSm , respectively. He concluded hissupplement by saying that "these systems are reducible if h is a multiple of4" [27, p. 464]. Note that the algebra K2 coincides with the algebra 'C andthat the algebras K3 and K3 coincide with the algebra 'H.

§2.13. Linear representations of simple Lie groups

In his paper Projective groups, under which no plane manifold is invariant[37], published in 1913, Cartan constructed the theory of linear represen-tations of complex simple Lie groups. This theory is the foundation of anumber of mathematical theories that have important applications to mod-ern physics.

A linear representation of a group G is a homomorphic mapping of thisgroup into a subgroup of the group GLN of real matrices of order N orthe group CGLN of complex matrices of order N. A linear representationis said to be reducible if in a linear space of representation, i.e., in a linearspace, whose matrices of linear transformations form a representation of thegroup, there is a subspace which is invariant under these transformations. Alinear representation Sp is said to be completely reducible if the linear space ofrepresentation decomposes into a direct sum of invariant subspaces. In theseinvariant subspaces, representations V 1 , 921 ... , SOk of the group G occur.In this case, the representation (p is called the direct sum of these represen-tations and is denoted by c 1 ED 92 ED ' ' ' ED Spk . The title of the Cartan paperindicates that he considered irreducible linear representations. He used theterm "projective groups" for groups of linear transformations since matricesof linear representations can also be considered as matrices of collineationsof projective spaces. First, Cartan showed that all linear representations of


semisimple Lie groups are completely reducible. It follows from this thatthe study of general linear representations of these groups is reduced to theirirreducible linear representations.

A linear representation of a Lie group G induces a linear representationof the Lie algebra of this group. One type of linear representation of simpleLie groups was already considered in Cartan's thesis. In the same manneras in that case, one can show that for any irreducible linear representationof a simple Lie group the linear transformations X - [HXJ, where X andH are the matrices representing an arbitrary element x of the Lie algebraand an element h of the Cartan subalgebra of this algebra, have the sameeigenvectors, and the corresponding eigenvalues are linear combinations withrational (no longer integer) coefficients of the basis roots; as these roots, onecan take "simple roots". These linear combinations are called the weightsof the linear representations. Since for any two weights, as for any linearcombinations of simple roots, the notion "greater than" can be defined, it ispossible to distinguish the maximal weight among all weights of a linear rep-resentation. This maximal weight is called the dominant weight of this linearrepresentation. Cartan showed that a linear representation of a semisimpleLie group is completely determined by its dominant weight.

If two linear representations cp and V of a group G are given in M- andN-dimensional spaces with vector coordinates xi and ya , i= 1, 2 , ... ,M, a = 1121... , N, then the products xiya also undergo linear trans-formations forming a linear representation of the group G in an (MN)-dimensional space with vector coordinates z`a . This representation is calledthe Kronecker product of representations cp and and is now denoted by(P ® . Cartan showed that if the dominant weights of representations (P and#,r are the forms cvl and cv2 , then the dominant weight of the representationcp ® is the form cv 1 + cv2 . For a linear representation cp of a group G inM-dimensional space, it is possible to define the kth exterior power sp[kl alinear representation of the group G in the (k) -dimensional space of skew-symmetric tensors aiIi2"'ik . Cartan showed that if the dominant weight of arepresentation rp is a form co1 and its following weights in decreasing orderare the forms c02 , c03 , ... , cvk , . . . , then the dominant weight of the kthexterior power (p [k] is the sum of the forms a)

1+(02 + - + cvk .

Cartan also showed that all linear representations of a complex simpleLie group G are Kronecker products of exterior powers of several basicrepresentations whose number is equal to the rank of the group, and each ofthese basic representations corresponds to a certain simple root of the groupG . The groups of matrices corresponding to these basic representations werecalled the "fundamental groups" by Cartan. Since the term "fundamentalgroup" has several different meanings (later Cartan used this term for thetransitive group of transformations of a homogeneous space), we will applyCartan's term "fundamental" not to the groups but to their representations,

§2.13. LINEAR REPRESENTATIONS OF SIMPLE LIE GROUPS 71

i.e., we will call fundamental representations of a simple Lie group those ofits representations from which it is possible to obtain all its representations.The dominant weights of these fundamental representations are called thefundamental weights. These weights are linear combinations (2.31) of simpleroots ai whose coefficients are the entries of the inverse matrix A of theCartan matrix of a simple Lie group. Thus, to each fundamental weight n` ,

there corresponds a simple root ai . On the other hand, the dominant weightof any linear representation of a simple Lie group is a linear combinationwith integer coefficients of the fundamental weights, i.e., of the points ofthe weight lattice of this group (and, therefore, they are linear combinationswith rational coefficients of simple roots). If the dominant weight of a linearrepresentation of a simple Lie group is a linear combination min` , this linearrepresentation is represented by the Dynkin graph where next to each dot ai ,

the integer mi is written.In particular, for a simple Lie group in the class An) fundamental rep-

resentations are its representation (p1 by matrices of order n + 1 from thegroup CSLn+1 and the exterior powers °k = 91 ) k = 2 , 3 , ... , n , of thisrepresentation by matrices of order (k+l). For a simple Lie group in theclass Bn , fundamental representations are its representation 9, by matricesof order 2n + 1 from the group Cotn+1 , the exterior powers °k = 9lkI ) k =2, 3, ... , n - 1 , of this representation by matrices of order (2n1)

, and arepresentation yrl by the matrices of order 2n , which later received the name"spinor representation". For a simple Lie group in the class Cn , fundamentalrepresentations are its representation (p 1 by matrices of order 2n from thegroup CSy2n and irreducible representations 9k of order (k) - (k"2) fromthe exterior powers 17lkl of the representation (p, , k = 2, 3, ... , n. Fora simple Lie group in the class Dn

)fundamental representations are its rep-

resentation cpl by matrices of order 2n from the group CO2,, the exteriorpowers °k = (P [ Ik] k = 2 ) 3 , ... , n - 2 , of this representation by matricesof order ('), and representations by the matrices of order 2n-1 . Similarlyto the representation yr1 of the group Bn , the representations yr1 and "2presently are called spinor representations.

We will call the dominant weights of linear representations (Pk of thegroups An) Bn , Cn the forms nk and the dominant weights of linear rep-resentations yr and yrl , yr2 of the groups Bn and Cn the forms nn andn n-1 , n n , respectively.

The roots ai of the adjoint representation x --+ [ax) were consideredin Killing's paper [Ki12) and in Cartan's thesis; the maximal roots coincidewith the dominant weights of these representations. For the groups An , theyare n ` + nn; for the groups Bn and Dn , they are n 2 and for the groupsCn , they are 27r'. Note that the dots of the Dynkin graphs marked by the


numbers 1 and 2 which correspond to these representations coincide withthose dots to which the additional dots of the extended Dynkin graphs areattached. The dominant weights of adjoint representations of the exceptionalsimple Lie groups have the same property.

Cartan's book Lectures on the theory of spinors [ 164), devoted to the spinorrepresentations yr1 and w2 , was written in 1938 when it was discovered thatsimilar representations of the group 04 of pseudo-orthogonal matrices (theLorentz transformations defining the transitions from one inertial coordinatesystem to another in the space-time of special relativity) are closely connectedwith electron spin discovered in the 1930s. The vectors of spaces of theserepresentations are called the spinors.

Before describing spinor representations of the groups Con , we describesimilar representations of real groups On of orthogonal matrices. We notedearlier that in 1878, in the paper [C12], Clifford defined the algebras Knwith 2n-1 units. In 1886, Rudolf Lipschitz (1832-1903), in his dissertationResearch on the sums of squares [Lip], discovered an important connectionbetween these algebras and the groups On . The simplest way to describe thisconnection is the following consideration. In the algebra Kn , as well as inthe algebra H of quaternions which is its particular case, one can define an"involution" a -+ a with the properties: a = a , a + /3 = a + /3 , and a,8 _zip. If we write an element a of this algebra in the form a = > a"'e i ...4

then the element a has the form > a'' `k ei , . ...i . Then the coefficient ofk -1 1

1 in the product as is equal to the sum of squares of all coordinates of theelement a. If we call this coefficient the square of the modulus I a+ and takeas the distance between elements a and /3 the modulus Ifi - al of theirdifference, then in the algebra Kn the metric of the Euclidean space R211-1

will be defined. Next, note that the algebra Kn is isomorphic to a subalgebraof the algebra Kn+

1generated by the units with even numbers of indices.

Consider now the following transformation of the algebra Kn+1 :

(2.40)/ = a-] a,

where a is an element of the algebra Kn , represented as a linear combinationof units of the algebra Kn+

1with even numbers of indices, and is an

element of the algebra Kn+1 of the form = x'ei , and assume that theelement a is such that the element ' of the algebra Kn+

1is also of the

form 'x'e,. Then the elements a form a group which is homomorphic tothe group 0n , and the kernel of this homomorphism is a subgroup of thisgroup consisting of the elements 1 and -1 .

The coordinates a, a'i , ... , a' 1'2 *'k of elements a of the algebra Kn

with even numbers of indices satisfy the condition I,,..,R

(a' ''2 _..'k) 2 = 1

§2.14. REAL SIMPLE LIE GROUPS 73

and the equations

aa11121314 = 3!!aI`1 `21a`3`41

(2.41)as it r2c3raisi6 t! i a Cis i2l a 13i415i6l

...................................aa1112-ilk = (2k 1)!!a[il i2]ai3i4...i2k]

where (2k - 1)!! = 1 3.5 . . . . (2k - 1) and [ ] is the alternation symbol.The surface (2.41) in the projective space PN , where N = 2n-1 - 1 (thoseequations first appeared in the work [Lip] of Lipschitz), is called the Lips-chitzian and is denoted by SZn . At present, we say that this group doublycovers the group On and call this group the spinor group of the group On .Thus, the spinor group of the group On is a subgroup of the group of invert-ible elements of the algebra Kn . But from the structure of the algebra Kndiscovered by Clifford it follows that the result of complexification of the al-gebra K2k+1 , i.e., the tensor product K2k+1 ® C) is isomorphic to the algebraC2k of complex matrices of order 2k , and the result of complexification ofthe algebra K2k , i.e., the tensor product K2k ® C , is isomorphic to the directsum C2k-1 ® C2k-1 (in explicit form, this result was first obtained by RichardBrauer (1901-1977) and Weyl in their paper Spinors in n dimensions [BrW]in 1935). It follows from this that the order of the matrices of the spinorrepresentation w1 of the group C02k+1 is equal to 2k and that the ordersof the matrices of the spinor representations y1 and yi2 of the group C02kare equal to 2k-1

§2.14. Real simple Lie groups

Cartan found the classification of real simple associative algebras imme-diately after finding the classification of complex simple associative algebras.However, he was able to solve the similar problem for real simple Lie groupsonly more than 20 years after solving this problem for complex simple Liegroups in his thesis. This happened in 1914 shortly after Cartan constructedthe theory of linear representations of complex Lie groups in the paper Realsimple finite continuous groups [38]. As in his thesis, in this paper Cartancharacterized real simple Lie groups by the nondegeneracy of their "Killing-Cartan form" (2.20), which he denoted in this paper by '(e) . However,in contrast to complex simple Lie groups for which this form can always bereduced to a sum of squares, for real simple Lie groups, if the group is com-pact, this form can be reduced to the sum of negative squares whose numberis equal to the group dimension, and if the group is noncompact, it can bereduced to the sum of a certain number of positive and a certain number ofnegative squares. Cartan characterized real simple Lie groups by an integera called the character which is equal to the difference between the numberof positive and negative squares in the canonical form of the form (e) .


The Cartan metric in real simple and semisimple Lie groups is definedby the metric in their Lie algebras in which the square of the length of thevector e is equal not to y/(e) but to -VI(e) . Therefore, the square of thelinear element in the Cartan metric in simple and semisimple real Lie groupsis equal to

(2.a2) ds 2 = -v(dx) = -camrc,'',,dxadxI ,

and this form is positive definite for compact groups and indefinite fornoncompact groups. Thus, compact real simple Lie groups in their Cartanmetrics are real Riemannian manifolds Vr , and noncompact real simple Liegroups in their Cartan metrics are real pseudo-Riemannian manifolds V,".The character b of a noncompact real simple Lie group is connected withits dimension r and the index l (the number of negative squares in thecanonical form of the form (2.27)) of the pseudo-Riemannian Cartan metricof this group by the relation b = 21 - r, and the character b of a compactreal simple Lie group is equal to the product of its dimension and the number

1 .

In 1929, after constructing the theory of symmetric Riemannian spaces,he returned to the problem of classification of noncompact real simple Liegroups and solved it by much simpler methods in the paper Closed and opensimple groups and Riemannian geometry [116] (here "closed groups" and"open groups" are compact and noncompact Lie groups, respectively). Inthe introduction to this paper, Cartan wrote that based on the geometrictheory constructed by him, "it will now be possible to reduce significantly thecalculations which I have performed. Thus, the extensive memoir in whichI determined all real forms of simple groups can now be reduced to twentyfrom the original 90 pages" [116, p. 2]. By the "extensive memoir" Cartanhad in mind his work [38].

If a complex simple Lie group is a subgroup of the group of CGLN ofcomplex matrices of order N, the compact real group having this group asits complexification is the intersection of this group and the group of complexunitary matrices of the same order. This intersection is called the "unitaryrestriction" of this group. Instead of discussing Lie groups defined up to alocal isomorphism, it is more convenient to discuss the Lie algebras of thesegroups.

Essentially, Cartan's reference to the theory of symmetric Riemannianspaces was that in the theory he had found all involutive automorphisms ofthe Lie algebras of compact simple Lie groups. If in the Lie algebra G ofa Lie group G, an involutive automorphism is given, i.e., an automorphismA - A.1 of this algebra such that (Aj ).1 = A , then in the Lie algebra Gone can take a basis {e,} whose elements ea, remain invariant under theautomorphism (ea) = ea and the elements e1 are multiplied by -1 (e-el) . Moreover, the Lie algebra is decomposed into the direct sum

§2.14. REAL SIMPLE LIE GROUPS 75

(2.43) G=HOE

of two linear subspaces with the bases {ea} and {e1}.Decomposition (2.43) is often called the "Cartan decomposition". In the

basis e1, the structure equations (2.12) of the algebra G have the form

(2.44) [eaep] = cC1 fl [eaeiI = cafe j , [e1e j] = 1 e.

from which we can see that the subspace H is a subalgebra of the Lie algebraG .

To each Cartan decomposition, there corresponds a new Lie algebra whosebasis can be obtained from the basis e1 of the Lie algebra G by multipli-cation of the basis elements e1 by the imaginary unit i. If we denote theproducts ie1 by the same letters el , then the structure equations (2.12) ofthe new Lie algebra have the form

(2.45) {e,,,e,8] = Ca,q ey [eae1] = calei , [eiej] = -Ci Jea

This new algebra is denoted by

(2.46) 'G = H (D iE.

The Lie group 'G defined by the new algebra 'G obviously has the samecomplexification CG as the group G. However, the group 'G is no longercompact. Since the number of the base vectors ea with positive inner squareis equal to dim H and the number of the base vectors ie1 with negative innersquare is equal to dim E, the character 4 of the noncompact group 'G isequal to the difference dim E - dim H. Thus, we can find all noncompactgroups 'G with the same complexification CG as the given compact groupG. At present, the transition from the Lie algebra (2.43) to the Lie algebra(2.46) and from the corresponding compact group G to the noncompactgroup ' G is called the Cartan algorithm.

Note that the Cartan algorithm can be applied not only to Lie algebras butalso to the associative and nonassociative algebras discussed above. In partic-ular, applying this algorithm to the field C and its involutive automorphisma d, we obtain the algebra 'C of split complex numbers, and applyingthis algorithm to the field H of quaternions and its involutive automorphisma - i -1 ai, we obtain the algebra 'H of split quaternions. Next, applyingthis algorithm to the alternative skew field 0 of octaves and its involutiveautomorphism, under which the units i and j are not changed and the unitl is multiplied by -1, we obtain the alternative algebra '0 of split octaves.Finally, applying this algorithm to the algebra Kn and its involutive auto-morphism under which the units e, are not changed and the units e,, are

multiplied by -1 , we obtain the algebra Kn K.


The compact real form of the group CSLn+1

is the group CS Un+1

ofcomplex unimodular unitary matrices. The Lie algebra of this group consistsof complex skew-Hermitian matrices (a), i.e., complex matrices satisfyingthe condition a,' _ -aj. This condition can be obtained from the unitaritycondition UUT = I. Moreover, for the Lie algebra of the group CS Un+1 ,the condition Tr U = 0, which is obtained in a similar manner from theunimodularity condition of matrices U.

If we denote by E, the diagonal matrix with diagonal entries e1 fromwhich 1 entries e. are equal to -I and the other entries Ei are equal to 1,

and b J the matrix 0 I of order 2n, where I is the identit matrixy (-1 0 yof order n, then all involutive automorphisms of the Lie algebra of the groupCSUn+i can be written in the form:

A -+ E1AE1,

(2.48) A -. A,

(2.49) A - - JA J.

Compact real groups in the classes Bn and D. are the groups o2n+1

ando2n of real orthogonal matrices whose Lie algebras are the algebras of realskew-symmetric matrices of the same orders. All involutive automorphismsof the Lie algebra of the groups o2n+

1and o2n can be written in the form

(2.47), and for the group o2n it can be also written in the form

(2.50) A - -JAJ.

On many occasions, Cartan considered a compact group in the class Cnas the intersection of the groups CSp2n and CSU2n . However, in the paperOn certain remarkable Riemannian forms of geometries with a simple funda-mental group [107] (1927), which will be discussed in more detail in Chapter6, Cartan indicated that a compact group in the class Cn can be representedby quaternion unitary matrices of order n, i.e., by unimodular quaternionmatrices "keeping invariant a quaternion positive definite Hermitian form"xJ aiJ.x` where "the quaternions a,J . satisfy the condition aiJ. = a!J." [107, p.392]. We will denote this group in the class Cn by the symbol HUn (the uni-modularity of matrices of this group follows from their unitarity). At present,a compact group in the class Cn is represented only in this way. Chevalleywas the first to represent systematically a compact group in the class Cn bymatrices of the group HUn . He did this in his book The theory of Lie groups[Chv l j (1946). The Lie algebra of the group HUn consists of quaternionskew-Hermitian matrices of the same order, i.e., of quaternion matrices (as)

j. = -aj. All involutive automorphisms of this Liesatisfying the condition aJ

52.14. REAL SIMPLE LIE GROUPS 77

algebra have the form (2.47) and

(2.51) A - -iAi.

Applying the Cartan algorithm to the group C Un+1

and the involutiveautomorphisms (2.47), (2.48), and (2.49) of its Lie algebra, we obtain re-spectively: the group CS UnI+ 1 of complex unimodular matrices satisfyingthe condition

(2.s2) UEUr=E,1 1

the group 'CS Un+1

of split complex unimodular unitary matrices which isisomorphic to the group SLn+I of real unimodular matrices, and a group iso-morphic to the group HSL(n+ 1)/2 of quaternion unimodular matrices. The

characters of the groups CS Un+ 1, CS Unl+ 1 , 'CS Un+ 1 = SLn+ I, and HSLn+

1

are equal to -n(n + 2), 41(n -1+ 1) - n(n + 2), n, and -n - 2, respec-tively.

Applying the Cartan algorithm to the group °2n+1

and the involutive au-

tomorphism (2.47) of its Lie algebra, we obtain the group Qn+1,1 of realmatrices satisfying the pseudo-orthogonality condition:

(2.53) UEIUT = E1.

The characters of the groups °2n+1

and Qn+1 are equal to -n (2 n + 1) and21(2 n -1 + 1) - n (2 n + 1) , respectively.

Applying the Cartan algorithm to the group HUn and the involutive au-tomorphisms (2.47) and (2.51) of its Lie algebra, we obtain the group HUnIof quaternion matrices satisfying the pseudo-unitarity condition (2.52) andthe group 'HUn of antiquaternion unitary matrices which is isomorphic tothe group Sp2n of real symplectic matrices, respectively. The charactersof the groups HU , HUnI , and 'HUn = Sp2n are equal to -n(2n + 1),81(n -1) - n(2n + 1), and n, respectively.

Applying the Cartan algorithm to the group °2n and the involutive au-tomorphisms (2.47) and (2.50) of its Lie algebra, we obtain respectively:the group 02n of real pseudo-orthogonal matrices and the group HSgn ofquaternion symplectic matrices, i.e., quaternion matrices satisfying the con-dition

(2.54) UiUT = ii.

The characters of the groups °2n , 02n , and HSqn are equal to -n(2n - 1) ,

21(2n - 1) - n (2n - 1), and -n, respectively.In the same papers, Cartan also found all real simple Lie groups in the

exceptional classes. He showed that there are two simple Lie groups in theclass G2 with the characters -14 and 2, three simple Lie groups in theclass F4 with the characters -52, -20, and 4, five simple Lie groups in


the class E6 with the characters -78, -26, -14, 2, and 6, four simple Liegroups in the class E7 with the characters -133 , -25, -5 , and 7, and threesimple Lie groups in the class E. with the characters - 248 , -24, and 8.In particular, Cartan showed that a compact simple group in the class G2is isomorphic to the group of automorphisms of the alternative skew field0 of octaves. Applying the Cartan algorithm to this group and the uniqueinvolutive automorphism of its Lie algebra, we obtain a noncompact Liegroup in the same class which is isomorphic to the group of automorphismsof the alternative algebra '0 of anti-octaves.

In 1914, in the paper [38], Cartan found all real simple Lie groups, sub-sequently. In the same year, in the paper Real continuous projective groups,under which no plane manifold is invariant [39], he constructed all irreduciblelinear representations of these groups.

For real simple Lie groups, simple roots of the Lie algebras can be realor imaginary. Thus, the system of simple roots of these groups can be rep-resented by the Dynkin graphs where real simple roots are represented bywhite dots, imaginary simple roots are represented by black dots, and pairsof imaginary conjugate simple roots are represented by white dots joined bycurved double arrows. These graphs are called the Satake graphs becauseSatake in the paper on representations and com pactificati ons of symmetricRiemannian spaces [Sat] (1960) used them for characterization of symmet-ric spaces with compact simple fundamental groups. (As noncompact realsimple Lie groups, these spaces correspond to involutive automorphisms ofcompact simple Lie groups with the same complex forms.)

The Satake graphs can be also defined for compact real simple Lie groups:these graphs coincide with the Dynkin graphs, but all dots of these graphs areblack. For noncompact real simple Lie groups, all simple roots of which arereal (such groups are called split or anticompact), the Satake graphs coincidewith the Dynkin graphs, and all dots of these graphs are white. Figure 2.10represents the Satake graphs for noncompact simple Lie groups in the classesAn , Bn , Cn , and Dn , and Figure 2.11 (see page 80) represents the Satakegraphs for non-compact simple Lie groups in the classes G2 , F4 , E6 , E7 ,

and E8 . Here, the noncompact simple Lie groups are denoted by the samesymbols which Cartan used for symmetric spaces with compact simple fun-damental groups defined by the same involutive automorphisms.

§2.15. Isomorphisms of real simple Lie groups

There are isomorphic groups from different classes among real simple Liegroups. All these isomorphic groups were found by Cartan in the paper [38]where he wrote {he designated the rank of the group by 1}:

"I. The real groups of the type (A) (1 = 1) have the characters 8 = 1or 8 = -3 and the same ones have groups of the type (B) [and (C)].

1 0 6 = I. There are isomorphisms between:

§2.15. ISOMORPHISMS OF REAL SIMPLE LIE GROUPS 79

n a3 n 1 a -Ia"

al a2a3 an-3 an-2 an-1 an

a) Al k) CII b - - -0-a1 °,

a3 a4 as a.-t anb) All Q - -1) DIa

an-1

al a2 a1 a1+1 an-3 an-2

an

c) AIIIa

d) AIIIb

an-2 a(n+I)/

e) AIV ----*-al a2 23 an-2

a(n-1)/2

an-1

al a2 a1 a1+1 an-2 an-1 anf) BIa C>---<>-

(11 1

a12 a13 an-2 an-1 ang) Bib 0 - - - -h) BII

al

0 (X2 (X-1

an

alm)DIb 0

aln) DIc 0

a2 a3

a2 a30---o

an-I

al a2 (X3an-3 an-2

p) DIIIa --- 0

a1 a2 (13 a4 an-3 an-2a1 a2 a3 an-2 a an q) DIIIb - - -

i)CI --

al a2 a3 a21 a RI an aj) CIIa

FIGURE 2.10

an-I

an-I

an-1

an-1

an

The special homogeneous linear group in two real vari-ables;The linear group of two complex variables x1 , x2 of theHermitian form x1 x1 - x2x2 ;The linear group of the real quadratic form x1 + x2 - x3 .

2° d = -3. There are isomorphisms between:The linear group of the Hermitian form x1x1 + x2x2 ;The group X' = AX of one quaternion variable [X] andone quaternion parameter [A] ;The linear group of the real quadratic form xl + x2 + x3 .

II. The real semisimple groups, which are obtained from complexgroups formed by subgroups of rank 1 of the type (A) , have thecharacters d = 2, 0, -2, and -6, and the same ones have groupsof the type (D) .


a1 a2 a1 a2 a3 a4 a1 a2 a3 a4

a) G c b) FI C) F11

f) EIII ba1 (X 2 a3 a4 a5

a6

h) EVa1 a2 a3

aa1 23 :4g} EIV

0---c i) EVIas a6

a1 a2 a3 a4 a5 a6

1a4

a7

j) EVIL 0 0

a7

a1 a2 a3 a4 a5 a6 a7

k) EVIII

k) EIX

FIGURE 2.1 1

a7

1 ° 8 = 1 . There is an isomorphism between:The linear group formed by the special linear group oftwo variables x1 , x2 and by the special linear group ofthe variables x3 , x4 ;

The linear group of the quadratic form xl + x2 - x3 - x4 .

2° 8 = 0. There is an isomorphism between:The special linear group of two complex variables x1 , x2 ;

The linear group of the quadratic form xl +X2+ x3 _X4 .

30 a = -2. There is an isomorphism between:The linear group formed by the special linear group oftwo real variables x1 , x2 and by the linear group of theHermitian form x323 + x424

;

The linear group of the quadratic form xI x2 + 23x4 andof the Hermitian form xlx1 - x222 + x323 - x424 of fourcomplex variables.

° 8 = -6. There is an isomorphism between:The linear group formed by the linear group of the Her-mitian form x12, + x222 and by the linear group of theHermitian form x323 + x424 ;

a

a6

a50

a1 a2 a3 a4 a5 a6

The linear group of the real quadratic form xl + x2 +x3+x4.

§2.15. ISOMORPHISMS OF REAL SIMPLE LIE GROUPS 81

III. The groups of the type (B) and (C) (1 = 2) can have the char-acters 6 = 2, -2 , and - 10 .

1 ° 6 = 2. There is an isomorphism between:The linear group of the real quadratic form xi + x2 +x2_ 2_ 2

3 4 5'The linear group of the real skew bilinear form [xlx2] +[x3x4] .

20 6 = -2. There is an isomorphism between:The linear group of the real quadratic form xl + x2 +X3 + x4 - xs ;The linear group of the skew bilinear form [x1 x2] + [x3x4]and the Hermitian form x191 + x2x2 + x3x3 - x4x4 offour complex variables x1, x2 , x3 , and x4.

3° 6 = -10. There is an isomorphism between:The linear group of the real quadratic form xi + x2 +X2+X2

The linear group of the skew bilinear form [x1 x2] + [x3x4]and of the Hermitian form x1x1 + x2x2 + X3-X3 + x4x4 .

IV. The real groups of type (A) or of type (D) (1 = 3) can have thecharacters 6 = 3, 1, -3, and -15 .

1 ° 6 = 3. There is an isomorphism between:The special linear group of four real variables (projectivegroup in the space);The linear group of the real quadratic form xl + x2 +

2 2 2 2x3 - x4 - xs - x6 .2° 6 = I. There is an isomorphism between:

The linear group of the Hermitian form x1.1 + x2x2 -x3x3 - x4x4 ,

The linear group of the real quadratic form xl + x2 +X3 + x4 - xs - x6 .

3° 6 = -3. There is an isomorphism between:The linear group of the Hermitian form x1.1 + X2-t2 +x3.3 - X4-X4 ;The linear group of of the quadratic form x1 x2 + x3x4 +xsx6 and of the Hermitian form x1; - x2x2 + x3x3 -x4.4 + x5.5 - X6-X6-

40 6 = -5. There is an isomorphism between:The group of X' = AX + B Y) Y' = CX + D Y of quater-nion variables X, Y and parameters A, B , C , D ;The linear group of the real quadratic form xl + x2 +x3+x4 +x2 -x2.

X5 65°

6 = - 15. There is an isomorphism between:


0 al al ala) Al = BII b) DIb c) DIc d) DIM

0 0 a2 da2 0 a2al

g) AI DI

aI a2 a3

1)

a3Dlllak) DIa

al a2 aa4

h) All

aI a2

J) AIII

FIGURE 2.12

e) BI

f) BII

CI

CIl

The linear group of the Hermitian form x1 xl + x2x2 +x3.23+x424;The linear group of of the real quadratic form xl + x2 +X

3+ x4 + xs + x6 ." [38, pp. 353-355]

These isomorphisms are clearly seen on the Satake graphs of the real simplegroups (Figure 2.12) analogous to the Dynkin graphs on Figure 2.3; here, asin Figure 2.3, to the Satake graphs of isomorphic simple groups, the graphsof isomorphic semisimple groups in the class D2 and the direct product oftwo simple groups in the class A 1 = B1 = C1 are added (as we will see,in one case a noncompact group in the class D2 is a simple group which isisomorphic to a complex simple group in the class Al = B1 = CO.

For real noncompact simple Lie groups, in addition to isomorphisms anal-ogous to isomorphisms between complex simple Lie groups, there is one moreisomorphism between two noncompact simple Lie groups in the class D4,namely, between groups of the types DI and DIII . This isomorphism is aconsequence of the triality principle in the spaces with fundamental groupsin the class D4 . The Satake graphs of these groups are represented in Figure2.12k.

§2.16. Reductive and quasireductive Lie groups

Semisimple Lie groups are particular cases of reductive Lie groups, whoseLie algebras are the direct sums of simple Lie groups without the requirementof noncommutativity of these simple groups. The term "reductive groups"was introduced by Armand Borel (b. 1923) and Tits in their paper Reductive

X2.16. REDUCTIVE AND QUASIREDUCTIVE LIE GROUPS 83

groups [BoTJ (1965 ). All commutative groups and all compact Lie groups, aswell as the group GL,, of all nonsingular matrices of order n, are reductivegroups.

If a reductive group G is a group of automorphisms of a commutativegroup R, the semidirect product G x R, i.e., the set of pairs (g, r), g EG, r E R with multiplication

(2.55) (g1, r1)(g2, r2) = (g1g2, r1 +g1r2),

where g1 r2 is the result of application of the automorphism g1 to the el-ement r2, is called the quasireductive group. The most important exampleof a quasireductive group is the group of affine transformations in the affinespace E7z .

An important class of quasireductive Lie groups, which are closely con-nected with Cartan's work, are quasisimple Lie-groups-semidirect productsG x R defined above where G is a semisimple Lie group. The Lie algebras ofthese groups can be obtained from the Lie algebras of semisimple Lie groups,represented in the form (2.43), by transition to the Lie algebra

(2.56) °G = H+cE,

where e is a dual unit of the algebra ° C of dual numbers. The transitionfrom the Lie algebra (2.43) to the Lie algebra (2.56), which is similar to theCartan algorithm transferring the Lie algebra (2.43) to the Lie algebra (2.46),is called the quasiCartan algorithm. Examples of quasisimple Lie groups arethe groups of motions of the Euclidean space Rn and the pseudo-Euclideanspaces R1 (if n = 4 and 1 = 1 , this is the nonhomogeneous Lorentz groupwhich is important in theoretical physics). The general definition of qua-sisimple Lie groups was formulated by Katsumi Nomizu (b. 1924) in thepaper Invariant affine connections on homogeneous spaces [No, p. 50] (1954)and by Marcel Berger (b. 1927) in the paper Non-compact symmetric spaces[Beg3, p. 93] (1957) (the main content of their works will be discussed inChapters 6 and 7). For a Lie group °G obtained from a simple Lie group bythe quasiCartan algorithm, Gel'fand (who called simple Lie groups obtainedfrom one another by the Cartan algorithm "dual groups in the sense of Car-tan") and his coauthors used the name "trial group in the sense of Cartan"(see [BGN] (1956)).

If in the definition of a quasireductive group we change a reductive groupG for a quasireductive group, we obtain a biquasireductive Lie group. Tri-quasireductive and r-quasireductive groups as well as biquasisimple, tri-quasisimple, and r-quasisimple groups can be defined in a similar manner.Cartan considered biquasisimple groups in the papers on manifolds with anaffine connection and the general relativity theory [66) (1922) and On a de-generacy of Euclidean geometry [ 147a] (1935 ).

Transitions from reductive Lie groups to quasireductive groups and from(r - 1)-quasireductive groups to r-quasireductive groups are particular cases


of contraction of Lie groups which were defined in connection with problemsof theoretical physics by the famous physicist Eugene Paul Wigner (b. 1902)and his student Erdal Inonu (b. 1926), a son of a President of Turkey (IsmetInonu) and presently (199!) himself a politician of that country, in the paperOn the contraction ofgroups and their representations [IW] (1953) {see also thebook Contractions and analytic prolongations of classical groups. An analyticapproach [Grog (1990) by Nikolai A. Gromov).

§2.17. Simple Chevalley groups

In 1954, in the paper On certain simple groups [Chv3], Chevalley definedanalogues of simple Lie groups over fields that are different from the fields Rand C, namely, over the Galois fields Fq , the fields Qp of p-adic numbers,and the fields Q(a, ...) of algebraic numbers. These groups are definedas the groups of automorphisms of the Lie algebras over the correspondingfields with the same integer structure constants as the Lie algebra having thesame name over the fields R and C. The analogues of simple Lie groupsdefined in this manner are called algebraic groups or Chevalley groups. Com-plete classification of simple Chevalley groups was given by Tits in the paperClassification of algebraic semisimple groups [TiS] (1965).

Simple Chevalley groups over the Galois fields Fq or their quotient groupsby their centers are finite simple groups. These groups are denoted in thesame way as the corresponding complex simple Lie groups. The finite groupsAn) Bn, Cn, and Dn are the groups FgSLn+l , FgO2n+1 , FgSp2n , and FgO2nor the quotient groups of these groups by their centers. If the Galois groupsof the characteristic equations of simple Lie groups do not coincide with theirWeyl groups, there are also "2-twisted" simple groups An(2) Dn2) E(2) andthe "3-twisted" simple group D(3) The simple Chevalley groups are char-acterized by the same Dynkin graphs as the corresponding Lie groups. The2-twisted simple Chevalley groups are characterized by the Satake graphs rep-resented in Figures 2.1 Od, 2.1 Om, and 2.11 e. The 3-twisted simple Chevalleygroup is characterized by the Satake graph represented in Figure 2.13.

The orders (the number of elements) of the finite simple groups corre-sponding to the finite Chevalley groups can be expressed by the single for-mula:

Iq

N fl(qa,+] - 0,U

where N is the number of positive roots of the corresponding compact simpleLie group, u is the number of the elements of the center of a finite simpleChevalley group, and the numbers al coincide with the exponents (2.34) and(2.35) of corresponding simple Lie groups. The orders of the twisted finite

§2.18. QUASIGROUPS AND LOOPS 85

FIGURE 2.13

Chevalley groups 42), Dn2) E62) ,and D43) are equal respectively to

ugNr1i(9r+i _(_l)r+,))

(2.58)qNN 1) fl,(q2' _ ls

6 s zu9 (q - 1)(q' + 1)(9' - 1)(9 - 1)(9' + 1)(9 - 1),u9'N(9$ +94 + 1)9'6 - 1)9'2 - 1)

(see Tits's talk [Ti4, pp. 213-214]).Note that while only one noncommutative division algebra, namely, the

field H, can be defined over the field R, it is possible to define differentdivision algebras of dimension m2 over the fields QP and Q(a, ...) (whena field is extended to an algebraically closed field, these algebras becomealgebras of matrices of order m over this field). Thus, for the fields QP andQ(a , ...) ,there are Chevalley groups of the class An whose Satake graphsdiffer from the Satake graphs of the groups All (Figure 2.1Ob) by the factthat each black point of the graph is replaced by m - 1 black points.

§2.18. Quasigroups and loops

Recently, in geometry as well as in theoretical physics, algebraic systemswith nonassociative operations have become of greater importance. First ofall, quasigroups and loops should be named among such systems. A quasi-group is a set Q where a binary operation is defined which to any two el-ements x and y assigns a third one, z , x o y = z , and is invertible withrespect to each of the factors on the left-hand side of this equation. Ingeneral, this operation is not associative. If there is a two-sided unit in aquasigroup, such a quasigroup is called a loop. A group is a particular caseof a loop it is a loop whose binary operation is associative. Weakening indifferent ways the associativity condition, one can obtain the most importantclasses of loops: the Moufang loops defined by Ruth Moufang (1905-1977),who also introduced the term "quasigroup", the Bol loops defined by GerritBol (1906-1987 ), and the monoassociative loops.

If a set of elements of a loop is a manifold and its algebraic operation canbe expressed by differentiable or analytic functions, a loop is called smoothor analytic, respectively. Such a loop is a nonassociative analogue of a Liegroup, and it is possible to construct an analogue of the Lie algebra for sucha loop. For analytic Moufang loops, such algebras were defined by Anatolii


1. Mal' cev (1909-1967) in the paper Analytic loops [Mall (1955). At present,these algebras are called Mat cev algebras. For the Bol loops, such algebraswere constructed by Lev V. Sabinin (b. 1932) and Pavel O. Mikheev in thepaper On analytic Bol loops [SM] (1982). In contrast to Lie algebras andMal 'cev algebras, they have not only a binary but also a ternary operation.For general loops, such algebras were constructed by Maks A. Akivis (b. 1923)in the paper The local algebras of a three-dimensional three-web [Ak7] (1976)(see the end of Chapter 7 of this book on the connection of quasigroups andwebs).

It is well known that a Lie algebra completely defines a local Lie group. Asimilar theorem for Moufang loops was proved by Evgenii N. Kuz'min (b.1938) in 1971 and for Bol loops by Sabinin and Mikheev in 1982. For generalloops and quasigroups, a theorem of this kind is not valid. However, thereexist certain classes of quasigroups defined by a certain number of constants.In 1985, Alexander M. Shelekhov (b. 1942) proved that mono-associativeloops form one of these classes. However, there is no complete descriptionof local algebras connected with such loops.

Moufang loops are the closest to Lie groups. At present, the theory ofMoufang loops has been extensively developed. In particular, the theory ofsimple smooth Moufang loops which is similar to the theory of simple Liegroups was constructed by A. S. Sagle in the paper Simple Mal'cev algebrasover fields of characteristic zero [Sag] (1962). In this paper, Sagle showedthat simple smooth Moufang loops are hyperspheres `a l = I of alternativealgebras 0, 'G and 0 ® C of octaves, solit octaves, and complex octaves.

The current status of the theory of quasigroups and loops is described inthe book Quasigroups and loops: Theory and applications [CPS] (1990) editedby Orin Chein (b. 1943), Hala O. Pflugfelder and Jonathan D. H. Smith (b.1949) (see also the textbook Quasigroups and loops: Introduction [Pfl] (1990)by H. O. Pflugfelder). In the book [CPS] we note the chapters Local differ-entiable quasigroups and webs [G1b2] by Vladislav V. Goldberg (b. 1936),Quasigroups and differential geometry [MS] by Sabinin and Mikheev, andTopological and analytic loops [HS] by Karl H. Hofmann and Karl Stram-bach.

CHAPTER 3

Projective Spaces and Projective Metrics

§3.1. Real spaces

In the titles of papers [37) and [39) Cartan called linear representations ofsimple Lie groups "projective groups", i.e., groups of projective transforma-tions (collineations) of projective spaces CBn and Bn . We saw earlier thatcomplex simple groups in the classes An, Bn , Cn , and Dn are represented bythe groups of collineations of the spaces CBn, the groups of motions of thenon-Euclidean spaces CS2n

, the groups of symplectic transformations of thesymplectic spaces CSy2n-1 , and the groups of motions of the non-Euclideanspaces CS2n-1 , respectively.

Real simple Lie groups admit similar geometric interpretations in the realforms of these spaces: in the real projective space Bn and in the real non-Euclidean spaces the elliptic spaces Stn and Stn-1 , the hyperbolic spacesS;

2n and S;2n-1 , and the symplectic space Sy2n -1 . All these spaces are de-fined in the same way as the spaces CBn, CS2n , CS2n-1 and CSy2n-1 were

defined. In addition, note that for the spaces SA the left-hand side of thequadric equation ai .x`xj = 0 is a positive definite quadratic form, i.e., thisequation can be reduced to the form >11(x')2 = 0 , and for the spaces SNthe left-hand side of the quadric equation is a nondegenerate form of indexl , i.e., this equation can be reduced to the form - I,(x)2 + Ei(x`)2 = 0 ,where the number of negative terms is equal to 1. The spaces defined in thismanner, where the classical groups can be interpreted, are called the classicalspaces. Cartan's book Lectures on complex projective geometry [ 134) (1931)was devoted to geometries of many of these spaces.

Historically, the first of these spaces which is different from the Euclideanspace was the hyperbolic Lobachevsky space S1 . The geometry of the spaceS3 was discovered by Nikolai I. Lobachevsky (1792-1856), who presentedit for the first time in his paper on the principles of geometry [Lob 1 ] (18 29);by Janos Bolyai (1802-1860), who presented his discovery in the form of anAppendix [Boy) to the book of his father in 1832; and by Gauss, who arrivedat the same geometry before Lobachevsky but did not publish his discoveryduring his lifetime.

87

88 3. PROJECTIVE SPACES AND PROJECTIVE METRICS

We have already mentioned that hyperbolic geometry was widely recog-nized by mathematicians only in the 1870s, the years of Cartan's childhood.This recognition was made possible by a series of important discoveries ofthe 19th century. In the middle of this century the treatment of projectiveproperties of figures became independent projective geometry, which, in thebook Geometry of position [Sta] (1847), Christian von Staudt (17 98-1867)freed from definitions connected with Euclidean geometry. In 1859, Cayleyshowed, in A sixth memoir upon quantics [Cay2], that the Euclidean planecan be considered as the projective plane where, in addition, a line ("the lineat infinity of the Euclidean plane") and a pair of imaginary conjugate pointson it (the pair of "cyclic points" in which this line intersects all circles) aregiven, and the "elliptic plane" , i.e., a sphere with antipodal points beingidentified, can be considered as the same projective plane where, in addition,an imaginary conic is given. Cayley thus exclaimed: "Metrical geometry isthus a part of descriptive geometry, and descriptive geometry is all geometry,and reciprocally." [Cay2] (see [Cay, vol. 2, p. 592]).

At the same time, in the papers of August Ferdinand Mobius (1790-1866),the treatment of affine properties of figures became afjine geometry, and thetreatment of circular transformations in the plane generated by inversionswith respect to circles became conformal geometry (also called Mobius ge-ometry and inversive geometry). At the time that the geometries of projec-tive, affine, and conformal planes were created, the geometry of the three-dimensional projective, affine, and conformal spaces arose. After the pub-lication of Grassmann's paper The science of linear extension [Gra] (1844),the geometries of the multidimensional Euclidean space Rn , the multidimen-sional hyperbolic space S1 , the multidimensional elliptic space Sn , the mul-tidimensional projective space Pn, the multidimensional affine space En ,

and multidimensional conformal space Cn appeared.The recognition of Lobachevsky's hyperbolic geometry came in 1868, when

Eugenio Beltrami (1835-1900) constructed an interpretation of the hyper-bolic plane S1 in a circle of the Euclidean plane, and in 1870, when Kleinshowed that the plane S1 can be realized as a part of the projective plane P2bounded by a real conic. In Klein's interpretation, the motions of the planeSl are represented by projective transformations preserving the conic (if thisconic is a circle, Klein's interpretation coincides with Beltrami's). The mul-tidimensional spaces Sn and Sl can be realized as the projective space P" ,

where an imaginary quadric is given, and as a part of the space P" boundedby an oval quadric, respectively. The motions of the spaces Sn and S1 arerepresented by projective transformations of the space Pn preserving thesequadrics.

In 1882, Poincare proposed another interpretation of the Lobachevskianplane in a circle of the Euclidean plane. In this interpretation, motions of theLobachevskian plane are represented by circular transformations preserving

§3.1. REAL SPACES 89

the circumference of the circle. (In this interpretation, a circle can be replacedby a half-plane.) Similarly, the space S1 can be represented as the interiorof a hypersphere in the space Cn , and, in,this interpretation, the motions ofthe space S1 are presented by conformal transformations of the space Cnpreserving this hypersphere.

Along with the Lobachevskian space S1 represented as the interior ofan oval quadric of the space pn, one can also consider the exterior of thisquadric which is called the ideal domain of the space S1 (in this case, thespace defined by Lobachevsky is called the proper domain of the space Sr).The space S1, considered as a set of the proper and the ideal domain andthe oval quadric which divides these domains, is a particular case of thehyperbolic space S;n the space Pn in which a quadric of index 1 is givenwhose equation can be reduced to the form Eici(x i

)2 = 0 , where i =

0, 1, ...,n, e,=-1 for i<1,and e,= 1 for i>I. This quadric iscalled the absolute of the space Sl , and motions of this space are projectivetransformations of the space Pn preserving the absolute.

If in the definition of the space S1 we replace the quadric aiJ.x`x3 = 0by a linear complex alJ.p`J = 0 (aiJ. -aid of straight lines, where pigx`yJ - y1 x3 are the Plucker coordinates of a straight line X Y (joining thepoints X(x') and Y(y')), we obtain the space Sy" . In the 19th century, thisspace was called "the space of linear complex". Later, after Weyl proposedcalling the "group of linear complex" the symplectic group, this space wasgiven the name symplectic space. In the space Sy" , the "null-system" u, =aiJ.x3 (a1J. _ -aJ.1) sending each point X(x') into the hyperplane uix` = 0passing through this point plays the role which the polar transformationsui = aiJ.x3 (a1J . = aJ.1) play in the spaces S1 . The linear complex ai p 'J = 0

J

is called the absolute linear complex of this space and consists of isotropicstraight lines which are transformed into the (n - 2)-planes passing throughthese lines. Since the null-systems are nondegenerate (their determinantsdet(a13) are different from zero) only for odd values of n, the dimensionsof symplectic spaces are odd.

After Einstein's special relativity was discovered (1905), the notion of apseudo-Euclidean space Rl appeared. A pseudo-Euclidean space Rl is anaffine space En in which an inner product of vectors xy is defined that canbe reduced to the form xy = >1 eixl yi , i = 1, ... , n, and the numbers e,have the same values as they have for, quadrics of index 1 + 1 . The space-timeof the special relativity is the space R4 .

If we extend the space R1 by adding the point at infinity which, underinversions in hyperspheres of this space, corresponds to the centers of thesehyperspheres, and by adding the "ideal points" corresponding, under these in-versions, to the points whose distances from these centers are equal to zero,we obtain the pseudo-conformal space C, . As we saw for the conformal


space Cn , conformal transformations of the space C1 are its transforma-tions that are generated by inversions in its hyperspheres.

In 1887, Poincare proposed another model of the Lobachevskian plane ona two-sheeted hyperboloid with identified antipodal points. As the distancebetween two points of the hyperboloid Poincare took a number which isproportional to the cross ratio of the rays going to these points and of twoasymptotes lying in the plane of these two rays.

From the modern point of view, this definition of distances on a hyper-boloid can be formulated in the following manner. If we write the equationof a two-sheeted hyperboloid in the form Fi e (x`)2 = -q2 and introducein the space the metric of the space R with inner product xy = >i e,x` y i ,

the hyperboloid will be a sphere of pure imaginary radius in the space R, ,

and the Poincare metric on this hyperboloid will coincide with the metricof this sphere. The models of Beltrami-Klein and Poincare in circles areclosely connected with the last Poincare model. Namely, if we project theupper half of a sphere of pure imaginary radius qi of the space R1 fromits center onto a tangent plane to that sphere (which is a Euclidean plane),we obtain the Beltrami-Klein model (Figure 3.1), and if we project the sameupper half from its "south pole" onto its equatorial plane (which is also a Eu-clidean plane), we obtain the Poincare model in a circle (Figure 3.2), whichessentially is a stereographic projection of the sphere onto this plane. Theinterpretation of the hyperbolic plane S2 on a sphere of imaginary radiusexplains an important similarity of hyperbolic geometry and usual sphericalgeometry. As was noticed by Lobachevsky himself, trigonometric formu-las in the hyperbolic plane can be obtained from formulas of the sphericaltrigonometry if we consider the lengths of the sides of a triangle to be pureimaginary (i.e., these formulas are trigonometric formulas on a sphere ofimaginary radius), and motions of the plane S2 are rotations of the spaceR 3 . The fact that the space-time of the special relativity can be considered asthe space R4 and the group of Lorentz transformations (which is importantin this theory) is isomorphic to the group of rotations of the space R 1 is thecause for the deep connections between hyperbolic geometry and the specialrelativity (in particular, the formula for addition of velocities in the lattertheory is equivalent to the law of cosines in hyperbolic geometry).

The elliptic space Sn can also be realized as the geometry on a hypersphereof the space Rn+ 1 with identified antipodal points and as the geometry onthe hyperplane at infinity of the space Rn+ 1 . The imaginary quadric of thishyperplane in which it intersects all hyperspheres of the space Rn+ 1 plays therole of the absolute of the space Sn . Similarly, the hyperbolic space S7 can

,be interpreted as a geometry on the hyperplane at infinity of the space R"+1and each of the domains in which the space S7 is divided by its absolutecan be interpreted as a hypersphere of real or pure imaginary radius with

§3.1. REAL SPACES

FIGURE 3.1

FIGURE 3.2

91

identified antipodal points in the space R"+ 1 . If the radius of a hypersphere,on which the space Sn or one of the domains of the space Sl is interpreted,is equal to r or q i , then the distance co between the points X (x') andY(y`) of the space Sn or one of the domains of the space Sl is respectivelydefined by the following formulas:

COSZ(Ei Ciyi)2

r E,(Xi)2 Y:,(Yl

2(3.2) costW . (>1c1xy)

>

rEi Ei(Xi)2 Er Er(Yi)2


cosh2 =

(1e,x'y')2-q Ei ei(xi)2 Ei ei(yi)2

The numbers 1/r2 and -11q 2 are called the curvatures of the spaces Snand Sl , respectively.

Note that in the space S" of curvature 1 /r2 the area of a triangle ABCwith angles A, B , and C is equal to

(3.4) S= r2(A+B+C-ir)

and the area of such a triangle in the space Sl of curvature -1/q2 is equalto

(3.5) S = q 2 (7r - A - B - C))

where the angles A, B, and C in formulas (3.4) and (3.5) are measured inradians.

The spaces Rn , En , Bn, Sn , S1 , and Cn were considered by Klein in his

Erlangen program [Kle] (1872). In this program, Klein also formulated cer-tain "transfer principles" which enable one to interpret one space within an-other. These principles are based on the isomorphisms of the groups of trans-formations of these spaces (Cartan called geometries of such spaces equivalentgeometries) and on the interpretation of the space Cn on the absolute of thespace Sr'. . The latter interpretation is based on the fact that if one takesthe angle between two hyperspheres as the distance between them (if the hy-perspheres are tangent to one another, this angle is equal to zero, and if thehyperspheres have no real intersection, this angle is imaginary), then the setof hyperspheres of the space Cn is isometric to the ideal domain of the spaceS1 +i of curvature 1. Furthermore, the points of the space Cn that can beconsidered as the hyperspheres of radius zero are represented by the pointsof the absolute of the space S1 +1 , imaginary hyperspheres are representedby the points of the proper domain of the space Si + 1 , and the conformaltransformations of the space Cn are represented by motions of the spaceSi + 1 . Similarly, one can prove that the space C;n is realized on the absoluteof the space S f ' , and the conformal transformations of the space C! arerepresented by motions of the space

The groups of rotations of the spaces Rn and Rl , the subgroups of affinetransformations of the space E" keeping fixed one point of this space andpreserving the volumes of parallelotopes (such transformations are calledcentro-affine), the groups of motions of the spaces Sn and Sl , the group ofprojective transformations of the space Bn, the group of conformal trans-formations of the space Cn, and the group of symplectic transformationsof the space Sy2n-1 (the groups of projective transformations of this spacepreserving its absolute linear complex) are all simple Lie groups except the

§3.2. COMPLEX SPACES 93

groups of rotations of the spaces R4 and R2 and the group of motions ofthe spaces S3 and SS , which are semisimple groups.

Properties of the corresponding Lie groups were substantially used by Car-tan in his works on differential geometry of the spaces Rn , En , Pn , and Cn.

The extended Cartan translation [46) of the Fano paper [Fa] (1907), pub-lished in the German edition of Encyclopaedia of Mathematical Sciences, wasdevoted to the connections between Lie groups and various spaces.

Cartan's translation [46) had the title Theory of continuous groups and ge-ometry. We have already mentioned that in 1914 only 21 pages of this paperwere published; because of the beginning of World War I, the rest of paperwas not published at that time. The complete text of the paper was publishedonly after Cartan's death in his Euvres Completes [207). (As was the casewith Cartan's extended translation of Study's paper [Stul] on complex num-bers and their generalizations, Cartan's translation contains many additions;Fano's paper had 100 pages, and Cartan's translation had 134 pages).

§3.2. Complex spaces

In the 19th century, along with real spaces, complex spaces obtained by thetransition from real coordinates to complex ones were the subject of study.The complex projective space CP" was especially widely used in algebraicgeometry. In addition, the complex Euclidean space CRn , the complex non-Euclidean space CSn , and occasionally, the complex conformal space CCnrealized on the absolute of the space CSn+ 1 , and the complex symplecticspace CSy2n-1 were considered.

We have already noted that the group of projective transformations of thespace CPn is a complex simple Lie group in the class An) and complex sim-ple Lie groups in the classes Bn , C , and Dn are represented by subgroupsof groups of projective transformations of complex projective spaces. WhenCartan gave the title "Projective groups, under which no plane manifold isinvariant" to his paper [37) on linear representations of complex simple Liegroups, he meant exactly this latter representation.

We have already mentioned that the group of motions of the space CS2n

is a complex simple group in the class Bn ; the group of symplectic transfor-mations of the space CSy2n-1 is a complex simple group in the class Cn ;and the group of motions of the space CS2n-1 is a complex simple groupin the class Dn (for n = 2, the latter group is a semisimple group locallyisomorphic to the direct product of two complex simple groups B1).

In his extended translation of Fano's paper, along with complex spacesthat were the subject of study in the 19th century, Cartan considered somenew complex spaces by means of which certain real simple Lie groups canbe realized. First of all, he introduced the Hermitian elliptic space CSnand the Hermitian hyperbolic space CSi , the geometries of which were firststudied by Guido G. Fubini (1879-1943) in the paper On definite metrics of


a Hermitian form [Fub2] (1903), and by Study in the paper Shortest pathsin the complex domain [Stu4) (1905). The space CSn can be defined as thespace CFn in which a positive definite real metric is given with the distanceco between points X and Y defined by the formula

2 60(3.b) cos

r xY Ei Y'yifi

which differs from formula (3.1) by substitution of the Hermitian formsEi x'z' for the quadratic forms Ei(x')2 . The space CSl is defined as thespace CFn with the distance co between points X and Y defined by theformula

2

(3.7) COSr E.x' ' E.yiyi

which differs from formula (3.2) by substitution of the Hermitian formsF,i eixlT' for the quadratic forms Ei Ei(x') 2

.

The groups of motions of the spaces Cr and CSl are represented bymatrices of a compact group in the class An (the matrix group CSI +,) andof a noncompact group in the same class (the matrix group CSUn1+1).

Cartan considered in detail the geometry of the spaces CS3 and CS3 in1

his above-mentioned Lectures on complex projective geometry [ 134) (1931).He showed that the line CS1 is isometric to a sphere of radius r of the

2

space R3 (while the complex projective line CF1 can be considered as theextended complex plane, the complex Hermitian elliptic line CS1 can beconsidered as the Riemannian sphere).

In the above-mentioned paper [Stu4], Study also defined the complex Her-mitian Euclidean space CRn as the space CEn in which a Hermitian innerproduct is defined, and this product can be reduced to the form Eix i

y ' ;

the Hermitian inner square Eixi Y' is defined as the square of the modu-lus lxi of a vector x = {x'}, and the distance between the points X andY is defined as the modulus ly - x1 of the vector y - x. The space CRn ,

which is also called the unitary space, is often used in linear algebra. Thisspace is isometric to the Euclidean space R2n . The group of motions ofthe space CRn consists of transformations (2.8) where U = (u?.) are ma-trices of the group CU. and the products of these transformations and thetransformation 'x' = z' . Substituting in the definition of the space CRnthe inner product Eicix iy c for the inner product Eix'y-, we obtain thecomplex Hermitian pseudo-Euclidean space CR7 , which is isometric to thespace R2! . The group of motions of this space consists of transformations(2.8) where U = (u) are matrices of the group CUn and the products ofthese transformations and the transformation 'x' = x' .

§3.3. QUATERNION SPACES 95

§3.3. Quaternion spaces

Among Cartan's supplements to Fano's paper [Fa], we must mention theintroduction of the quaternion projective space HP" , which can be obtainedfrom the complex projective space CPn with the substitution of quaternionprojective coordinates (defined up to multiplication by a quaternion factorfrom the right) for complex projective coordinates. The group of projectivetransformations of the space HPn , i.e., the group HSEn+1 of quaternionunimodular matrices, is one of the noncompact groups in the class A2n+

1 .

In the paper On certain remarkable Riemannian forms of geometries witha simple fundamental groups [ 107] (1927), Cartan used the representation ofa compact simple Lie group in the class C by the group H U of quaternionunitary matrices. It follows from this representation that the latter group canbe considered as the group of motions of the quaternion Hermitian Euclideanspace HSn-1 , i.e., the quaternion space HPn-1 in which a real metric is de-fined by the same formula (3.6) as in the space CSn-1 I. Similarly, one canuse formula (3.7) to define a real metric of the quaternion Hermitian h y p e r -b o l i s s p a c e HS -1 . In the same manner as the isometry of the line CS1I toa sphere of radius r/2 in the space R3 was proved, one can prove the isom-etry of the line HS1 to a hypersphere of radius r/2 in the space R5 . Thespace HSn was defined in the paper Symmetric spaces and their geometricapplications [Roll by Boris A. Rosenfeld (b. 1917), which was published asa supplement to the collection of his translations of Cartan papers titled Ge-ometry of Lie groups and symmetric spaces [206] (1949). In the same paper,Rosenfeld proved that the space Cr is isometric to a paratactical congru-ence of straight lines of the space S2n+ 1 if one takes as the distance betweenlines of the congruence their unique stationary distance and that the spaceHSn is isometric to a paratactic congruence of the space CS2n+ 1 . Further-more, in this paper Rosenfeld defined the Hermitian hyperbolic spaces CSIand HSI . This material was presented in detail in Chapter VI of Rosenfeld'sbook Non-Euclidean geometries [Ro3] (1955) (see also [Ro71). The complexand quaternion Hermitian elliptic and hyperbolic spaces CSn , CSI ) HSn ,

and HSI can be defined as the space CPn or the space HPn whose fun-damental groups are subgroups of the group of projective transformationspreserving the Hermitian hyperquadric xJ a1).x' = 0 (a1J . = aJi) , and there-fore commuting with the polarity relative to this hyperquadric defined by the

i . .formula u, = xJ aJAs Hilbert showed in his book Foundations of geometry [Hil] (1899), in

spaces over nonassociative fields the Desargues configuration does not takeplace, and this is the reason that the geometry of such spaces (the planeOpt is one of them) is a non-Desarguesian geometry. In the same bookHilbert proved that, if n > 2, the Desargues configuration follows from theincidence axioms of the projective geometry, and this is the reason that a


non-Desarguesian geometry is impossible in a space of dimension greaterthan two.

In the paper Alternative fields and the theorem on complete quadrilateral(D9) [Mou] (1933), Moufang proved that, in a projective plane over an al-ternative skew field, the configurational theorem on complete quadrilateralholds. Using this paper of Moufang, Guy Hirsch (b. 1915), in his paperThe projective geometry and the topology of f ber spaces [Hir] (1949), definedthe plane OP2 by topological methods and proved that the straight lines ofthis plane are homeomorphic to eight-dimensional spheres. In the paper Theprojective plane of octaves and spheres as homogeneous spaces IBorl] (1950),A. Borel defined the plane OS2 as the plane OP2 (defined by Hirsh) withthe metric of the symmetric space V 16

. In the paper Octaves, exceptionalgroups and the geometry of octaves IFrdlJ (1951-1985), Freudenthal definedthe planes OP2 and OS2 algebraically by means of the octave coordinatesin these planes.

Reducing the group of projective transformations of the space HP' toits subgroup consisting of projective transformations commuting with the"null-system" u . = zj a13 (alb = -aj!) , we obtain the quaternion Hermitiansymplectic space HSyn defined by Ludmila V. Rumyantseva (b. 1937) in thepaper Quaternion symplectic geometry [Ru] (1963). In the space HSyn , onecan always choose a coordinate system in which the "absolute null-system" ofthis space will be of the form uk = xk i. (The complex Hermitian symplecticspaces defined in a similar manner coincide with the spaces CSn and CSlsince multiplication of a skew-Hermitian matrix (alb) by i gives a symmetricHermitian matrix.)

Note that, as was shown by David Hilbert (1862-1943) in his book Foun-dations of geometry [Hil] (1899), in spaces over noncommutative skew fieldsthe Pappus-Pascal configuration does not take place, and this is the reason thegeometry of such spaces (the space HPn is one of them) is a non-Pascaliangeometry.

§3.4. Octave planes

A compact simple group in the class G2, which is the group of automor-phisms of the alternative skew field 0 of octaves, is the transitive group ofrotations of the six-dimensional sphere that is the intersection of the hyper-sphere Ic l = 1 of octaves of modulus one with the hyperplane a = - a .

Furthermore, the metric of the space R8 is introduced into the field 0 ofoctaves: in this metric the distance between two octaves a and 8 is themodulus I / - a I of their difference. Identifying antipodal points of this hy-persphere, we obtain the elliptic space S6 whose fundamental group is thegroup indicated above. Such a space is called the group of motions of theG-elliptic space and is denoted by Sg6.

§3.5. DEGENERATE GEOMETRIES 97

A noncompact simple group in the class G2 which is the group of auto-morphisms of the algebra '0 of split octaves admits a similar representationas the group of rotations of the six-dimensional sphere of the space R3 andthe G-hyperbolic space Sg3 . A. Borel, in the paper Octave projective planeand spheres as homogeneous spaces [Bor] (1950), and Freudenthal, in the pa-per Octaves, exceptional groups and octave geometry [Frd 1) (1951), showedthat a compact simple group in the class F4 is the group of motions of the

Hermitian elliptic plane OS2 which is defined in the same way as the planesCV and HS2 , and that one of the noncompact simple groups in the classE. is the group of projective transformations of the projective plane OP2 .

In contrast to the planes CP2 and HP2 , the plane OP2 cannot be definedby means of arbitrary triplets x0 , x i

, and x2 of elements of the field 0defined up to multiplication by an arbitrary element a of this field sincenonassociativity of the field 0 implies (x'a)/3 34 x`(a/3). However, pointsof the plane OP2 can be defined by means of triplets x0 , x 1 , and x2 froman associative subfield of the field 0 defined up to multiplication by anarbitrary element of this associative subfield. Freudenthal defined points ofthe planes OP2 and OS2 by means of Hermitian symmetric octave matricesx' = z' satisfying the condition x`ixjk = x`kxjj , from which it followsthat all octaves x`3 belong to an associative subfield and thus it is possibleto find three octaves x' in this skew subfield such that x`1 = x`x1 .

Note that in the paper Two point homogeneous spaces [Wan] (1952), WanHsien Chung showed that any compact metric space in which, for any twoequidistant pairs of points, there is a motion transforming one of these pairsinto the other, is isomorphic to a hypersphere of the space Rit+1 , the spacesS'1 , C? , HSn and the plane OS2 .

In a series of papers under the common title The relations between E7and E. to the octave plane [Frd2) (1954-1963), Freudenthal also defined theoctave Hermitian symplectic space 03y5. However, since it is impossible todefine projective spaces of dimension higher than two over nonassociativefields, the space Osy5 cannot be defined as the space OP5 with the funda-mental group being a subgroup of the group of projective transformations ofthis space. The space 09y5 can only be defined as a set of planes OP2 thatare the analogues of two-dimensional isotropic planes of the space Sy 5 .

§3.5. Degenerate geometries

In Chapter 2, we defined the quasi-Cartan algorithm transforming a Liegroup G with the Lie algebra G = H ® E into a Lie group 'G with the Liealgebra °G = H ®EE where z is the dual unit of the algebra °C of dualnumbers a + be (E2 = 0) . If G is a semisimple Lie group, the Lie group°G is called a quasisimple group. Application of this algorithm r times to asemisimple Lie group leads to an r-quasisimple Lie group. The quasi-Cartan


algorithm can also be applied to both associative algebras and the alternativealgebras: if such an algebra A possesses an involutive automorphism andcan be represented as the direct sum A = B ® C, where B is a subalgebraconsisting of the elements that are invariant under this automorphism and Cis a linear subspace consisting of the elements that are anti-invariant underthis automorphism (i.e., under this automorphism, they are multiplied by-1), then the quasi-Cartan algorithm transfers this algebra A into the algebra°C = B ® X. The algebra 0C itself is obtained in this manner from the fieldC and the algebra 'C of split complex numbers which have undergone theinvolutive automorphism a a . Applying this algorithm to the algebras Hand 'H and their automorphism a -i iai-1 , we obtain the algebra 0H ofsemiquaternions, and, applying this algorithm to the algebras 0 and '0 andtheir automorphism, under which the field H is an invariant subalgebra, weobtain the alternative algebra °O of semioctaves.

In Chapter I of the memoir On manifolds with an affine connection andgeneral relativity theory [66] (1923) devoted to relativity theory, Cartan con-sidered the transformations of the space and time coordinates of classicalGalilei-Newton mechanics which he wrote in the form:

x' = alx+ bly+clz +g1t+h1,y = a2x+b2y+c2z+ g2t+h2,z' =a3x+b3y+c3z+g3t+h3,tt+h,

where the matrix with entries al , bl , and c, is an orthogonal matrix oforder three. Cartan assumed that x, y, z, and t are the coordinates of apoint of a four-dimensional space whose fundamental group is the group oftransformations

x' =alx+a2y+a3z+h1,yb1x+b2y+b3z+h2,z' =c1x+c2y+c3z+h3,t' = glx+g2y + g3z + t + h.

This space coincides with an isotropic hyperplane of the pseudo-Euclideanspace R i . The hyperplane is tangent to the isotropic cone of R i . This isthe reason that the space R

1

is called the isotropic space and is denoted byI" . The rotations of this spaces (motions (3.9) with h1 = h2 = h3 0)transform the vector {0, 0, 0, 1} into itself, and the direction of the axisOt is invariant under these rotations. The space R

ican be considered as

the affine space E4 , in whose hyperplane at infinity a point (in the directionof the axis Ot) and an imaginary hypercone of second order (defining in thishyperplane the geometry of the co-Euclidean space *R3 , dual to the spaceR3) are defined. The group of motions (3.9) of this space is a biquasisimpleLie group which can be obtained by applying the quasi-Cartan algorithm to

§3.5. DEGENERATE GEOMETRIES 99

the groups of motions of the spaces R4 and R4 R. Later, the space la wasconsidered by Karl Strubecker (1904-1991) in the paper Differential geometryof isotropic space [Str] (1941). Alexander P. Kotelnikov (1865-1944) in thepaper The principle of relativity and the Lobachevskian geometry [KotA3](1926) considered the space with the group of motions (3.8). He thought thatthis space is the space-time of Galilei-Newton mechanics. Since this space isthe affine space Ea , with the geometry of the space R3 in its hyperplane atinfinity, it is called the Galilean space and is denoted by F4

. The group ofmotions of this space is also a biquasisimple Lie group. In the same chapterof the memoir [66], Cartan considered a four-dimensional manifold whosetangent spaces are the spaces j4

. Such a manifold can be called the "spacewith an isotropic connection".

Cartan's note On a degeneracy of Euclidean geometry [147a] (1935) wasdevoted to a two-dimensional isotropic geometry. In it, Cartan consideredthe geometry of the "isotropic plane" 12 , i.e., a plane of the space R i tan-gent to an isotropic cone. The geometry of such a plane coincides with thegeometry of the Galilean plane r2 . Cartan's note was the exposition of histalk at a session of the French Association for the Development of Sciencein Nantes. Despite the fact that similar talks by Cartan at other sessions ofthis association were included in Cartan's cEuvres Completes [207) and [209),this note is missing from them. The note began as follows: "The geometryin an isotropic plane differs deeply from the geometry of the classical plane:in this plane, the lines, that in a non-isotropic plane play the role of circles,are parabolas tangent to the line at infinity at the same point" [147a, p. 128].Cartan wrote the motions of this plane in the following form:

fx' =x+a,y'=cx+hy+b.

If h = 1, these transformations are analogues of transformations (3.9) forthe two-dimensional case. If h I , these transformations are analogues ofsimilarities of the plane R2 .

The "isotropic plane" is the affine plane E2 , in whose line at infinity apoint is defined. The latter can be considered as a result of coincidence ofthe imaginary cyclic points of the plane R2 or the real cyclic points of theplane R i . (While in the planes R 2 and R i circles are conics passing throughthe cyclic points of these planes, in the plane I2 obtained by the passage tothe limit from these planes, the role of circles is played by conics tangent tothe line at infinity at the point of coincidence of cyclic points, i.e., parabolaswith diameters directed toward this point.)

The group of motions of the plane r2 is a biquasi-simple Lie group ob-tained by the quasi-Cartan algorithm from the groups of motions of the planesR2 and R, . In the note [147a], a number of problems on an isotropic planewere solved.


The title of the note [147a] shows that Cartan considered the passage fromthe geometry of the Euclidean plane to that of the plane I2 as a "degeneracyof Euclidean geometry". Note that this degeneracy is not the only possibil-ity. The idea of more general "degenerate geometries" obtained from thenon-Euclidean spaces Sn and Si was suggested in Klein's lectures on non-Euclidean geometry (1910), and the complete enumeration of all such geome-tries was given by Duncan Maclaren Young Sommerville (1879-1934) in thepaper The classification of geometries with projective metrics [Som] (1910).A year later, in the paper Euclidean kinematics and non-Euclidean geome-try [Bla l ] (1911), Wilhelm Blaschke (1885-1962) considered an importantcase of a degenerate elliptic geometry the geometry of the quasielliptic spaceS 1' 3 . In 1912, in the paper Construction of the entire geometry on the basis ofthe projective axioms alone [Mu], Ch. Muntz arrived at the same geometriesas Sommerville. Blaschke defined the metric in the quasielliptic space S 1 ' 3

as an analogue of the Cartan metric for the group of motions of the Euclideanplane R2 . Blaschke also introduced the term "quasielliptic space". In 1966Freudenthal analogously proposed the term "quasisimple group". The latterterm was the cause of the terms "quasisimple algebra", " r-quasisimple groupand algebra" and "quasi-Cartan algorithm".

The quasielliptic spaces Sm' n whose groups of motions are representedby matrices of the quasi-simple group On+1 can be defined as the space pnin which a degenerate imaginary quadric is given. This quadric is a coneof second order with a plane (n - m - 1)-dimensional vertex. This coneis called the absolute cone, and its equation can be reduced to the formEa

(xa) 2 = 0, a = 0, 1 , ... , m. In addition, in the plane vertex of thiscone (this vertex is called the absolute plane, and its equation has the formx = 0), a nondegenerate imaginary quadric is given. This quadric is calleda

the absolute quadric, and its equation can be reduced to the form >u(xM)20 , u = m + 1, ... , n . If the projective coordinates x` of the points of thisspace are normalized by the condition >a(x2)2 = 1 , the distance w betweenthe points X (x') and Y(yi) is defined by the relation cos w = Ea xaya . Ifthe line X Y intersects the absolute plane and w = 0, then the distancebetween the points X and Y is defined as the number determined bythe relation: q12 =

>(yU - x11)2. It is easy to see that the space S°' n ,

whose absolute cone is the hyperplane x0 = 0 taken twice, coincides withthe Euclidean space Rn . (In this case the absolute quadric coincides withthat imaginary quadric in which all hyperspheres of this space intersect oneanother, and if n = 2, it coincides with the cyclic points of the plane R2 .)

The space Sn -1, n coincides with the co-Euclidean space *Rn correspondingto the space Rn according to the principle of duality of the space pn. (Inthis case the absolute cone is an imaginary cone with a point vertex, and therole of the absolute quadric is played by the vertex of this cone taken twice.)The absolute cone of the space S1, n degenerates into a pair of imaginary

§3.6. EQUIVALENT GEOMETRIES 101

conjugate hyperplanes. In particular, for the Blaschke quasielliptic spaceS 1 ' 3 , the absolute consists of a pair of imaginary conjugate planes, theirintersection line, and a pair of imaginary conjugate points on this line. (TheBlaschke metric in the group of motions of the plane R2 is defined as follows:the distance co between two motions A and B is defined as the angle ofrotation of the motion BA-1 if this motion is a rotation about a point; ifthis motion is a translation and co = 0, the distance is defined as thelength of the vector of translation.)

If we substitute the space Sm, ' n-m0-1 for the space Sn-m°-1 in the abso-lute plane of the space Sm°' n , we obtain the biquasielliptic space Sm°' m" n

with a biquasisimple group of motions. In particular, the space S0 `1_ 'n

is the isotropic space 1n considered by Cartan in the paper [66] for n = 4and in the paper [ 147a] for n = 2. The space S°' 1, n = F" was consideredby A. P. Kotelnikov in the paper [KotA3] mentioned above. Note that inthe paper Projective geometry of the Galilean space-time [Sil] (1925), LudwikSilberstein (1872-1942) considered not the space 14 or the space I,4 butthe space R4 . Repeating this procedure a few times, we obtain the r-quasi-elliptic space Sma' M I mr-' ' n whose group of motions is an r-quasisimpleLie group. A particular case of this space is the flag space S°' 1 ' .... n-1 , n = En

whose absolute is a "flag" consisting of rn-dimensional planes of all dimen-sions m from 0 to n - l each contained in all p-dimensional planes (p > m)of this flag. (The flag plane F2 also coincides with the plane 12 consideredby Cartan.)

Substituting in the definition of the space S"' '' a real cone of index 10and a real quadric of index 11 for an imaginary absolute cone and an imagi-nary quadric, we obtain the quasihyperbolic space S! whose particular cases

0,are the pseudo-Euclidean space Rl and its dual space * R1 . The r-quasi-hyperbolic spaces sl °1 m""Mr_, P. n are defined in a similar manner. Thegroups of motions of the spaces Sm' n and Sm' n can be obtained by ap-

;0,r1plying the quasi-Cartan algorithm to the groups of motions of the spacesSn and S! , and the groups of motions of the spaces Sma' m, , ..' , m,_, , n and

Sm°' m, , m,_, ' n can be obtained from the same b applying the uasi-1° , 11)- I I

jr groups by qCartan algorithm r times.

The general theory of the spaces S"', Sl / , 5m0 ' m ' "' ' mr _ i ' n , andm m m n

a> >

Sl° °l , ' '' ,''r-11 was presented in the paper Projective metrics [YRY] (1964)

rby Isaac M. Yaglom (1921-1988), Rosenfeld, and Evgeniya U. Yasinskaya(b. 1929) and in Rosenfeld's book Non-Euclidean spaces [Ro7] (1969) (seealso his book A history of Non-Euclidean geometry [Ro 8 ] (1988)).

§3.6. Equivalent geometries

Chapter 2 of Fano's paper [Fa] was entitled "Relationships of differentgeometries from group-theoretical point of view". In Cartan's extended


translation [46] of this paper, this chapter received the shorter title "Equiva-lent geometries". This is the term Cartan used for geometries of spaces withisomorphic fundamental groups. In Klein's "Erlangen program", the repre-sentation of objects of one geometry by geometric objects of another wascalled the "transfer principle". The first of these principles was the "Hessetransfer principle" suggested by Otto Hesse (1811-1874) in the paper On atransfer principle [Hes] (1866) which gave the name to these principles. TheHesse transfer principle is based on the stereographic projection of a conic inthe projective plane P2 onto a line P 1 in this plane and on the isomorphismbetween the group of projective transformations of the plane P2 preservinga conic and the group of projective transformations of the line P1 .

Since the first of these groups can be considered as the group of motionsof the plane Sl , any geometric object of the plane S'1 is represented bya certain geometric object of the line P1 . In particular, straight lines ofthe plane S1 are represented by pairs of points of the line P 1 . Thus theHesse transfer principle is based on the isomorphism of the simple groupsA 1 and B1 . Another transfer principle the "Plucker transfer principle"suggested by Julius PlUcker (1802-1868) in his paper New geometry of spacebased on considering a straight line as a spatial element [Plu] (1868)-is wellknown. This principle is based on the representation of the straight lines ofthe space P3 by the points of the space P5 whose projective coordinates arethe Plucker coordinates p`3 = x`yj - xiy` , i, j = 0, 1, 2, 3, where x` andy i are the projective coordinates of the points X and Y of the line X Y .Since the coordinates p`3 are connected by the quadratic relation p01 p23 +p 02 p 31 + p 03p 12 = 0, the straight lines of the space P 3 are represented bythe points of the quadric of index 3 of the space P5 . Moreover, the groupof projective transformations of the space P3 is isomorphic to the group ofprojective transformations of the space P 5 preserving this quadric. Thus,the Plucker transfer principle is based on the isomorphism of the simple Liegroups A3 and D3.

In his paper [Fa], Fano gave only a few geometric interpretations knownat the beginning of the 20th century. In the section Equivalent geometriesof the extended translation [46] of Fano's paper, Cartan gave geometric in-terpretations of all isomorphisms from real simple Lie groups in the paper[38].

The groups indicated by Cartan as the groups Al = B1 = C1 (a = 1)are the group of collineations of the line P 1 and the groups of motionsof the plane S2 and of the line CS1 1 . The isomorphism of the first twogroups defines the Hesse transfer of P 1 on the absolute conic of S1 . Theisomorphism of the last two groups defines the Poincare interpretation of S2in the complex plane.

The groups Al = B1 = C1 (o = -3) are the groups of motions of the


line CS1 , the quaternion group JAI = 1 , and the group of motions of theplane S2 (or the group of rotations of a sphere in the Euclidean space R3).The isomorphism of the first and the third groups defines the metric of theRiemannian sphere in the complex plane. The isomorphism of the last twogroups defines the representation X' = AX A-1 ,MAN = 1 , of the group of ro-tations of a sphere in the space R3 . Cartan formulated these representationsas follows:

"(a) The projective geometry of the real line is equivalent to thehyperbolic non-Euclidean geometry of the plane and to the hyperbolicHermitian geometry of the line", and"(a) The elliptic non-Euclidean geometry of the plane is equivalentto the elliptic Hermitian geometry of the line" [46, p. 1834].

The groups D2 = A, x A, (J = 2) are the groups of motions of the spaceS2 and the direct product of two groups of collineations of the line P 1 . Thisisomorphism defines the interpretation of the manifold of straight lines ofthe space S2 by pairs of points of two lines P 1 and by points of two planesS1 , and also the interpretation of the line 'CP 1 over the algebra 'C of splitcomplex numbers a + be, e2 = + 1 , on the absolute ruled quadric of thespace S2 .

The group D2 (J = 0) is not semisimple but simple. It is isomorphic notto the direct product of two real groups A, but to the complex group A, .Being the group of collineations of the line CP 1

, the group D2 (J = 0) is thegroup of motions of the space S1 . The isomorphism of these groups definesthe Kotelnikov-Study transfer of the manifold of straight lines of the spaceS1 on a sphere of the complex Euclidean space CR3 .

The groups D2 = A, x A, (a = -2) are the group of symplectic transfor-mations of the line HSy 1 and the direct product of the group of collineationsof the line P 1 (which is isomorphic to the group of motions of the plane S )and the group of motions of the plane S2 . This isomorphism defines the in-terpretation of the line HSy 1 by the points of the planes S2 and S2

I.

The groups D2 = A, x A, (J = -6) are the group of motions of the spaceS'3 and the direct product of two groups of motions of the line CS' or oftwo planes S2 . This isomorphism defines the Fubini-Study transfer of themanifold of straight lines of the space S2 on the points of two spheres of theEuclidean space R3 and the Kotelnikov-Study transfer of the same manifoldon a sphere of the split complex Euclidean space 'CR3 .

Cartan formulated these interpretations as follows:

"(b) The real projective geometry of a real ruled quadric of the spaceE3 (Cartan denoted any n-dimensional space by En) is equivalentto the union of the real projective geometries of two lines",


"(b) The hyperbolic non-Euclidean geometry of the space E3 (or thereal projective geometry of a real non-ruled quadric) is equivalent tothe projective geometry of the complex line", and(b") The elliptic non-Euclidean geometry of the space E3 is equiv-

alent to the elliptic non-Euclidean geometry of two planes, or to theelliptic Hermitian geometry of two lines, or to the Euclidean geome-try of two spheres" [46, pp. 108-110].

Cartan omitted the case (b") of equivalence of "the projective geometryof a quadric x1 x2 + x3x4 =0 and of a hyperquadric (Hermitian quadric)x1 X 1 + x2x2 + x3X3 + x4x4 =0 of the space E3 and the geometry of theunion of two non-Euclidean planes one elliptic and one hyperbolic".

The groups B2 = C2 ((5 = 2) are the group of symplectic transforma-tions of the space Sy3 . This isomorphism defines the interpretation of themanifold of straight lines of the space Sy 3 in the space S2 .

The groups B2 = C2 ((5= -2) are the groups of motions of the space S1

and of the line HS1H. This isomorphism defines the Poincare interpretationof the space Sl in the quaternion 4-space.

The groups B2 = C2 ((5 = -10) are the groups of motions of the spaceS4 and of the line HS' . This isomorphism defines the isometry of the lineHS' and a hypersphere in the space R5 .

Cartan formulated these interpretations as follows:"(c) The real projective geometry of the quadric z 2 + z2 + z3 - z4 =0 of the space E4 or the geometry of cycles of the plane (Lie's "highergeometry" of oriented circles) is equivalent to the real projective ge-ometry of a linear complex","(c) The real hyperbolic non-Euclidean geometry of the space E4or the real conformal geometry of the space E3 is equivalent to theprojective geometry of the linear complex p12 + p34 = 0 and of thehyperquadric x1 X, + x2x2 + x3X 3 - x4x4 0 of the space E3", and"(c ') The real elliptic non-Euclidean geometry of the space E4 isequivalent to the projective geometry of the linear complex p1 2 +p34 = 0 and of the hyperquadric x11 + x2x2 + x3X3 + x4x4 = 0"[46, pp. 110-111].

The groups A3 = D3 ((5 = 3) are the group of collineations of the spaceP3 and the group of motions of the space S3 . This isomorphism definesthe Plucker transfer of the manifold of straight lines of the space P3 on theabsolute quadric of the space S3 .

The groups A3 = D3 ((5 = 1) are the groups of motions of the spacesCS23 and S5 S. This isomorphism defines the interpretation of the manifoldof straight lines of one of these spaces in the manifold of lines of the other.

The groups A3 = D3 ((5 = - 3) are the group of motions of the spaceCS3 and the group of symplectic transformations of the plane HSy2 . This

I


isomorphism defines the interpretation of the manifold of straight lines ofthe space CS-33 in the plane H3y2 .

The groups A3 = D3 (J = -5) are the group of motions of the space Siand the group of collineations of the line HP 1 . This isomorphism definesthe interpretation of the line HP I in the absolute quadric of the space S5 .

The groups A3 = D3 (J _ -15) are the group of motions of the spacesCS3 and S5 . This isomorphism defines the interpretation of the manifoldof straight lines of one of these spaces in the manifold of lines of the other.

Cartan formulated these interpretations as follows:

" (d) The real projective geometry of the quadric zi

+ z2 + z 3 - z 2 -z5 - z6 = 0 of the space E5 is equivalent to the general real projec-tive geometry of the space E3 ","(d' ) The real projective geometry of the quadric z

i+ z2 + z3 +

z4 - z5 - z6 = 0, or the geometry of oriented spheres of the spaceE3, is equivalent to the Hermitian geometry of the hyperquadricxl z 1 + x222 - x323 - x424 = 0 ,"(d") The real hyperbolic non-Euclidean geometry of the space E5 ,or the real conformal geometry of the space E4 , is equivalent to theprojective geometry of the quaternion line","(d"') The real elliptic non-Euclidean geometry of the space E5 isequivalent to the elliptic Hermitian geometry of the E3", and

"(d"") The hyperbolic Hermitian geometry of the space E3 is equiv-alent to the projective geometry of the fundamental quadric x1 x2 +73x4 + 75x6 = 0 and of the fundamental hyperquadric XI X I + x272 +x373 - x424 + x5 - x6X6 = 0 of the space E5" [46, pp. 111-112].

Note that instead of the quaternion spaces IFS, HS" , and HS y" (thefirst of which appeared only in [107] (1927)), in his translation of Fano'spaper, Cartan used complex (2n + 1)-dimensional spaces with a quadric ora linear complex and with a "hyperquadric" (Hermitian quadric).

The interpretation of the complex projective line CF' in the absolute ofthe space Si was formulated by Klein in his "Erlangen program". The in-terpretation of the manifold of straight lines of the space S3 in a pair ofspheres of the space R3 was proposed by Fubini in his dissertation Cliffordparallelism in elliptic spaces [Full (1900) and by Study in the paper On non-Euclidean and line geometry [Stu2] (1902). Similar interpretations of themanifold of straight lines of the space S2 in a sphere of the split complexspace 'CR3 and of the manifold of straight lines of the space Si in a sphereof the space CR3 were proposed by A. Kotelnikov in his doctoral thesis Pro-jective theory of vectors [KotA2] (1899) and by Study in his book Geometryof Dynames [Stu3] (1903). Cartan devoted a special section in his Lectureson complex projective geometry [ 134] to the interpretation of the quaternion


line HP' in the absolute of the space Si which was considered in detail inStudy's paper An analogue of the theory of linear transformations of a com-plex variable [StuS] (1923-1924). In the paper [ 1341, Cartan also consideredthe interpretation of the spaces S5 and S2 as manifolds of paratactic con-gruences of lines in the spaces CS3 and CS3 .2

Note also that in the same way that the conformal space C3 is interpretedon the absolute of the space S1 , the group of the Lie "higher geometry ofspheres" (defined in Lie's paper [Lie!] (1872) on a line and spherical com-plexes) which is the group of transformations of the manifold of orientedspheres of the space R3 (points and planes are considered as spheres of zeroand infinite radius), is isomorphic to the group of motions of the space S2and the manifold of spheres of this geometry is interpreted in the absoluteof the space S2 (Lie's imaginary transformation of the manifold of lines ofthe space P3 in the manifold of spheres of R3 is based on the imaginarytransformation of the absolutes of the spaces Sz and S3 ).

The interpretation o f the manifold of straight lines of the space CS3 i n thesame type of manifold of straight lines of the space S2 forms the foundationof the "twistor program" of Roger Penrose (b. 1931) presented in his paperTwistor theory, its aims and achievements [Pen] (1975). The twistors are thespinors of the group of motions of the space S2 . They are vectors of thespace C4 representing the points of the absolute of the space CS3. Thus,the points of this absolute represent rectilinear generators of the absolute ofthe space S2 , and similarly, the points of the absolute of the space S2 are

represented by rectilinear generators of the absolute of the space CS3. But2

the absolute of the space S2 represents the pseudoconformal space C, whichis obtained as the extension of the space R4 the space-time of the specialrelativity. Thus, the points of the absolute of the space S2 can be consideredas the space-time points of the Universe of special relativity. This explainsthe title The complex Universe of Roger Penrose of the paper [Gi] (1983) bySemen G. Gindikin (b. 1937) which is devoted to this interpretation.

In addition to the "transfer principles" based on the isomorphisms of sim-ple Lie groups, in the papers [Fa] and [46], Fano and Cartan also gave "trans-fer principles" based on the isomorphisms of quasi-simple Lie groups whichare obtained by passage to the limit from simple groups. The first of theseprinciples was the interpretation of the elliptic Hermitian line °CS 1 over thealgebra 0C of dual numbers a + be, e2 = 0, in the manifold of straight linesof the Euclidean plane R2 presented in detail by I. M. Yaglom in his bookComplex numbers in geometry [Ya2] (1968). The second principle was theinterpretation of the dual projective line °CP 1 in the geometry of Laguerretransformations in the real plane, i.e., the geometry of the manifold of ori-ented circles of the plane R in which those nonpoint transformations are

§3.7. GENERALIZATIONS OF THE HESSE TRANSFER PRINCIPLE 107

considered that transfer circles into circles and straight lines into straight linesand preserve the tangential distances between circles (the segments of thecommon tangents between the points of tangency). If we take the tangentialdistance as the distance between circles, this manifold becomes isometric tothe pseudo-Euclidean space R , , and the group of Laguerre transformationsis isomorphic to the group of motions of this space. Finally, the interpreta-tion of the dual projective plane °CB2 in the manifold of straight lines ofthe space R3 was considered. In this interpretation, the motions of the spaceR3 are represented by the motions of dual elliptic plane °CS2 . The first ofthe above-mentioned interpretations is based on the isomorphism of quasi-simple groups which are obtained by passage to the limit from the isomorphiccomplex groups, and the second and the third interpretations are based onthe isomorphism of quasi-simple groups which are obtained by passage to thelimit from the complex groups D2 and B1 x B1 . The latter interpretationwas studied in detail by A. Kotelnikov in his master's thesis Twist calculusand some of its applications to geometry and mechanics [KotAl] (1895) andby Study in his Geometry of Dynames [Stu3I (1903).

§3.7. Multidimensional generalizations of the Hesse transfer principle

After presenting a few "transfer principles" based on isomorphisms of sim-ple Lie groups, Fano formulated a few generalizations of the "Hesse transferprinciple". First, he gave the generalization of the Hesse principle suggestedby Wilhelm Franz Meyer (1856-1934) in the book Apolarity and rationalcurves [Me] (1883). According to this generalization, the points of the pro-jective line B1 are represented by the points of the "unicursal normal curve"of the space P', i.e., by the algebraic curve defined by the parametric equa-tions

(3.10) x`=t`, i=0, 1,... n,

where t' is the ith power of the parameter t.

The group of projective transformations of the space Bn preserving thiscurve is isomorphic to the group of projective transformations of the lineB1 . For n = 2, this interpretation coincides with the Hesse transfer prin-ciple. Fano also gave another generalization of the Hesse transfer principleaccording to which the conics ai .xtxi = 0 of the plane B2 are representedby the points of the space B5 with coordinates ai1 In this representation,the degenerated conics that are decomposed into pairs of straight lines arerepresented by the points of the algebraic hypersurface det(ai) = 0 of thisspace, and the degenerated conics that are twice taken straight lines are rep-resented by the points of the two-dimensional algebraic surface of fourthorder

(3.11) z`' = x'xJ


in the same space. This surface was studied by Cayley in 1868 and byGuiseppe Veronese (1854-1917) in the paper The two-dimensional normalsmooth surface of fourth order in a five-dimensional space and its projectionsonto a plane and onto the usual space [Ver] (1884). Veronese's term "normalsmooth surface" (superficie omaloide normale) stresses that he consideredthis surface as a generalization of the unicursal normal curve. The samemapping, with the reference to Veronese, was considered by Corrado Segre(1863-1924) in the paper Geometry of conic sections in the plane and on itsrepresentation in the form of complexes of straight lines [SeC 1 ] (1885). In thistitle C. Segre emphasized the analogy between the representation of conic sec-tions of the plane P2 in the space P5 and the representation in the samespace of linear complexes of the space P3 which follows from the "Pluckertransfer principle": the linear complexes of the space P3 are defined by the

t .p i1 = 0 in the Plucker coordinates, and thus, in the space P5 ,equation aJthey are represented by cross-sections of the quadric of index three by hy-perplanes and also by the points of the space P5 that are the poles of thesehyperplanes with respect to the quadric. Fano indicated that this representa-tion can be generalized to the representation of quadrics of the space Pn inthe form of points of the space pN where N = ((n + 1)(n + 2)/2) - 1 , andthe quadrics that are twice taken hyperplanes are represented by the pointsof the surface (3.11) of the space PA' which, at present, is also called theVeronese hypersurface or the Yeronesian and is denoted by Yn Y.

Fano mentioned one more generalization of the Hesse transfer principleunder which a set of points of several projective spaces of different dimen-sions is represented by the points of an algebraic surface in a certain projec-tive space PAT . He indicated that the simplest cases of this representationwere considered by C. Segre in the paper Varieties representing pairs of pointsof two planes or spaces [SeC2] (1891). In this paper, Segre considered the rep-resentation of two pairs of points of two planes P2 or two spaces P3 in theform of the points of the algebraic surface

(3.12)!a 1 az =xy

of the space P8 or P15 . At present, surface (3.12) in the space pmn+m+n

whose points represent the pairs of points of two spaces Pm and Pn, where

in general m n , is called the Segre surface or the Segrean and is denotedby Im ' n .

When Cartan translated this section of Fano's paper [Fa], in his transla-tion [46] he added the section "The Hesse principle applied for generationof the projective groups that are isomorphic to a given projective group". Inthis section, he considered the Segreans (3.12) of general type in the spacepmn+m+n and indicated that the subgroup of the group of projective trans-formations of this space preserving the Segrean is the Kronecker product ofthe groups of projective transformations of the spaces Pm and Pn.

§3.8. FUNDAMENTAL ELEMENTS 109

The Veronesians and the Segreans as well as the surfaces that are obtainedby their projections onto planes (the "quasi-Veronesians" and the "quasi-Segreans") have important applications in differential geometry. We will seein Chapter 5 that the Veronesians and the quasi-Veronesians are the indica-trices of curvature of p-dimensional manifolds in the space Rn , and thatthey also appear in the theory of p-dimensional manifolds in the space Bn.We will see in Chapter 6 that the Veronesians define the absolutes of one typeof symmetric Riemannian spaces and the Segreans are the local absolutes ofseveral types of symmetric Riemannian spaces. We will also encounter inChapter 7 an application of the Segreans to the theory of multidimensionalwebs. Application of the Segreans and the quasi-Segreans to different prob-lems of differential geometry is discussed in the two papers of Rosenfeld,M. A. Polovceva, T. I. Yuchtina, et al.: The Segreans and the quasi-Segreansand their application to the geometry of straight lines and planes [RPRY](1989) and The metric and symplectic Segreans and quasi-Segreans [RKSY](1989).

§3.8. Fundamental elements

Geometric objects of the spaces CBn, CS2n , CSy2n-1 , and CS2n-1 areconnected with the fundamental linear representations of complex Lie groupsin the the classes An) Bn , Cn , and Dn . These objects help to give geometricinterpretations to these groups. Probably, because of the connection of thesegeometric objects with the "fundamental groups" of Cartan, Tits, who furtherdeveloped the theory of these groups, called these objects the fundamentalelements.

The linear representation (p1 of a complex simple Lie group in the classAn that can be considered as the group of projective transformations of thecomplex projective space CBn is a representation of this group in the spaceCBn itself, i.e., the coordinates of vectors of this representation coincidewith the projective coordinates of points of the space CBn . The coordinatesof vectors of the linear representation Ok of this group coincide with theGrassmann coordinates

(3.13) p 1011 __.ik-I = k!xIi°xiI ... xik-1I

of (k - 1)-dimensional planes (for k = 2, these coordinates coincide withthe Phicker coordinates p h of straight lines) of the space CBn . Thus, thefundamental elements of the space CBn are points, straight lines, and m-dimensional planes (m = 2, 3, ... , n - 1).

Grassmann coordinates (3.13) satisfy the equations

(3.14)pLf°ll...iMPJ0JJ,,..Jn _ 0.

In the projective space B N where N = (,) - 1 , surface (3.14) representsthe manifold of m-dimensional planes of a complex or real n-dimensional


projective space. This manifold is called the Grassmann manifold and is de-noted by Gn, m . This is the reason that surface (3.14) is called the Grassman-nian and is denoted by 1T, ,,,, . We will see in Chapter 6 that the Grassmanni-ans (as the Segreans) are the local absolutes of several symmetric Riemannianspaces.

The linear representation (p1 of complex simple Lie groups in the classesBn and Dn that can be considered as the groups of motions of the com-plex elliptic spaces CS2n and CS2n -1 are representations of these groupsin the spaces CS2n and CS2n-1 themselves; i.e., the coordinates of vectorsof these representations coincide with the projective coordinates of points ofthe spaces CS2n and CS2n-1 . The coordinates of vectors of the linear repre-sentations 9k of these groups also coincide with the Grassmann coordinatesp'O'l '!k-1 (for k = 2, with the Plucker coordinates p il of straight lines) ofthe spaces CS2n and CS2n-1 . As to the spinor representation yrl of thegroup Bn and yr1 and w2 of the group Dn , in his Lectures on the theory ofspinors [164] (1938), Cartan showed that the coordinates of vectors of theserepresentations, the so-called spinors, can be considered as the coordinates ofthe isotropic spaces of maximal dimension of the complex Euclidean spaces

CR2n+1 and CR2n whose groups of rotations are complex simple Lie groupsof the classes Bn and Dn D. But the isotropic subspaces of maximal dimen-sion of the spaces CR2n+ 1 and CR2n cut on the hyperplanes at infinity ofthe spaces CR2n+1 and CR2n the plane generators of maximal dimensionsof these absolutes. These plane generators form one family for the space Stn

and two families for the space Stn-1 . Thus, the fundamental elements of thespaces CS2n and CS2n -1 corresponding to the spinor representations are theplane generators of maximal dimensions of absolutes of these spaces. Thus,as the fundamental elements of the spaces CS2n and CS2n-1 correspondto the linear representations (pl , 92 , and qk , one must consider not arbi-trary points, straight lines, and (k - 1)-dimensional planes of these spacesbut only the points and rectilinear and plane generators of the absolutes ofthese spaces. Therefore, the fundamental elements of the spaces CS2n and

CS2n-1 are the points and the rectilinear and m-dimensional plane genera-tors of the absolutes of these spaces (m = 2, 3, ... , n - 1 for the group Bnand m=2, 3 , ... , n-3, n- l for the group Dn).

The linear representation of a complex simple Lie group in the class Cnthat can be considered as the group of symplectic transformations of thespaces CSy2n-1 is a representation of this group in the space Csy2n-1 it-self, i.e., the coordinates of vectors of this representation coincide with theprojective coordinates of points of the space Csy2n-1 . The coordinates ofvectors of the linear representations °k of this group coincide with the Grass-mann coordinates of (k - 1)-dimensional isotropic planes (for k = 2, withthe Plucker coordinates of isotropic lines) of the space Csy2n- i , i.e., suchstraight lines and planes that lie entirely in those (2n - k - 1)-dimensional

§3.8. FUNDAMENTAL ELEMENTS 111

planes that correspond them relative to the null-system of the space CSy2n-1 .

The values of the dimensions of the spaces of the representations (Pk 'whichare equal to (2k) - (J2)' are explained by the fact that the Grassmann co-ordinates p`°`' "'1k = k!xl`°x`' x`k-11 of the (k - 1)-dimensional isotropicplanes are connected by the linear relations a! i p1°`' A-' = 0 whose par-

o]ticular case is the equation ai .ply = 0 of a linear complex of straight linesin the space CP3 . Thus, the fundamental elements of the space CSy2n-1

are its points (the absolute null-system of this space maps each point of thespace into a hyperplane passing through this point), isotropic lines, and m-dimensional planes (m = 2, 3, ... , n - 1).

Since to each simple root of a simple Lie group there corresponds a fun-damental linear representation of this group, to this root there correspondsalso a fundamental element of the corresponding classical space. Thus, thesets of the Dynkin graphs shown in Figures 2.2 and 2.5 not only repre-sent simple roots of simple Lie groups but also the fundamental elementsof the corresponding spaces. In particular, to the graph dots represent-ing simple roots of a simple Lie group in the class An) there correspondpoints, straight lines, and rn-dimensional planes (rn = 2, 3, ... , n - 1) ofthe space CPn ; to the graph dots representing simple roots of a simple Liegroup in the class Bn , there correspond points, rectilinear generators, andrn-dimensional plane (rn = 2, 3, ... , n - 1) generators of the absolute ofthe space CS2n ; to the graph dots representing simple roots of a simpleLie group in the class Cn, there correspond points and isotropic and m-dimensional (rn = 2, 3, ... , n - 3, n - 1) planes of the space CSy2n-1 ;

and to the graph dots representing simple roots of a simple Lie group inthe class Dn , there correspond points and rectilinear and m-dimensional(rn = 2, 3, ... , n - 3, n - 1 and n - 1) plane generators of the absolute ofthe space CS2n-1 (here, the two numbers n - 1 are related to two familiesof plane generators of maximal dimension; in this case, (n - 2)-dimensionalplane generators are not fundamental elements of the space CS2n-1 theyare determined by two plane generators of maximal dimension belonging todifferent families).

Similarly, starting from linear representations of real simple Lie groups,we can find the fundamental elements of the real spaces Pn, Sn , S! , andSy2n-1 ; we understand the latter space as the space Pen-1 where a linearcomplex a . p` = 0 of straight lines is given. For the spaces Pn, Snn ,

Sy2n-1,

and Snn-1 , whose fundamental elements are defined in the same way asthe fundamental elements of the corresponding complex spaces CPn ,

CS2n ,

CSy2n-1, and CS2n-1 , all these elements are real. For the spaces Stn andStn-1 , all these elements are imaginary, and, for the spaces Stn and Stn -1

for 0 < 1 < n, there are both real and imaginary elements.


In the lecture on the general probl em of deformation (55] (19 20) presentedat the International Congress of Mathematicians in Strasbourg, Cartan notedthat the manifold of straight lines of the projective space P3 can be mappedby means of an imaginary transformation onto the conformal space C4 . Thistransformation maps the group of projective transformations of the space P3to the group of conformal transformations of the space C4, which allows usto apply the theory of deformations of surfaces of the space C4 to the theoryof deformations of families of straight lines of the space P3 . The latterCartan conclusion was based on the "Plucker transfer principle" by meansof which the manifold of straight lines of the space P3 is represented bythe absolute of the space S3 , and the imaginary transformation mentionedabove is a transformation of the space Sj into the space S1 whose absoluterepresents the space C4. Similar to the imaginary transformation used byLie in his paper [Liel ] (1872), here Cartan essentially introduced the ideaof pseudo-conformal space C, and its interpretation on the absolute of thespaces S

1

n+ 1 and S!+n+1 1 ; in this interpretation, conformal transformationsof the space Cl are represented by motions of the space Sl+ 11 . Since thepoints of the absolutes of the spaces Sl+ are the fundamental elements ofthese spaces, the points of the spaces Cn and Cl are also the fundamentalelements of these spaces.

The geometry of fundamental elements was further developed in the the-sis on the topology of certain homogeneous spaces [Eh!] (1934) of Cartan'sstudent Ehresmann. In his thesis, Ehresmann found topological invariants ofmany manifolds of fundamental elements. In the book Unitary representa-tions of the classical groups [GN2] (1950), Israel M. Gel'fand (b. 1913) andMark A. Naimark (1909-1980) used many of these manifolds for construc-tion of linear representations of noncompact Lie groups by means of uni-tary operators of the Hilbert space. Stationary subgroups of "fundamentalelements" were studied by Vladimir V. Morozov (1910-1975) in his unpub-lished dissertation On nonsemi-simple maximal subgroups of simple groups(Kazan, 1943). These groups are maximal nonsemisimple subgroups of sim-ple Lie groups.

The theory of "fundamental elements" was substantially developed in thepapers of Tits who, along with these "fundamental elements", consideredmore general geometrical elements whose stationary subgroups are parabolicsubgroups of simple Lie groups (i.e., these subgroups contain a maximal solv-able subgroup of these groups which is called the Borel subgroup). At present,these geometric elements are called the parabolic elements. (In the paperFigures of simplicity and semi-simplicity [Ro6] (1963), Rosenfeld called the"fundamental elements" the "figures of simplicity", and he called the moregeneral parabolic elements the "figures of semisimplicity".) Tits consideredthe "parabolic elements" in the papers On certain classes of homogeneousspaces of Lie groups [Ti!] (1955) and On the geometry of R-spaces [Ti 3 ]

§3.9. THE DUALITY AND TRIALITY PRINCIPLES 113

(1957). The R-spaces here stand for the manifolds of parabolic elements.At present, these spaces are called the parabolic spaces. According to Tits,two fundamental elements are incident if the intersection of their stationarysubgroups is a parabolic subgroup.

For the space CPn , parabolic elements are "flags" consisting of planes ofdifferent dimensions enclosed one inside the other (straight lines and pointsare considered to be l - and 0-dimensional planes).

For the spaces CSn and CSy2n-1 , the parabolic elements are the "flags"consisting of the plane generators of the absolute or of isotropic planes. Thisis the reason that the parabolic spaces are also called "flag manifolds" (see,e.g., the paper by Wolf, The action of a real semi-simple group on a complexflag manifold [Wo 1] (1969)). A general survey of the geometry of parabolicspaces has been presented in the paper Parabolic spaces [RZT] (1990) byRosenfeld, Mikhail P. Zamakhovsky (b. 1942), and Tamara A. Timoshenko(Stepashko) (b. 1949).

Later on, departing from the geometry of " R-spaces", Tits constructed amore general geometry of buildings (immeubles) of "spherical types" (see hisbook Buildings of spherical types and finite BN-pairs [Ti6] (1980)) and also ageometry of buildings of Euclidean (affine) types corresponding to analoguesof usual and affine Weyl groups (see [Ti7] and [Ti8]).

§3.9. The duality and triality principles

In the paper The duality principle and the theory of simple and semi si mpl egroups [82] (1925) which we discussed earlier, Cartan posed the question oftransformations of simple and semisimple Lie groups "preserving the groupstructure" and of representations of these transformations in spaces for whichthese Lie groups are groups of transformations. In this paper, Cartan provedthat the only simple Lie groups whose Weyl group is a subgroup of the Galoisgroup of the characteristic equation of the group are groups in the classesAn) Dn, and E6 , and for them the quotient groups of the Galois groups bythe Weyl groups are isomorphic to the multiplicative group t l , -1 I and,for the group D4 , to the group of permutations of three elements. Thepresence of these quotient groups explains the symmetries of the Dynkingraphs discussed earlier and shown in Figures 2.3, 2.4, and 2.6. These graphshave the bilateral symmetry for the classes An and E6 in the vertical axis ofsymmetry, for the class Dn (n 4) in the horizontal axis of symmetry, andthe trilateral symmetry for the class D4 . Cartan connected these symmetrieswith the transformations of the spaces under which the fundamental elementsof these spaces corresponding to simple roots of the groups are transformedinto the fundamental elements corresponding to other simple roots of thesegroups.

For the first of these symmetries, points of the projective space CPn cor-respond to hyperplanes of these spaces, straight lines correspond to (n - 2)-dimensional planes, and m-dimensional planes correspond to (n - m - 1)-


dimensional planes. This correspondence is expressed by the classical dualityprinciple of the space CPA , and the latter was the reason for the title of theCartan paper [82]. This duality can be realized by means of the correlation

i .xf which maps the point X (xinto the hyperplane u xf = 0 withui = aJ

the tangential coordinates u! .

For the second of these symmetries, (n - 1)-dimensional plane generatorsof the absolute of the space CS2n-1 from one family correspond to (n - 1)-dimensional plane generators of the same absolute from another family. Inthe space CS2n-1 , there are transformations interchanging the plane gener-ators of these two families. Cartan considered the correspondence betweenplane generators of these two families as an analogue of the duality principleof the space Cpn.

There is also the duality principle in the real space Pn , and there is thecorrespondence between plane generators of two families of generators in thespaces Stn-1 and Snn -1

We mentioned earlier the Freudenthal interpretation of one of the non-compact groups in the class E6 in the form of the group of projective trans-formations of the octave plane OP2 . It follows from this interpretation thatthe complex simple group E6 admits the interpretation in the form of theprojective plane (0.0 C)p2 over the algebra of complex octaves (the tensorproduct of the algebras 0 and C). This is the reason that the third symme-try indicated by Cartan coincides with the duality principle of the projectiveplanes OP2 and (0 0 C)P2 .

As to the trilateral symmetry of the graph of groups in the class D4 , Car-tan connected it with the isomorphism of stationary subgroups of the pointsof the absolute of the space CS7 and of the three-dimensional plane genera-tors from different families of this absolute and also with coincidence of thefundamental linear representations (p 1 ,

1, and w2 (the matrices of all of

these three representations are matrices of order eight). In this connection,Cartan wrote: "We can say that the duality principle of projective geometryis replaced here by the triality principle" [82, p. 373).

By this "triality principle", the points of the absolute of the space CS7can be replaced by three-dimensional plane generators of the absolute fromboth families of these plane generators. But the points and three-dimensionalplane generators of the absolute of the space CS7 are the fundamental ele-ments of this space corresponding to three simple roots of the group of itsmotions, and these elements are transferred one into another by the rotationof the graph of simple roots of this group through 1200 (see Figure 2.6).Similar triality principles hold in the real spaces S7 and SS . Note that, inthe same paper, Cartan connected this "triality principle" with the algebra ofoctaves: if one introduces the metric of the space R8 in the algebra 0 ofoctaves taking as the distance between two octaves a and 8 the modulusIf - al of their difference, then the metric of the space CR8 arises in the

§3.9. THE DUALITY AND TRIALITY PRINCIPLES 115

algebra 0 ® C. Thus, any point of the absolute of the space CS7 can berepresented as a complex octave of zero modulus. But the three-dimensionalplane generators of this absolute can be also represented by octaves of zeromodulus since the equations ac = 0 and a = 0 (where a and are com-plex octaves of zero moduli and the complex octave a represents a point ofthe absolute) define the three-dimensional plane generators of different fam-ilies of the absolute of the space CS7. On the other hand, if the complexoctaves a and a of zero moduli are octaves such that their product a/3 = yis not zero, then this product is a nonzero complex octave of zero modu-lus. If the complex octaves a and fi represent two three-dimensional planegenerators from different families of the absolute, then the complex octavey represents their unique common point. If the complex octaves a andfi represent a three-dimensional plane generator from one of the familiesand a point of the absolute, then the complex octave y represents a three-dimensional plane generator from another family which intersects the firstgenerator at this point of the absolute. Cartan's research on the connectionbetween the triality principle in the space S7 and octaves was continued byE. A. Weiss in the paper Octaves, Engel's complex and the triality principle[Wes] (1938 ).

In the joint paper On R iemanni an geometries admitting an absolute paral-lelism [92] (1926), Cartan and Schouten considered a similar triality principlein the real space S7 which is obtained by identifying antipodal points on thehypersphere Jal = 1 in the algebra 0 with the metric of the space Rg . Theydefined in this space two continuous families of transformations of this hy-persphere mapping the octave into the octave ' and the octave 11 intothe octave ,j . These transformations are called the absolute parallelisms,and there are (+)-parallelisms

(3.15)

and (-)-parallelisms

(3.16)

n(c'a) = *c'a)

(a/C I

)"I = *-I)IIeach of which depends on seven parameters. They noted that "the pointsof S7 and the (+)- and (-)-parallelisms can be considered as elements ofS7"and that "thus we have triality in S7 by means of which we can definethe distance between two (+)- or two (-)-parallelisms, etc." [92, p. 944].Parallelisms (3.15) and (3.16) are defined in such a way that the segmentscq and 'c' q are considered to be parallel to one another.

Cartan also considered the triality principle in the space CRg , which isequivalent to that in the space CS7, in his Lectures on the theory of spinors[164] where he formulated this principle for isotropic vectors of the spaceCR8 representing the points of the absolute of the space CS7 and for "semi-spinors" of first and second kinds defining the four-dimensional isotropic


planes of the space CR8 representing the three-dimensional plane generatorsof the absolute of the space CS 7 . Cartan did not mention octaves here buthe mentioned the Brioschi formulas which express the product of two sumsof eight squares in the form of a similar sum and are equivalent to the octaveidentity: kI31 = lall/3l.

Finally, in the unpublished manuscript Isotropic surfaces of a quadric in aseven-dimensional space [ 1771, Cartan, without mentioning the triality prin-ciple, in fact considered an application of this principle in the space SS . Inthis manuscript, he considered the real projective space P7 and the quadricfl(x) = x°x7 + x Ix6 + x2x5 + x3x4 = 0 in this space. This quadric is the ab-solute of the SS . The collineations ("homographies"), preserving this quadric(i.e., the motions of the space S4 ), were called the "absolute homographies"by Cartan. The hyperquadric considered by Cartan has two families of three-dimensional plane generators. Cartan called these generators the "generatorspaces". Cartan proved that, in general, two "spaces" from different fami-lies always intersect each other at a point, and if they have one more com-mon point, they have a two-dimensional intersection. Cartan called these2-planes of intersections "isotropic planes". Through any isotropic plane,there passes a unique pair of "generator spaces" from different families. Ingeneral, two "generator spaces" of one family do not have common points.If they have a common point, then they have a common "isotropic line".The main goal of this manuscript was the study of "isotropic surfaces", i.e.,two-dimensional surfaces all of whose tangent planes are isotropic planes.Cartan's terms "isotropic line", "isotropic curve" (a curve whose tangentsare isotropic lines), and "isotropic surface" indicate that he considered a hy-perquadric of the space P7 as the pseudo-conformal space Cb although henever used this term. Cartan applied the equations of "generator spaces"which are equivalent to the equations a = 0 and a = 0, where a and

are elements of the alternative algebra '0 of split octaves satisfying theconditions f 2 = 0 and 0(a) = lal2 = 0. In the manusript, Car-tan also gave the formula (equivalent to the formula a/3 = y) for three splitoctaves, one of which represents a point of the absolute and the other tworepresenting two "generator spaces" from different families.

§3.10. Spaces over algebras with zero divisors

We have already discussed many times the spaces over the field C of com-plex numbers considered by Cartan, namely, the spaces CP", CE", CS" ,CS", CSy2n-' , and others as well as the analogues of some of these spacesover the algebra 'C of split numbers and the algebra 0C of dual numbers,over the skew field H of quaternions, the alternative skew field 0 of oc-taves, and some other algebras. Many of these spaces are used for geometricinterpretations of simple Lie groups.

§3.10. SPACES OVER ALGEBRAS WITH ZERO DIVISORS 117

The projective line R2P 1 over the algebra R2 of real matrices of secondorder was first considered by Niccolo Spampinato (1892-1971) in the paperon geometry of the line space considered as a hypercomplex S1 [Spa] (1934).In the paper The manifold S5 of lines considered as the hypercomplex S2 con-nected with a regular complex algebra of order 4 [Cab] (1936), his studentCarmela Carbonaro considered the projective plane R2P2 over the same alge-bra. Spampinato and Carbonaro also studied interpretations of the projectiveline R2P1 and the projective plane R2P2 in the form of the manifolds ofstraight lines of the real projective spaces P3 and P5 and indicated that theresults obtained by them can be generalized to higher dimensions. For thisgeneralization, one must define the n-dimensional projective space R,n+ 1 P"over the algebra Rm+

1of real matrices of order m + 1 . Each point of this

space is defined by n + 1 coordinates x' that are the matrices (xa`) definedup to multiplication x' -- x`a by a nonsingular matrix (a) of the samealgebra. In this representation, each point x (x') is represented by an m-dimensional plane of the real space pmn+m+n defined by the points x,, with

coordinates xa` . (When the coordinates x' are multiplied by the matrix(a), the points x are replaced by their linear combinations y = x afl .)

An attempt to construct a general theory of spaces over rings with zerodivisors was made by Dan Barbilian (1895-1961) (who is also known asthe poet "Ion Barbu") in the paper The axiomatics of projective plane ringgeometries [Bab] (1940-1941). Because of this, the spaces over rings andalgebras with zero divisors are often called the "Barbilian spaces". Barbilianwas first to notice that, although in general through two points of these spacesthere passes a unique straight line, there are pairs of points in these spacesthrough which there passes more than one straight line. Barbilian called thepoints in the first case the "points in clear position" and the points in thesecond case the "points in spectral position". At present, the pairs of pointsthrough which there passes more than one straight line are called the adjacentpoints, and the pairs of straight lines having more than one common pointare called the adjacent lines.

In the paper Symmetric spaces and their geometric applications [Ro 1 ](1949), Rosenfeld defined the spaces 'Cpn and 'Cr over the algebra 'C ofsplit numbers and the spaces 'HP" and 'HS" over the algebra 'H of splitquaternions (by the isomorphism 'H = R2 , the space 'HP" coincides withR2P"). He proved that the spaces 'CS" and 'HS" admit the interpretationin the form of the manifolds of "0-pairs" (a 0-pair is a point and a hyper-plane) of the space p" and in the form of the manifolds of straight lines ofthe space Sy2n+1 . He also proved that the groups of motions of the spaces'CS" and 'HS" are isomorphic to the fundamental groups of the spaces P"and S y 2n+1 , respectively.

In the interpretation of the space R2P" in the form of the manifold of


straight lines of the space P2n+1 . adjacent points are represented by inter-secting lines and adjacent lines by the manifolds of straight lines in three-dimensional planes belonging to a four-dimensional plane, and thereforethese 3-planes have a common 2-plane. The terms "adjacent points" and"adjacent lines" were introduced by Wilhelm Klingenberg (b. 1924) in thepaper Projective and affine planes with adjacent elements [Kli] (1954). In1957, in the papers Projective spaces over algebras [Jay 1 ] and Non-Euclideangeometries over algebras [Jav2], Maqsud A. Javadov (1902-1972) definedthe spaces R,+1 Pn and the Hermitian elliptic spaces Rm+1 Sn over the al-gebra R,+1 and found the interpretations of these spaces in the manifoldsof m-dimensional planes in the spaces Pmn+m+n and S'nn+m+n . A detailedexposition of the geometry of projective and non-Euclidean spaces over al-gebras is given in Chapter VI of Rosenfeld's book Non-Euclidean geometries[Ro3] (1955) (see also his book [Ro7]).

§3.11. Spaces over tensor products of algebras

Projective, Hermitian elliptic and Hermitian hyperbolic spaces were alsodefined over the tensor products of algebras C, H, and 0 and their ana-logues. The Hermitian elliptic spaces over the tensor products A ® B are-ndenoted by (A ® B)S , and the Hermitian hyperbolic spaces over the same

Nnalgebras are denoted by (A ® B)S, (these notations are explained by the factthat in the definitions of these spaces, the involution a - a is used in thetensor product A ® B, and this involution consists of the involution a - ain the algebra A and the same involution a -i a in the algebra B). Notethat the tensor product C ®C the elements of this algebra were called thebicomplex numbers by Cartan-is isomorphic to the direct sum C ® C and

nthe space (C ® C)S admits the interpretation in the form of the pair ofspaces CSn . The tensor product H ®C the elements of this algebra werecalled the biquaternions by Hamilton, the inventor of the algebra H -is iso-morphic to the algebra C2 of complex matrices of second order, and the

N n

space (H ® C)S admits the interpretation in the form of the manifold ofstraight lines of the space

CS2n+ 1. The tensor product H ® H is isomorphic

=nto the algebra R4 of real matrices of fourth order, and the space (H ® H)Sadmits the interpretation in the form of the manifold of three-dimensionalplanes of the space San+3 .

In the papers [Ro2] (1954) and [Ro4] (1956), Rosenfeld defined the Her-N2 -'2 -2

mitian elliptic planes (0 ® C)S , (0 ® H)S , and (0 ® 0)S and showedthat the groups of motions of these elliptic planes are compact simple Liegroups in the classes E6, E7 , and E8 .

Later he also defined the Hermitian hyperbolic planes over the same tensorproducts and the Hermitian elliptic planes (which can be obtained from the

§3.11. SPACES OVER TENSOR PRODUCTS OF ALGEBRAS 119

Hermitian elliptic planes over the tensor products 0 ® C , 0 ® H , and 0 (& 0by replacing one or both factors in these tensor products by the correspondingalgebras 'C, 'H, and '0) and showed that the groups of motions of theseplanes are noncompact simple Lie groups in the same classes. In the earliermentioned paper [Frd2] on the connection between the simple Lie groups E7and E8 with the octave plane, Freudenthal noticed that the groups of motionsof two-dimensional Hermitian elliptic planes over the fields R, C, H, and0, the groups of projective transformations of two-dimensional projectiveplanes over the same fields, and the groups of symplectic transformations offive-dimensional symplectic spaces over the same fields can be represented asthe first three rows of the following "magic square":

B1 A2 C3 F4

A2 A2xA2 A5 E6C3 A5 D6 E7

Fa E6 E EsThis was the reason that Freudenthal used the term metasymplectic geome-

tries for the geometries whose fundamental groups are the groups indicated inthe fourth row of the square (3.17). We will denote the spaces correspondingto these geometries over the fields R, C, H, and 0 by Ms, CMs, HMs,and OMs, respectively. These spaces are sets of so-called symplecta whichin turn can be considered as sets of two-dimensional isotropic planes of thesymplectic spaces Sy 5 , CSy 5 , 16y' , and 03y'. Thus, the main geo-metric objects of metasymplectic geometries are symplecta, two-dimensionalprojective planes, projective lines of these planes, and points of these linesand planes. In the paper Metasymplectic geometries as geometries on the ab-solutes of Hermitian planes [RoS] (1983), Rosenfeld and Stepashko showedthat the Freudenthal metasymplectic geometries are represented in the ab-

2, -2 -2solutes of Hermitian elliptic planes OS , ('0 ® C)S , ('0 (& H)S , and

Co (& O)S and that the similar metasymplectic geometries 'CMs, 'HMs,and 'OMs are represented in the absolutes of the Hermitian elliptic planes

('0 , ('0 (& H}S , and ('0 (&O}S . The Freudenthal "magic square"represents also the geometric interpretations of compact Lie groups formingthis square. In this case all the groups of this square are the groups of mo-tions of the Hermitian elliptic planes over the fields R, C, H, and 0 andtheir tensor products.

Table 3.1 (next page) is the table of the real simple Lie groups (up to alocal isomorphism). In the first column of this table, the class of the sim-ple Lie group is indicated, in the second column the character of the realsimple group, in the third column the spaces for which these groups are thefundamental groups, and in the fourth column the number of the figure rep-resenting the Dynkin graph of a compact group or the Satake graph of anoncompact group.


An 6 = -n(n +2)n-1)/2

CS" = (H®C)S Fig. 2.2 (a)6 = n P" = `CSn Fig. 2.10 (a)

6 = -n-2 HP(n-1)/2 = (H ®'C)S(n 1)/2Fig. 2.10 (b)

6=41(n-1+1) CSj Fig. 2.10 (c, d)- n(n + 2)

6 = 2n - n2 CSi Fig. 2.10 (e)

Bn 6 = -n(2n + 1) Stn Fig. 2.2 (b)6=21(2n-1+1) Sr" Fig. 2.10 (f, g)

- n(2n + 1)6 = 3n - 2n2 Sr" Fig. 2.10 (h)

Cn 6 = -n(2n + 1) HS"-1 Fig. 2.2 (c)6 = 2n Syl "-1 = 'HS"-1 Fig. 2.10 (i)6=81(n-1) HSi-1 Fig.2.10(j,k)

- n(2n + 1)

Dn 6 = -n(2n - 1) Stn-1 = 'HSy"-1 Fig. 2.2 (d)6 = 21(2n -1) Stn-1 Fig. 2.10 (1, m, n)

- n(2n - 1)6=5n-2n2-2 S,"-1 Fig. 2.10 (o)a = -n HSyn-1 Fig. 2.10 (p, q)

G2 6 = -14 Sg6 Fig. 2.5 (a)6 = 2 Sg3 Fig. 2.11 (a)

F4 0 = - 52 OS2

Fig. 2.5 (b)6 = -20 0Si Fig. 2.11 (c)6 = 4 'OS2 = Ms Fig. 2.11 (b)

E6 6 = -78=2

(O ® C)S Fig. 2.5 (c)

6 = -26 OP2 = (O (& 'C)S1 Fig. 2.11 (g)

6 = -14=2

(0 (& C)S1

Fig. 2.11 (f )

6 = 2 ('O ® C)S = CMs Fig. 2.11 (e)

a = 6 'OP2 = ('O C)S1 ='CMs Fig. 2.11 (d)

E7 0 = -133 (O (& H)S Fig. 2.6 (d)

0 = -25^-2

OSp5 = (0 0'H)S Fig. 2.11 (j)

6 = -5 (O ® H)S1 = ('O ® H)S = HMs Fig. 2.11 (i)

0 = 7--2

'OSps = ('O H)S = 'HMs Fig. 2.11 (h)

E8 6 = -248=2

(O ® O)S Fig. 2.5 (e)

6 = -242

(O®'O)^-S

= OMs Fig. 2.11 (1)

0 = 8 2('O O'O)S = (0 (& O)S1 ='OMs Fig. 2.11 (k)

TABLE 3.1

Table 3.2 gives the table of isomorphisms of real simple Lie groups (upto a local isomorphism). The construction of Table 3.2 is similar to that ofTable 3.1.

§3.12. DEGENERATE GEOMETRIES OVER ALGEBRAS 121

A1 = BI= Cl 6=-3 CS' = SZ Fig. 2.3 (a)b = -1 CSC =P 1 ='CS1 = S Fig. 2.12 (a)

D2 =B, x B, 8 = -6 S3 =S 2 XS 2 = CSZ Fig. 2.3 (b)b = 0 S3 =

CS2

Fig. 2.12 (b)b = 2 SZ = S XS

i= CS Fig. 2.12 (c)

b = 6 HSyl = SZ X S2 Fig. 2.12 (d)

B2 = C2 81= -10 S4 = HS' Fig. 3 (c)J = -2 S =HST Fig. 2.12 (f)b = 2 Sz = Sy3 ='HS1 Fig. 2.12 (e)

A3 = D3 b = -15 S5 = CS3 Fig. 2.3 (d)b = -5 S = HP Fig. 2.11 (b)8 = 1 SZ = CS-2 Fig. 2.12 (h)J = 3 S3 = P3 = IC-s3

Fig. 2.12 (J)

J = -3 HSy2 =CSC Fig. 2.12 (i)

D4 b = -4 I HSy3 = SZ Fig. 2.12 (k)

TABLE 3.2

§3.12. Degenerate geometries over algebras

Applying to the groups of motions of the spaces CSn and CS1quasi-Cartan algorithms similar to the algorithm by means of which fromthe groups of motions of the spaces Sn and Sl we obtained thegroups of motions of the quasielliptic spaces Sm, n , the quasihyper-bolic spaces S !

, ,n , the r-quasielliptic spaces Sm0 , in ''nr-1 ' n , the r -quasi-

01

hyperbolic spaces Sin l ' 'Mr-1,

n , we obtain the complex Hermitian, quasi-

elliptic, quasihyperbolic, r-quasielliptic and r-quasihyperbolic spaces CS11,n

,01

M o ,...,m_ n MOP m ,...,in,-,,nCS o ' r 1 and CS ..1' . In like manner, from the spaces

n n mn m,n mo,irr1,...,irrr_1,nHE and HS1 we obtain the spaces HS HS1of1

, HS

andHSmo , m 1 , ...H7'1°'''' In etc.1011 ..lr

On the other hand, applying to the groups of motions of the spaces Crand HSn the quasi-Cartan algorithms corresponding to the involutive auto-morphisms a -* aaa where a is the motion 'x' = x' and 'x' = ix' i-1 ,

we_obtain the groups of motions of the Hermitian elliptic spaces °CSn and°HSn over the algebra °C of dual numbers and the algebra °H of semi-quaternions. The spaces, similar to those we have defined over the fields Cand H, can be also defined over the algebras °C and 'H.

The quasi-Cartan algorithm applied to Lie groups is a particular case ofthe "contraction of Lie groups" defined by Wigner and Inonii in the earliermentioned paper [IW] (1953). A complete classification of all quasisimple


Lie groups and of their geometric interpretations was given by Rosenfeld andLudmila M. Karpova (b. 1934) in the paper Flag groups and contraction ofLie groups [RK] (1966). (In this paper, the authors called quasisimple Liegroups "flag groups".)

The quasisimple Lie groups obtained by the method indicated above fromcompact simple groups are obtained by the same method from noncompactsimple groups enumerated by Cartan. Because of this, we will denote thesegroups by the same Cartan symbols which we used for notation of noncom-pact simple Lie groups.

In particular, the quasisimple Lie groups Al, All, AIII, and AIV aretn-1)12 m, n-

, (H ® oC)S, CS , andthe groups of motions of the spaces oCSn

CR'1 , respectively. The quasisimple Lie groups BI and BII are the groupsof motions of the spaces Sm' 2n (for m > 0) and R2n , respectively. Thequasisimple Lie groups CI and CII are the groups of motions of the spaces

0HSn-1 , HSm n-1(HR n 1 for m = 0), respectively. The quasisimple Lie

groups DI, DII, and DIII are the groups of motions of the spaces Sm' 2n-1

(for m > 0), R2n-1 , and oHSyn-1

, respectively. The quasisimple Lie groupG is the group of motions of the space Sgt' 6. The quasisimple Lie groupsFI and FII are the groups of motions of the planes 00- S2 over the algebra'0 of semioctaves (obtained by the quasi-Cartan algorithm from the field 0)and OR2 , respectively, etc.

As we have for simple Lie groups, to the isomorphic or locally isomor-phic quasisimple Lie groups there correspond "equivalent geometries" whosefundamental groups are such groups.

Tables 3.3 and 3.4 are the tables of real quasisimple Lie groups that are ob-tained by the quasi-Cartan algorithm from compact simple Lie groups (up toa local automorphism) and the isomorphisms of real quasisimple Lie groups.As in Tables 3.1 and 3.2, Tables 3.3 and 3.4 also indicate the spaces whosefundamental groups are the corresponding groups.

An

Bn

Cn

Dn

G2

F4

0 -n =(n-1)12 -1-1,n

E6

E7

E8

CS , (H®t'C)S , CSSI-1,2n

0HS,n-1 , HSI-1 n-1

1-1 S,2n-1 0HSyn-1Sg2,6

OR2 , °OS20 2

.... 2

(O ® QS , (O ® C)R , (°O (& C)S

(0®°H)S2 (O® H)R2 0 O ® H)S2

(0®°0)S O O R2

TABLE 3.3

§3.13. FINITE GEOMETRIES

Al = B1 = C1D2= B, x B,

B2 = C2

A3 =D3

D

CR' = O CS' = R2

R3 = °CS2 , 1S' ' 3 = 'CR2 , °HSy' = S2 x R2

R4=HR',S1'4=°HS'R5 = (H ®°C)S , S1,5

1,3= CS ,

S2' 2,5 = °CS3 , HSy2 = CR3°HSy3 = S1 17

4

TABLE 3.4

123

§3.13. Finite geometries

The simple Chevalley groups also admit geometric interpretations in spacesconstructed over the corresponding fields and over the algebras built overthese fields.

The finite Chevalley groups admit similar interpretations in spaces similarto those which were considered in this chapter but constructed over the finitefields Fq . In particular, groups in the class An can be interpreted as thegroups of collineations of the projective space FgPn , groups in the classBn as the group of motions of the non-Euclidean space FqStn , groups inthe class Cn as the group of symplectic transformations of the symplecticspace FqSy2n-1 , groups in the class Dn as the group of motions of the non-

Euclidean space F Stn-1 , groups in the class A(2) as the group of motionsq

of the Hermitian space F 2Sn , and groups in the classes Dn2) and D43 asq

the groups of motions of the non-Euclidean spaces F S -1 and F 3 S 3q () q ( )

The Chevalley groups whose Satake graphs are shown in Figure 2.13 ad-mit geometric interpretations in the form of groups of collineations of pro-jective spaces over division algebras constructed over corresponding fields.The geometry of the projective line P' and the plane 2 , whose group ofcollineations is isomorphic to that of the line P' over the field F2 of 2-adicnumbers, were considered by Jean-Pierre Serre (b. 1926) in the paper Trees,amalgams, SL2 [Se 1 ] (1977). The geometric interpretations of arbitraryChevalley groups are particular cases of the Tits "buildings" whose theory, aswe have already indicated, is presented in his book [Ti5] and in his papers[Ti6] and [Ti7].

CHAPTER 4

Lie Pseudogroups and Pfaffian Equations

§4.1. Lie pseudogroups

After solving in his thesis the problem of the structure of usual (finite-dimensional) Lie groups which Cartan called "finite continuous groups", Car-tan posed the similar problem for "infinite continuous groups", i.e., for in-finite-dimensional analogues of Lie groups. The following papers by Cartanwere devoted to this problem: the two-part paper On the structure of infinitegroups of transformations [21], [22] (1904), Simple continuous infinite groupsof transformations [23], [28] (1907 and 1909), and Subgroups of continuousgroups of transformations [26] (1908).

While the finite-dimensional Lie groups are connected with the theory ofordinary differential equations, their infinite-dimensional analogues are re-lated to the theory of partial differential equations. Cartan started to studythe latter as far back as 1899.

At present, infinite-dimensional analogues of Lie groups are called Liepseudogroups. The Lie pseudogroup considered by Cartan is a set of trans-formations of a space that contains the identical transformation (playing therole of the neutral element) and possesses the property that the result of suc-cessive realization of two transformations of this set (when this is possible)belongs to the same set. However, in contrast to usual Lie groups of trans-formations, in this case the successive realizations of transformations is notalways possible: each such transformation is given by functions defined incertain domains, and the domain of one of the transformations may not havecommon points with the domain to which another transformation maps itsdomain. This explains the fact that this set of transformations is not a groupand is the reason it is called "pseudogroup".

In the papers mentioned above, Cartan considered manifolds whose pointsare defined by complex coordinates and assumed that the transformationswhich he studied were given by analytic functions of these coordinates. Asin the case of the finite-dimensional Lie groups, Cartan considered only "in-finitesimal transformations". This explains why he did not encounter thecases when for two transformations the result of their successive realizationcannot be found. Because of this, Cartan used the term "groups" for sets ofsuch transformations.

125

126 4. LIE PSEUDOGROUPS AND PFAFFIAN EQUATIONS

As he did in the case of the finite-dimensional Lie groups, Cartan consid-ered only those transformations of manifolds for which there is no subdi-vision of manifolds into the classes transposed by the transformations un-der consideration. Such groups and pseudogroups of transformations arecalled primitive groups and pseudogroups. Cartan showed that every infinite-dimensional primitive pseudogroup of complex analytic transformations be-longs to one of the following six classes:

1° . The pseudogroup of all analytic transformations of n complex vari-ables.

2°. The pseudogroup of all analytic transformations of n complex vari-ables with a constant Jacobian (i.e., transformations that multiply all volumesby the same complex number).

3°. The pseudogroup of all analytic transformations of n complex vari-ables whose Jacobian is equal to one (i.e., transformations that preserve vol-umes).

4°. The pseudogroup of all analytic transformations of 2n > 4 complexvariables that preserve the double integral

n

ll zt AdZn+i

ff i=1

5°. The pseudogroup of all analytic transformations of 2n > 4 complexvariables that multiply the double integral (4.1) by a complex function.

6°. The pseudogroup of all analytic transformations of 2n + 1 complexvariables that multiply the form d z° + El1 z'd zn+' by a complex function.

The pseudogroup 4° is called the symplectic pseudogroup since its transfor-mations preserve the exterior form El , d z` A d zn+r'

, and the latter definesthe "symplectic geometry" in the hyperplanes at infinity of the tangent spacesCE2n to the manifold under consideration. This pseudogroup is also calledthe Hamiltonian pseudogroup since the mechanical system with generalizedcoordinates q' and generalized impulses pi , whose motion is described bythe Hamiltonian equations, can be viewed as a space where the exterior closeddifferential form to= d q i A d pl is given (da = 0). The pseudogroup 6° iscalled the contact pseudogroup since in this case 2n + 1 complex coordinates

0can be viewed as 2n + 2 coordinates z , z 1, ... , zn , w° , w 1 , . . . , wnconnected by the relation 11(z° , z 1 , ... , zn , w° , W 1 , ... )Wn) = 0 whichestablishes the correspondence between the points z (z° ) z 1 , ... , zn) of ann-dimensional space and hyperplanes of the space with coordinates w° , w 1

,

see , wn . Such transformations are called contact transformations (or transformations of tangency). The theory of contact transformations was devel-oped by Lie.

Cartan showed that the pseudogroups 1 ° , 3°, 4° , and 6° are simple pseu-dogroups, or, in his terms, they are "simple infinite continuous groups", andthe pseudogroups 2° and 5° are "invariant subgroups" of the pseudogroups

§4.2. THE KAC-MOODY ALGEBRAS 127

3 0 and 4° . He called the classes 1 ° , 3° , 4° , and 6° the "four large classesof simple infinite continuous groups" and considered them to be analogousto the "four large classes of simple finite continuous groups" the infiniteseries of simple finite Lie groups.

There are similar classes of pseudogroups for primitive pseudogroups ofreal transformations defined by analytic functions of real variables.

In his extended French translation [46] of Fano's paper [Fa] Cartan de-voted a few sections to the real Lie pseudogroups. Along with Lie's andCartan's research described by Fano, Cartan included in [46] some resultswhich were obtained after the publication of [Fa]. We note one pseudogroupfrom pseudogroups considered in [46], namely, the pseudogroup of transfor-mations of the set of straight lines of the space R3 under which the normalcongruences (the congruences of normals to a surface) are transformed intothe same kind of congruences. Cartan noted the importance of these trans-formations for optics.

The infinite-dimensional pseudogroups were applied to geometry in thebook of Oswald Veblen (1880-1960) and John H. C. Whitehead (1904-1960),The foundations of differential geometry [V W] (1932), since the transforma-tions of coordinates of a differential-geometric manifold form precisely sucha pseudogroup. In this connection the series of papers by Victor V. Wagner(1908-1971) is very interesting. We note his papers On the theory of pseu-dogroups of transformations [Wag l j (19 50) and Algebraic theory of d iferen ti algroups [Wag2] (19 51).

§4.2. The Kac-Moody algebras

At present, several types of infinite-dimensional generalizations of Lie al-gebras and groups are being studied. Victor G. Kac (b. 1943) in his bookInfinite-dimensional groups with applications [Kac2] (1985) indicated that,although the general theory of infinite-dimensional theory of Lie algebrasand groups had not yet been constructed, "there are, however, four classesof infinite-dimensional Lie groups and algebras that underwent more or lessintensive study. There are, first of all, the ... Lie algebras of vector fieldsand the corresponding groups of diffeomorphisms of a manifold. Startingwith the work of Gel'fand-Fuks ... , there emerged an important directionhaving many geometric applications, which is the homology theory of infinite-dimensional Lie algebras of vector fields on a finite-dimensional manifold.There is also a rather large number of works which study and classify variousclasses of representations of the groups of diffeomorphisms of a manifold ...The second class consists of Lie groups (respectively Lie algebras) of smoothmappings of a given manifold into a finite-dimensional Lie group (resp. Liealgebra). In other words, this is a group (resp. Lie algebra) of matrices oversome function algebra but viewed over the.base field. (The physicists referto certain central extensions of these Lie algebras as current algebras.) ...The third class consists of the classical Lie groups and algebras of operators


a) A22)

b) A(2)2n

D (3)a

FIGURE 4.1 FIGURE 4.2

in a Hilbert or Banach space. There is a rather large number of scatteredresults in this area ... Finally the fourth class of infinite-dimensional Liealgebras is the class of so-called Kac-Moody algebras" [Kac2, pp. ix-x]. TheLie pseudogroups belong to the first of these classes. Here, Kac mentionedthe paper by Gel'fand and Dmitry B. Fuks (b. 1939), The cohomology ofthe Lie algebra of tangent vector fields on a smooth manifold (1969-1970)[Gel, vol. 3, pp. 290-306 and 323-329] (see also the book Cohomology ofinfinite-dimensional Lie algebras [Fuk] (1986)).

The Kac-Moody algebras introduced b y Kac i n the paper Simple graded Liealgebras of finite growth [Kac 1 ] (1968) and by Robert V. Moody (b. 1941)in the paper A new class of Lie algebras [Moo] (1968) are closest in theirproperties to the Lie algebras of the simple Lie groups. Kac's book [Kac2]is also devoted to the theory of these algebras. As for the Lie algebras ofsimple Lie groups, for the Kac-Moody algebras, the following notions canbe defined: the root systems, the Weyl groups which in this case are infinitediscrete groups generated by reflections, and also the systems of simple rootsand the Dynkin graphs. The Kac-Moody algebras are divided into threetypes: nontwisted algebras, 2-twisted algebras, and 3-twisted algebras. Thesenames and notation of these algebras are given by analogy with the namesand notation of the Chevalley groups mentioned earlier. The Dynkin graphsof the nontwisted Kac-Moody algebras coincide with the augmented Dynkingraphs (Fig. 2.8). Figures 4.1 and 4.2 show the Dynkin graphs of the 2- and3-twisted Kac-Moody algebras, respectively [Kac2, pp. 44-45].

The congruence of the Dynkin graphs of the nontwisted Kac-Moody alge-bras with the augmented Dynkin graphs of the simple Lie groups is connectedwith the isomorphisms of the Weyl groups of the nontwisted Kac-Moody al-gebras and the affine Weyl groups of the simple Lie groups. Note that thetwisted Kac-Moody algebras are denoted by Kac as A22 , A2n) , A2n-1) Dn+ 1 ,

E62) , and D43) , and by Moody (if one transfers on top the low indices 2and 3 which Moody places after the low index preceded by a comma) as

X4.3. PFAFFIAN EQUATIONS 129

A 2) , B C 2) , 2) , B, 2 , F (2) , and G23) [Moo, p. 229], respectively. This no-tation corresponds to the notation of those Lie groups whose Dynkin graphsare obtained by removal of one vertex of the Dynkin graph of the twistedKac-Moody algebras.

§4.3. Pfaflian equations

Cartan's first work on the theory of partial differential equations was hispaper On certain differential expressions and the Pfaff problem [14] {1899),which was followed by the papers On some quadratures, whose differential el-ement contains arbitrary functions [ 15] (1901), On the integration of systemsof exact equations [16] (1901), On the integration of certain Pfaffian systemsof character two [17] {1901), On the integration of completely integrable dif-ferential systems [ 18] (1902), and On the equivalence of differential systems[19] (1902).

In the first of these papers Cartan showed that every system of partialdifferential equations is equivalent to a system of differential equations:

(4.2) 6° = a°(x)dx' = 0 , a = 1 , 2, ... , s,

a so-called system of PfafJian equations. For example, the Laplace equation

az az{4.3) - o2ax2

ay

which is one of the fundamental equations of mathematical physics, withthe help of substitution (9z/(9x = u, a z /a y = v, can be reduced to thefirst-order system of partial differential equations:

au =av (9u (9V(4.4) ax ay ' a y ax'the so-called system of Cauchy-Riemann equations which the real and imag-inary parts of an analytic function w = u + iv = f(x + iy) of a complexvariable x + iy satisfy. The latter system is equivalent to the following sys-tem of differential equations:

(4.5) 01 = du -pdx - qdy = 0, 02 =dv + qdx - pdy = 0,

where p = (9u/ax = av/a y , q = (9u/( 9y = -av/ax. The appropriatenessof transition from systems of partial differential equations to Pfaffian equa-tions is explained by the fact that equations (4.2) are invariant with respectto an arbitrary change of both dependent and independent variables, while ina system of partial differential equations the choice of independent variablesis predetermined.

The Pfaffian equations are named after the mathematician and astronomerJohann Friedrich Pfaff (17 65-182 5) who considered such equations in 1814-1815. The term "Pfaffian equations" was introduced by Carl Gustav JacobJacobi (1804-18 51) who named the problem of integration of such equation


the "Pfaff problem". Papers of Feodor Deahna (1815-1841), August LeopoldCrelle (1780-1855), Jacobi, Ferdinand Georg Frobenius (1849-1917), Lie,and Darboux were devoted to the investigation of this problem. Appearingin Cartan's research in 1899, the Pfaffian equations were the subject of hisinvestigations and then the tool of research in many of his papers during hisentire life. After finding in his papers of 1899-1902 a new approach to theinvestigation of such systems, Cartan used them widely both in his geomet-ric papers and in his papers on the theory of Lie groups and mathematicalphysics.

System (4.2) of differential Pfaffian equations permits the following geo-metric interpretation. If one considers the variables to be the coordinates ofpoints of an n-dimensional manifold X' , then the differentials d x' can beconsidered as coordinates of the vector dx belonging to the tangent linearspace Tx (X n) of the manifold X n at its point x. If system (4.2) containss linearly independent equations, s < n, then it defines a linear subspaceOh (x) of dimension h = n - s in the space Tx(Xn). If the rank of system(4.2) remains constant and equal to s in the whole manifold Xn, then thissystem defines a subspace Ah (x) of the space Tx (X n) at each point x ofthe manifold X n . The set of subspaces Ah (x) of the tangent spaces TX (X n )

taken at each point x of the manifold X n is said to be a distribution ahAn integral manifold of Pfaffian system (4.2) is a smooth submanifold

Vk of the manifold Xn such that, at each of its points, it is tangent tothe subspace eh(x) determined at the point x by system (4.2). The di-mension k of the integral manifold Vk cannot exceed the dimension h ofAh (x) . System (4.2) always has one-dimensional integral manifolds the in-tegral curves, and in finding them, the system (4.2) is reduced to a system ofordinary differential equations.

§4.4. Completely integrable Pfaffian systems

Pfaffian system (4.2) is said to be completely integrable if it has integralmanifolds V h of maximal dimension h, and through every point of themanifold Xn there passes a unique integral manifold Vh , i.e., the integralmanifolds Vh of a completely integrable system (4.2) form a foliation in themanifold Xn.

Conditions for the system (4.2) to be completely integrable were found byFrobenius. In order to write down these conditions, one must construct thebilinear covariants of the system. Let x be a point of the manifold Xn andd

1x = {d1x'}, d2x = {d2x'} be two tangent vectors to this manifold at the

point x. Denote by 0°`(d1) = aa(x)d1x` and 0°`(d2) = aa(x)d2x` the valuesof these forms on these vectors. Differentiate the first of these expressionsalong the vector d

1x and the second one along the vector d2x. Then, the

§4.4. COMPLETELY INTEGRABLE PFAFFIAN SYSTEMS 131

bilinear Frobenius covariant is the difference of these two differentials:

(4.6) d26°(dj) - d, 0°(d2) = d2a'(x)d, x` - d, a'(x)d2x`

(on the right-hand side of expression (4.6) the terms containing the mixeddifferentials and d, d2x` cancel). Initially Cartan called the left-handside of expression (4.6) the exterior derivative of the form B' and denotedit by (Ofk)/. Later he started to call this expression the exterior differentialof the form 6° and denoted it by d6°. Cartan called the right-hand side ofexpression (4.6) the exterior product of the forms da and dx` . He initiallydenoted the exterior product of the forms coy and cot by the symbol cv,co2and later by the symbol [co, cv2] . At present, this product is denoted bycoy A w2, and relation (4.6) can be written in the form:

(a.7) ae° = aa° A ax'.

On any integral manifold Vk of system (4.2), equations (4.2) are satisfiedas well as the equations

(4.8) dOa = 0

obtained by exterior differentiation of system (4.2). By (4.7), equations (4.8)can be written in the form (8a'/(9xj)dx` A dxj = 0. But since dx' A dxj _-dxj n dx', one can also write these equations in the form:

(4.9) aaa aa' dx' n dx' = 0.Caxj axe

These equations impose conditions on the coordinates of any two vectorsd,x and d2x tangent to an integral manifold. If the vectors d, x and d2xsatisfy equations (4.9), then we say that they are in involution relative to thesystem of exterior forms (4.8).

Exterior differentiation of equations (4.8) leads to identities since

d(dB") . d (aai n dx` n dx' = (_' I dx` n dx' n dxk . 028xi aXjaXk

by the symmetry of the second mixed derivatives. Thus, the system of equa-tions (4.2) and (4.8) is closed with respect to the operation of exterior differ-entiation.

If the system of equations (4.2) is completely integrable, the integral man-ifolds of this system are of dimension h = n - s, and equations (4.8) mustnot impose any new relations on the coordinates of the tangent vectors inaddition to the relations imposed by equations (4.2). It is easy to see thatthis condition can be written in the form

(4.10) dO =O/AO , a, /3= 1 , 2 , ... ,s.f


In his paper of 1877 Frobenius proved that condition (4.10) is not onlynecessary but also sufficient for the complete integrability of system (4.2).

§4.5. Pfaffian systems in involution

In the papers on integration of systems of exact equations [16] (1901) andOn the structure of infinite groups of transformations [21], [221 (1904-1905),Cartan constructed the theory of systems of Pfaffian equations that are notcompletely integrable. Following Lie, who used the term "involutive systemsof equations", Cartan said the system of Pfaffian equations (4.2) were in in-volution if at least one two-dimensional integral manifold V 2 passes througheach integral curve V 1 of this system, at least one three-dimensional integralmanifold V3 passes through each of its integral manifolds V2 , etc., andfinally at least one integral manifold V" passes through each of its integralmanifolds VP-1.

Cartan found necessary and sufficient conditions for the system of Pfaffianequations (4.2) to be in involution. For this, he considered p-dimensionalelements consisting of a point x of the manifold X' and a p-dimensionalsubspace E" of the tangent space Tx (X n) to the manifold Xn at the pointx. This element is called the integral element and is denoted by Ip if all itsvectors satisfy the system of Pfaffian equations (4.2 ), i.e., I" belongs to Ah

,

and if, in addition, any two vectors of this integral element are in involutionrelative to the system of equations (4.8). It is obvious that if system (4.2) isin involution, then a two-dimensional integral manifold I2 passes througheach of its one-dimensional integral elements P, a three-dimensional inte-gral element I3 passes through the integral element I2 , etc., and finally anintegral element Ip passes through the integral element 9 -1 . This sequenceof enclosed integral elements I1 , I2 , ... , I" is called an integral chain. Anintegral chain is said to be regular if each of its integral elements is in generalposition, i.e., no more integral elements Ik pass through an element Ik-1

than through any neighboring (k - 1)-dimensional integral element. Cartanproved that a necessary and sufficient condition for the system of Pfaffianequations (4.2) to be in involution is the existence of a regular chain of inte-gral elements I1 , I2 , ... , I" for each point x of the manifold Xn.

When one is constructing an integral chain, there comes a time when thereis no integral element of dimension g + 1 passing through an integral elementof dimension g . In this case system (4.2) is in involution for all p < g , butit does not have this property for p > g. The number g is called the genreof system (4.2).

Cartan proved the existence theorem for solution of system (4.2) of genreg. Let I" be an integral element of system (4.2) of dimension p < g at apoint x of the manifold Xn ; then there exists an infinite set of

§4.5. PFAFFIAN SYSTEMS IN INVOLUTION 133

p-dimensional integral manifolds passing through a manifold Vp-1 and tan-gent to the element Ip at the point X0 , and for p = g there exists only onesuch manifold. The proof of this theorem is based on the classical Cauchy-Kowalevskaya theorem on the existence of a solution of a system of partialdifferential equations. Since the Cauchy-Kowalewskaya theorem is valid onlyin the class of analytic functions, the Cartan theorem is also valid only in thecase when all coefficients of equations (4.2) are analytic functions, and thedesired integral manifolds are analytic manifolds.

The Cartan theory not only gives the answer to the question of the existenceof integral manifolds of system (4.2) but also establishes arithmetic testsunder which there exist integral manifolds of a certain dimension p (p < g)of system (4.2) and indicates an arbitrariness with which these manifoldsexist.

These tests were formulated by Cartan in his paper On the structure ofinfinite groups of transformations [21), [22] (1904-1905). Here, applying thenotion of the "character of a system of Pfaffian equations" introduced byEduard von Weber in the paper On the theory of invariants of a system ofPfaffian equations [Web] (1898), Cartan determined the system of charactersof a system of Pfaffian equations (the von Weber character was the firstof Cartan's characters), and using these characters, he established necessaryand sufficient conditions for existence of a solution of a system of Pfaffianequations. We now show in more detail how Cartan established these tests.

Suppose one is looking for p-dimensional integral manifolds of the systemof Pfaffian equations (4.2) where p < g . System (4.8) of exterior differentialsof this system can be reduced to the form

{4.11) aa. A OJ + 2aa 0` A Eu + as cu A O = 0 ,i iu uv

where 0', i = 1 , ... , p, are Pfaffian forms that are independent on anintegral manifold, and Ou , u = 1, ... , q , are the remaining characteristicforms of system (4.2). Let rl be the rank of the system of linear equationswhich is obtained from (4.11) when one constructs an (i + 1)-dimensionalintegral element. The characters sl , I < i < p, of system (4.2) are thedifferences r. - r1-1 (note that sl + s2 + + sp_ 1 < q) , and the charactersp The number Q atpresent is called the "Cartan number", is equal to the number of parameterson which a p-dimensional integral element depends. The characteristic forms8u can be expressed from system (4.11) in the form of linear combinationsof the basis forms 0` : 8u = bu0`. If we denote by N the number ofindependent coefficients in these decompositions, then the Cartan test for theinvolutivity of system (4.2) is expressed by the relation: N = Q. Moreover,if the last nonvanishing character is s , then the solution of system (4.2)depends on sm functions of m real variables.


§4.6. The algebra of exterior forms

We have already mentioned the operation of exterior differentiation of alinear form and the operation of exterior multiplication of two such formswhich were introduced by Cartan. These operations are particular cases ofmore general operations applied by Cartan not only to linear forms but alsoto differential forms

(4.12) o)=ai. ..t dxl' Adx`21 2 p

called exterior forms of degree p. Here ai!l ...l is the tensor which is skew-symmetric

p

symmetric in all its indices, i.e., it changes the sign with any odd substitutionof indices and preserves the sign with any even substitution of indices, andA is the symbol of exterior multiplication which also indicates that the form(4.12) changes the sign with any odd substitution of the differentials andpreserves the sign with any even substitution of the differentials. For exteriorforms, the operation of exterior multiplication:

cvl AGo2 =(a1 i . dx" Adxl2 Adx' )1 2 p

A(b . dx'1 A dx'2 A ... A dx'q }(4.13)

=ar i2. ..i b dx'1dx" Adx'2 A...1 p l l2 q

Adx' Adx'1 Adx'2 A Adx'q

and the operation of exterior differentiation:

(4.14)dw = d (a 1 .. dx" A dx`2 n A dxtp}

12 p

=day 1 .. Adx`1 Adx`21 2 p

are defined.Moreover, if w1 and &2 are differential forms of degrees p and q re-

spectively, then the exterior differential of the product co1 A CO2 is equal to

(4.15) d((o1 n coz) _ (dcvl ) A cvz + (1)"aji n dcoz.

We have seen that Frobenius used the operations (4.13) and (4.14). Wealso encountered a particular case of the rule:

(4.16) d(d(o) = 0)

which essentially was known to Poincare and thus frequently called thePoincare theorem.

An exterior differential form w is called closed if d w = 0 and exactif there exists a differential form 0 such that to = d9 . By the Poincaretheorem, every exact differential form is closed, although not every closeddifferential form is exact.

§4.7. APPLICATION OF THE THEORY OF SYSTEMS IN INVOLUTION 135

Cartan often used the property that if the equation Oi A of = 0 holdswhere 8i and cvl are Pfaffian forms and the forms cv` are linearly indepen-dent, then the forms 8i are linear combinations of the forms cv` , and thecoefficients b. of these linear combinations are symmetric:

! = bit Cvt , bit = b}i .

At present this statement is called the Cartan lemma.The exterior forms constitute an algebra with respect to their addition and

multiplication. This algebra coincides with the Grassmann algebra.

§4.7. Application of the theory of systems in involution

In many of his investigations Cartan applied the theory of systems of Pfaf-fian equations in involution which he created. In the paper The Pfafan sys-tems with five variables and partial differential equations of second order [30](1910), this theory was applied to the investigation of a system of two partialdifferential equations of second order -a problem investigated by EdouardGoursat (1858-1936). Using this theory, in this paper Cartan investigateda system of Pfaffian equations with five variables to which these two equa-tions can be reduced, solved the equivalence problem relative to admissi-ble transformations for two such systems, and gave a detailed classificationof the systems of this type. In the paper on systems of partial differentialequations of second order with one unknown function and three independentvariables in involution [33] (1911), Cartan investigated the systems indicatedin the title that can be reduced to a system of four Pfaffian equations. Inthe paper on Backlund transformations [45] (1915), the theory of systems ininvolution was applied to the study of Backlund transformations by meansof which the known solutions of a system of partial differential equations canbe transformed into certain new solutions of this system. In the paper Onthe theory of systems in involution and its application to relativity theory [ 131 ](19 31), this theory was applied to the investigation of equations to whichcertain problems of general relativity can be reduced. Most applications ofthe theory of systems in involution are related to differential geometry of sub-manifolds of various homogeneous spaces, which we will consider in Chapter5.

In 1934 the theory of systems in involution constructed by Cartan forPfaffian equations was generalized for systems consisting not of only Pfaf-fian equations but also of exterior differential equations of different ordersby Erich Kahler (b. 1906) in his book Introduction to the theory of systemsof differential equations [Kah2]. In the book Exterior differential systems andtheir geometric applications [181] (1945), Cartan presented a systematic ex-position of both his own theory of solution of Pfaffian equations and Kahler'stheory. Following Kahler, in this book Cartan changed his original term "ex-terior derivative" to the presently accepted term "exterior differential" and


his original notation cv' of this operation to the presently accepted notationdo.).

The books Geometric theory of partial differential equations [Ra2 ] (1947)by Petr K. Rashevskii (1907-1983) and Cartan's method of exterior formsin differential geometry [Fin] (1948) by Finikov are devoted to the originalexpositions of the Cartan theory.

§4.8. Multiple integrals, integral invariants, and integral geometry

The calculus of exterior forms created by Cartan turned out to be veryuseful in the theory of multiple integrals as well as in the theory of integralinvariants and in integral geometry, which are both connected with the theoryof multiple integrals.

While the simple Riemann integral is invariant under a change of vari-ables, the double integral ffD f (x , y) dx d y , under the change of variablesx = x(u, v), y = y(u, v), will be transformed according to the followingformula:

f{x(u, v), y(u, v))J(u, v)dudv,(4.18) fir f(x, y)dxdy = J lDw

where J u v) = ax /a uax / 8 v is the Jacobian of the functions x =

{ aylau aylavx(u, v), y = y (u, v) with respect to the variables u and v , and D' is thedomain of the variables u and v (the functions x = x(u, v), y = y(u, v)are assumed to be differentiable, and the Jacobian J (u , v) is assumed tobe nonvanishing in the domain D'). Formula (4.18) shows that a doubleintegral is not invariant under a change of variables, i.e., the right-hand sideof this formula cannot be obtained by a simple substitution of the differentialsdx = and d y = (ay/ov)dv intoits left-hand side. The same is true for triple and other multiple integrals.However, the expression of a double integral can be made invariant if wewrite it in the form:

x, y) dx n dy,(4.19) ff f(i.e., use the exterior multiplication dx A dy in its integrand since dx A dy =J(u, v)du A dv. After this, formula (4.18) can be written in the form:

(4.20)

fLf(x, y)dx n dy = ff f{x(u, v), y(u, v))dx(u, v) ndy(u, v).'

S imilarly, for a surface integral d y d z+ Q d z d x+ R d x d y) to befj(Pinvariant under a change of variables, we should write it in the form

(4.21) fj(PdyAdz+QdzAdx+RdxAdy)

§4.8. MULTIPLE INTEGRALS AND INTEGRAL GEOMETRY 137

i.e., in the form of an integral of an exterior form. Integrals (4.19) and (4.21)are particular cases of the integral

(4.22) (J) = a1 1

.i (x)dx" A dx`2 A A dx`PJ!P JP t 2 p

along a p-dimensional submanifold V ° of an n-dimensional manifold X.Expression (4.22) remains invariant under any differentiable transformationof coordinates in the manifold X' . The classical formulas of Green, Gauss,and Stokes are particular cases of the general formula:

(4.23)y

N = dcv ,aVP VP

where a V P is the boundary of the submanifold V°, co is an exterior dif-ferential form of degree p - 1 , and dcv is the exterior differential of theform co which is an exterior form of degree p on the closed manifold theclosure of the manifold Vp. For the Green formula, p = 2, V2 is a planedomain, a V2 is its boundary, the form co is w = Pdx + Qd y , and

dcv= aQ - aP dxAd .(ax ay y

For the Gauss formula, p = 3, V 3 is a domain of a three-dimensional space,(9 V 2 is a surface boundary of this domain, the form co is:

Pdyndz+Qdzndx+Rdxndy,andand

dcv= az + aQ+ aR)dxndyndZ.

For the Stokes formula, p = 2, V2 is a domain on a two-dimensional sur-face, 8 VZ is its boundary, the form w is w = Pd x + Qdy + Rd z, and

dcv = (aQ - ay) dxndy+ (aR - aQ) dyndZ+ (aZ aR) dZndx.

Formula (4.23) is called the generalized Stokes formula.Cartan systematically presented the theory of integral invariants in his

book Lectures on integral invariants [64] (1922) where, applying the methodof exterior forms, he completed the construction of this theory created byPoincare.

Suppose a system of ordinary differential equations

(4.24)dx' , ..dt

= P (x1, x 2 n. )x , t}, i=1,2,... ,n,

is given, where P` (x 1,x2 , ... ) xn , t) are differentiable functions. An in-tegral invariant of this system is an integral fi,P cv along a submanifold V°of dimension p < n on which the parameter t has a constant value, and


this value is not changed when the points of the submanifold Vp move alongintegral curves of system (4.24). An integral invariant is called absolute if theproperty of invariance holds for any domain of integration, and it is calledrelative if this property holds only for closed domains.

Applications of this theory to mechanics are most important. The funda-mental differential equations of mechanics can be written in the form of theHamiltonian equations:

(4.25)dpi aH dq` aHdt - aql' dt ap1'

where q' are the generalized Lagrange coordinates of a system, pi are gener-alized momenta, and H = H(pt , qi , t) is a Hamiltonian function. A relativeintegral invariant of this system is the integral fa Up pi d qi , and its absoluteinvariant is the integral ff vP dpi A d q' . By the generalized Stokes theorem,we have the following relation:

(4.26) Pi dql - dpi A dq'.a vp ffvp

In Cartan's book these and some other integral invariants of mechanicsare investigated in detail. The general theory, which was developed duringthis investigation, was applied to the three-body problem, to light propaga-tion in a homogeneous medium, and to other problems of mechanics andmathematical physics.

As far back as 1896, in his paper The principle of duality and certain mul-tiple integrals in tangential and line spaces [10], Cartan considered multipleintegrals on families of straight lines and planes of the space R3 . These inte-grals are integral invariants relative to the groups of motions of the spaces R2and R3 . Such an invariant for a one-parameter family of straight lines inter-secting a given closed curve is the "perimeter" which is proportional to thecurve length. Cartan also defined an integral invariant for a two-parameterfamily of straight lines in the space (a rectilinear congruence). This integralvanishes if a congruence is normal (i.e., it is a congruence of normals to asurface). By means of this invariant, one can prove very simply the classicaltheorem of Etienne Malus (177 5-1812), which states that a normal congru-ence remains normal after any number of reflections and refractions. ThisCartan paper initiated a branch of geometry which is at present called in-tegral geometry. Before this paper, problems from this branch of geometrywere considered in probability theory. Such problems include, for example,the problem of throwing a disk, a square plate, and a needle, which weresolved by Georges Louis Buffon (1707-1788) in his Essay of moral arith-metic [Buf] (1777), and the "Crofton formulas" found by Morgan WilliamCrofton (1826-1915) in his paper On the theory of local probability [Cro]

§4.9. DIFFERENTIAL FORMS AND THE BETTI NUMBERS 139

(1868 ). Cartan was the first to solve problems of this type as pure geometricproblems.

Integral geometry was significantly developed in the 1930s. The essen-tial role in this development was played by the invariant measure in the Liegroups used by H. Weyl and later by Cartan himself in their research on thetheory of simple Lie groups. This measure allows one to define invariantmeasures in the manifolds of different geometric objects in the spaces whosetransformation groups are the groups indicated above. The term "integral ge-ometry", by analogy with the term "differential geometry", was suggested byBlaschke in his books Integral geometry I. Determination ofdensity _for linearsubspaces in E [Bla4, vol. 2, pp. 219-238] (1935) and Lectures on integralgeometry [Bla5] (1936). Following the book Integral geometry I. Blaschkeand his students and co-workers (Boyan Petkantschin, 0. Varga, Luis An-tonio Santalo (b. 1911), Wu Tayen, Hildegard Rohde, and others) wrotea long series of papers under the general heading Integral geometry [Blal].Altogether there were 33 papers in this series. They were related to integralgeometry in the Euclidean, non-Euclidean, affine, projective, and Hermitianspaces. These and many other investigations in integral geometry were sum-marized by Santalo in his books Introduction to integral geometry [San 1 ](1953) and Integral geometry and geometric probability [San2] (1976). Wenote also Chern's paper On integral geometry in Klein spaces [Chr 1 ] (1942 ),where the author introduced a general method for solving problems of thistype based on integration in Lie groups. New directions in integral geometrywere found by Rashevskii in the paper Polymetric geometry [Ra 1 ] (1941) (pa-pers by Boris V. Lesovoi (1916-1942 ), Measure of area in a two-parameterfamily of curves on a surface [Les] (1948), and I. M. Yaglom, Tangential met-ric in a two-parametric family of curves on a surface'ace [Ya I] (1949), are alsorelated to these directions) and in the book Integral geometry and representa-tions theory [GGV] (1962) by Gel'fand, Mark I. Graev (b. 1922), and NaumYa. Vilenkin (1920-199 1), where a series of problems of integral geometryconnected with the theory of representations of noncompact Lie groups bylinear operators in function spaces was solved (see also the book Groups andgeometric analysis [He12] (1984) by Sigurdur Helgason (b. 1927)).

§4.9. Differential forms and the Betti numbers

In the paper On the integral invariants of certain closed homogeneous spacesand topological properties of these spaces [118] (1929), Cartan consideredintegral invariants that are integrals of exterior invariant forms on compacthomogeneous spaces and that are invariant relative to transformations ofthese homogeneous spaces. He showed how to use these invariants to defineimportant topological invariants of these spaces the so-called Betti numbers.

The term "topology", i.e., the geometric discipline that studies the invari-ants of one-to-one continuous transformations whose inverses are also con-tinuous, came from the term Analysis situs or Geometria si/us. This term was


introduced by Gotfried Wilhelm Leibniz (1646-1716), who, in 1679, in hiswell-known letter to C. Huygens, expressed the idea that in addition to alge-bra, "we need still another analysis which is distinctly geometric or linear andwhich will express the situation (situm) directly as algebra expresses the mag-nitude". Under the influence of this idea of Leibniz, Euler in his "problem onthe seven Konigsberg bridges" used the term "geometry of position" (geome-tria situs) in the sense of what we now call topology. Afterwards the termthe "geometry of position" (geometric de situation, geometric de position,Geometric der Lage) was used in the sense of the theory of chess problemsby A. T. Vandermonde (1735-1796) and in the sense of projective geometryby Lazare Carnot (17 53 -18 23), Theodor Reye (183 8-1919), and von Staudt.Grassmann created a vector calculus in a multidimensional space also underthe influence of this idea. This term in the sense of topology was used byGauss, and Bernhard Riemann (1826-1866) in Theory of Abelian functions[Riel] (1857) gave to this term in the same sense the name Analysis situs-the "analysis of position". This term was used by Poincare for the title ofhis fundamental memoir on combinatorial topology.

The term "topology" appeared in 1847, as the translation of the Latinterm of Leibniz into Greek, in the paper of Gauss's student Johann Bene-dict Listing (1808-1882), Preliminary studies in topology [Lis]. However,this term was accepted only in the 20th century. Originally Cartan used theRiemann and Poincare term and, in spite of the fact that in the titles of hispapers [97] (1927) and [118] (1929) the terms "topology" and "topologicalproperties" appeared, in the title of the book [128] (1930) he again usedthe term "Analysis situs". In Theory of Abelian functions [Rie I], Riemannconsidered multivalent surfaces which represent multivalued functions of acomplex variable and are defined by algebraic equations F (x , y) = 0 con-necting the complex variables x and y. At present these surfaces are calledRiemannian surfaces. He subdivided such surfaces into simply connectedsurfaces (divided into two parts by any cut), doubly connected surfaces (thecuts that do not divide them into two parts make them simply connectedsurfaces), triply connected surfaces (the cuts make them doubly connected),etc., and to each closed two-sided surface he put in correspondence the "or-der of connection" determined by the number of cuts that are necessary tomake the surface simply connected. In the case of closed two-sided surfaces,this number of cuts is always even and if one denotes this number by 2p,then the "order of connection" is equal to 2p + 1 (for a sphere, p = 0,for a torus, p = I, and for a "sphere with p handles", it is equal to p).At present, the number p for Riemannian surfaces defined by the equationF (x , y) = 0 is called the genus of a plane algebraic curve F (x , y) = 0. Fora polyhedron with N. vertices, N1 edges and N2 faces, the number p isconnected with the Euler characteristic X = No - N1 + N2 by the relationX = 2 - 2p. In his Fragments related to Analysis situs, published posthu-mously, Riemann suggested a multidimensional generalization of his "orders

§4.9. DIFFERENTIAL FORMS AND THE BETTI NUMBERS 141

of connection" defined by his friend Enrico Betti (1823-1892) in his paperOn spaces of arbitrary numbers of dimensions [Bet] (1871). Betti introducedthe orders of connection of the mth type, in various dimensions.

The theory outlined by Riemann and Betti was developed by Poincare inhis memoir Analysis situs [Poi4J, which we already mentioned above. In thismemoir Poincare introduced the notion of homeomorphism of manifoldsthat are curves or surfaces in a multidimensional space (actually this spaceis an affine space E") and the Betti numbers of these manifolds coincidingwith the "orders of connection" of Betti. Poincare defined these numbers asfollows. To each p-dimensional manifold VI' , he put in correspondence the(p - I)-dimensional manifold a V" and the boundary of VP , and he calledthe manifold V ° homological to 0 if this manifold itself is the boundaryof a (p + 1)-dimensional manifold VP+ I : V" = a V". If V" = a Vp+ 1 ,

then the boundary of VI) is equal to 0, i.e., a V" _ 0. Distinguishing thepositive and negative orientation of manifolds, Poincare defined multiplica-tion of manifolds by integers where multiplication by -1 means change oforientation. Poincare also defined the sum of manifolds, their linear combi-nations with integer coefficients, and the linear independence of these linearcombinations. If a manifold Vn carries pm - 1 and only pm - 1 linearlyindependent closed m-dimensional manifolds, Poincare said that the "orderof connection" of the manifold Vm relative to the m-dimensional manifoldsis equal to pm . The numbers p1, p2 , ... , p,, _I defined in this way, andcoinciding with the Betti "orders of connection", Poincare called the Bettinumbers. Poincare also defined the commutative groups that are quotientgroups of groups of all closed linear combinations of submanifolds of thegiven manifold (at present they are called cycles) by the subgroup of thisgroup consisting of all linear combinations homological to 0. These groupsare the direct sums of a certain number of free cyclic groups Z (which areisomorphic to the additive group Z of integers) and a few finite cyclic groupsZ1 . The number of free cyclic summands of this group is equal to pm - 1 ,

i.e., one less than the Betti number defined by Poincare (at present, the num-bers pm - I themselves are called "Betti numbers" and are denoted by pm ),and the orders ti of finite cyclic summands Z1 of these groups are calledthe "torsion coefficients". Since these groups are closely connected with theBetti numbers, Poincare called these groups the Betti groups.

In the earlier mentioned paper, On the integral invariants of certain closedhomogeneous spaces and topological properties of these spaces [118] (1929),Cartan, developing Poincare's idea on the importance of integrals of exactdifferentials for topology (which Poincare expressed in his Analysis situs),showed that the Betti numbers of compact topological spaces can be cal-culated as the number of linearly independent integrals of the exact differ-ential forms of order p. In this paper Cartan introduced the polynomialsE,p,t', whose coefficients are the Betti numbers, and suggested callingthem the Poincare polynomials.


The essence of the connection of integrals of differential forms with theBetti numbers established by Cartan was explained by Georges de Rham(1903-1990) in the paper On the Analysis situs of manifolds of n dimension[Rh] (1931), where he defined the so-called de Rham cohomology groups.These groups are the quotient groups of the groups of closed differential formsof order p relative to their subgroups consisting of the exact differentialforms. De Rham also established the isomorphism of these groups and the"Betti groups".

In the memoir Analysis situs Poincare also defined the noncommutativegroup consisting of closed paths on a manifold that are defined up to a con-tinuous transformation of these paths into one another. This group is calledthe connection group of a manifold, or the Poincare group or fundamentalgroup. The study of this group forms the basis of the homotopy theory ofmanifolds.

§4.10. New methods in the theory of partial differential equations

The theory of partial differential equations which, in the beginning of the20th century, was developed in different directions by Vessiot and Cartan,underwent new developments during the last decades due to the synthesis oftheir methods and some new methods of contemporary mathematics. Amongthese new methods, homological algebra should be especially noted. Homo-logical algebra has grown, to a great extent, from Cartan's papers in homologytheory of compact simple Lie groups and symmetric Riemannian spaces. Inthis connection, we first note the following papers of H. L. Goldschmidt:Existence theorems for analytic partial differential equations [Gls 1 ] (1962),Prolongations of linear partial differential equations [Gls2] (1965), Integra-bility criteria for systems of non-linear partial differential equations [Gls3](1969), and On the structure of the Lie equations [Gls4] (1972), the thesis ofDaniel G. Quillen, Formal properties of over-determined systems of linear par-tial differential equations [Qu] (1964); and the paper over-determined systemsof linear partial differential equations [Spe 11 (1965) by Donald C. Spencer (b.1912). For investigation of systems of partial differential equations Spencerand A. K. Kumbera developed a special technique in the papers Deformationof structures of manifolds defined by transitive continuous pseudogroups [Spe2](1962-1965) and Lie equations: general theory [KuSp] (1972).

The theory of "contraction" of Lie algebras and groups and Wigner)mentioned in Chapter 2 was also generalized for Lie pseudogroups by D. S.Rim in the paper Deformation of transitive Lie algebras [Rim] (1966) and byWilliam Stephen Piper (b. 1940) in the paper Algebraic deformation theory[Pip] (1967). We also note the paper The classification of irreducible com-plex algebras of infinite type [GuQS] (1967), by Victor Guillemin (b. 1937),Quillen, and Shlomo Sternberg, where a new simpler proof was given forCartan's theorem on classification of irreducible Lie pseudogroups.

§4.10. NEW METHODS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS 143

The systematic presentations of new methods in the theory of partial dif-ferential equations that are developments of Cartan's methods are given inthe book Exterior differential systems [BCG] (1990) by R. L. Bryant, S. S.Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, and the bookSystems of partial differential equations and Lie pseudogroups [Poml] (1978)by Jean Pommaret (b. 1945) (see also his books Differential Galoistheory [Pom2] (1983) and Lie pseudogroups and mechanics [Pom3] (1988)).

CHAPTER 5

The Method of Moving Framesand Differential Geometry

§5.1. Moving trihedra of Frenet and Darboux

Numerous papers by Cartan and his successors on differential geometry ofclassical spaces are based on the application of the method of moving frames.This method is connected with the theory of finite continuous groups andthe theory of systems of Pfaffian equations in involution, both developed byCartan.

Cartan indicated that he adopted the method of moving frames from Dar-boux, who used it in his classical Lectures on the general theory of surfaces[Dal (1887) under the name of the method of moving trihedrons. In reality,this method was first used by Martin Bartels (1769-1836), a professor of theUniversity of Dorpat (now Tartu in Estonia). He is best known as a teacher ofyoung Gauss and, later, while at the University of Kazan, of Lobachevsky. Toeach point of a space curve, Bartels associated a trihedron, which at presentwe call the "Frenet trihedron", and obtained formulas that are equivalent tothe Frenet formulas. These formulas were published by his student Carl Ed-uard Senff (1810-1849) in the book Principal theorems of the theory of curvesand surfaces [Snf] (1831). He indicated that these formulas were obtainedby Bartels. The moving trihedron related to the rotating globe was also usedby another Bartels student-Petr I. Kotelnikov (the father of A. P. Kotel-nikov mentioned earlier) in Presentation of analytical formulas determiningthe perturbation of the rotational motion of the Earth [KOP] (1832). Later theFrenet formulas appeared in Joseph Serret's (1819-1855) On some formulasrelated to the theory of curves of double curvature [Srt) (1851) and in JeanFrederic Frenet's paper (1816-1900) On certain properties of curves of doublecurvature [Frn] (1852). However, Frenet's thesis, where these formulas weregiven, appeared in 1847. The axes of the Frenet trihedron are directed alongthe tangent to a curve, its principal normal (the straight line that is orthog-onal to the tangent and located in the osculating plane), and the binormal(the perpendicular to the osculating plane of a curve). If we denote the unitvectors parallel to these axes by e1 , e2 , and e3 , the Frenet formulas can bewritten in the form

145

146 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

de de2= -ke+ Ke d ei

(5.1 } 3 ,di=kel, dss

where s is the arc length of the curve, k is its curvature, and K is itstorsion. Note that Frenet found only six formulas equivalent to the first twoformulas of (5.1) and that Serret discovered all nine formulas equivalent toall formulas (5.1).

The Frenet formulas were generalized for an n-dimensional space Rn byCamille Jordan (1838-1922) in his work On the theory of curves in a space ofn dimensions [Jo2] (1874). With every point of a curve in a space Rn , Jordanassociated an n-hedron whose axes are directed along the tangent line to thecurve, the straight line in the osculating 2-plane of the curve orthogonal to thetangent line and, in the same way, the straight line in the osculating (i + 1)-plane of the curve orthogonal to the osculating i-plane, and finally the normalline is orthogonal to the osculating hyperplane of the curve. If we denote theunit vectors parallel to these axes by e1 , e2 , ... , en , the generalized Frenetformulas can be written in the form

(5.2) ds = klel ,de2

= -klel + k2e3 , ...

de,= -kl_lel_1 + klel , .ds

den.. , = -kn_len_1 ,ds

where s is the arc length of the curve and k1 , k2 , ... , kn_ 1 are its 1st, 2nd,... , (n -- 1)th curvatures.

In the theory of surfaces of the space R3 , the moving trihedrons werefirst used by Albert Ribaucour (1845-1893) in his Investigation of elassoidesor surfaces of zero mean curvature [Rib] (1882). (Ribaucour's "elassoides"are now called minimal surfaces; he called the application of the method ofmoving frames to the theory of surfaces the method of "perimorphie". )

The method of moving frames was systematically applied to the theory ofsurfaces by Darboux in his Lectures on the general theory of surfaces [Da].For studying curves on surfaces, Darboux considered trihedra whose vectorse 1 and e3 are parallel to the tangent line to the curve and to the normalline to the surface, and for studying the surfaces themselves, he consideredtrihedra whose vectors e 1 and e2 are parallel to the tangent lines to thecurvature lines of the surface, i.e., parallel to the two principal directions,and the vector e3 is parallel to the normal line to the surface. Darbouxconsidered the derivatives of the vectors of the first frame relative to the arclength of a curve on the surface. These derivatives have the form

(5.3)de1 = de2 de3

-- knee - kge3 ,d

= -knee + lcge3 = keel -- Kge2ds s ds

§5.2. MOVING TETRAHEDRA AND PENTASPHERES OF DEMOULIN 147

where kg is the geodesic curvature of the curve (if kg = 0, the curve isa geodesic line), k is the normal curvature of the surface along the givencurve, and Kg is the geodesic torsion of the curve on the surface.

In the second case Darboux considered the derivatives of the vectors ofthe frame relative to the arc lengths of the curvature lines. The coefficientsof the decompositions of these derivatives with respect to the vectors ofthe trihedron are the principal curvatures of the surface and the geodesiccurvatures and the geodesic torsions of its curvature lines.

§5.2. Moving tetrahedra and pentaspheres of Demoulin

For spaces different from the Euclidean space, the method of movingframes was generalized by the Belgian geometer Demoulin in his papers Onthe application of a moving tetrahedron of referenceerence to the Cayley geometry[Dem1] (1904) and Principles of the anallagmatic and line geometry [Dem2](1905). In the first of these papers, Demoulin considered non-Euclideanspaces with nondegenerate absolutes. Actually he considered only the geom-etry of the elliptic space S3 but indicated that the same theory is applicableto any "-Cayley space", i.e., to any space Sl . With any point of a curve ora surface of the space S3 Demoulin associated a moving tetrahedron whichis an autopolar tetrahedron with respect to the absolute of the space. For acurve, Demoulin placed one of the vertices of the tetrahedron at the pointof the curve and directed the edges of the tetrahedron emanating from thisvertex along the tangent line to the curve, its principal normal and binormal.In the case of a surface, Demoulin also placed one of the vertices of the tetra-hedron at the point of the surface and directed the edges of the tetrahedronemanating from this vertex along the principal directions of the surface andits normal Demoulin considered the derivatives of coordinates of the ver-tices of the moving tetrahedron of the curve relative to the length of the curveand partial derivatives of the vertices of the moving tetrahedron relative tothe lengths of its curvature lines and obtained formulas similar to the Frenetand Darboux formulas.

In the second paper, Demoulin considered the conformal space C3 , andwith every point of a curve or a surface, he associated a moving pentasphere,i.e. a system of five mutually orthogonal spheres defining a system of pen-taspherical coordinates in this space. The Darboux transfer maps these fivespheres onto the vertices of an autopolar simplex of the space Sl whose abso-lute represents the space C3 . Demoulin also considered the manifold of thestraight lines of the projective space P3 , and with each rectilinear generatorof a ruled surface or with each line of a congruence, he associated six linearcomplexes that are pairwise in involution. The Phicker transfer maps thesesix complexes into points of the space SS , and these points are the vertices ofa simplex which is autopolar with respect to the absolute of this space. Thisabsolute represents the manifold of straight lines of the space P . However,3


Demoulin did not notice that one of the spheres of the pentasphere whichhe considered must be imaginary, and actually, instead of the spaces S4 andS3 , he considered the elliptic spaces S4 and S5 .

Also, in the year 1905, Ernest J. Wilczynsky (1876-1932), in the paperGeneral projective theory of space curves [Will, constructed the theory ofcurves of the projective space P3 applying the moving tetrahedron of thisspace, and E. Vessiot, in the paper On minimal curves [Ves), applied themoving trihedron for study of imaginary isotropic curves of the space R3 .

At that time these curves were called "minimal curves". These curves havezero arc length, and, because of this, the usual Frenet formulas are not validfor them.

Finally, in the same year, 1905, Emile Cotton (1872-1950) published thepaper Generalization ofthe theory of movi ng. frame [Cot), where he introducedthe concept of generalization of the method of moving frames for arbitraryspaces that possess transformation groups.

§5.3. Cartan's moving frames

Developing the ideas of Darboux and Cotton, in 1910 Cartan publishedfirst the short note On isotropic developable surfaces and the method of moving.frames [29] and later the paper The structure of continuous groups of transfor-mations and the method of a moving trihedron [311. In the first note Cartanapplied the method of moving trihedrons to the theory of imaginary devel-opable surfaces of the space R3 whose rectilinear generators are isotropicstraight lines. In the second paper he connected the "method of a movingsystem of reference", which later received the name moving frame, with thestructure of Lie groups and the theory of Pfaffian equations.

With every homogeneous space X" where a transformation group G acts,one can associate a family of frames Ra with the property that the groupG acts simply transitively on it, i.e., each pair of frames defines a uniquetransformation S of this group that sends the first frame into the secondone.

For example, in the Euclidean space R" , the systems of orthogonal unitvectors e! , elegy = ale , with the origin at an arbitrary point x of the spacecan be chosen as these frames. Since, in any orthogonal coordinate system,the coordinates of the vectors of such a frame are elements of an orthogonalmatrix belonging to the group o" of dimension n (n - 1) /2 and the originsof these frames are determined by n coordinates, the frames {x, e.} of thespace R" depend on the same number, n(n - 1)/2 + n = n(n + 1)/2, ofreal parameters as the group of motions of the space R" . The frames inthe pseudo-Euclidean spaces R! can be chosen in a similar way, but in thiscase the orthonormality condition for the vectors of a frame has the forme.e1=e151p where ea -1, a= 1,... ,1, 611= 1, u=I+1,... ,n. Inany orthogonal coordinate system, the coordinates of the vectors of such a

§5.3. CARTAN'S MOVING FRAMES 149

frame are elements of a pseudo-orthogonal matrix belonging to the group Onthe frames of the space R, depend on the same number, n(n - 1)/2 + n =n(n + 1)/2, of real parameters as the group of motions of the space Rl .

In the affine space En , one can take the systems of linearly independentvectors ei with the initial point at an arbitrary point x of the space asthe family of moving frames. Since in any affine coordinate system, thecoordinates of the vectors of such a frame depend on n2 parameters, theframes of the space En depend on the same number, n2 + n , of parametersas the group of affine transformations of the space En .

As a model of the projective space Pn Cartan considered the linear spaceLn+1 in which collinear vectors are assumed to be equivalent. To each one-dimensional subspace of the space Ln+ 1 , there corresponds a "geometricpoint" of the space Pn , and Cartan called each vector of this subspace an"analytic point". The operations of addition and multiplication by real num-bers, typical for vectors, are applied to these "analytic points". Cartan em-phasized that the "analytic points" x and Ax determine the same "geometricpoint" x of the space Pn . Because of this, to define a projective frame inthe space Pn , one should take n+ 1 points e! , i = 0 , 1, ... , n. of gen-eral position and a unit point e. The vectors representing these points inthe space Ln+1 are connected by the relation e = E1 ei and are defined upto a common real factor. It follows from this that a projective frame of thespace Pn depends on the same number, n(n + 2), of parameters as the groupSLn+l of unimodular matrices which is locally isomorphic to the group ofprojective transformations of the space Pn.

The non-Euclidean spaces Sn and Sl can be considered to be the pro-jective space Pn where an absolute is given as a nondegenerate quadricQ(x, x) = 0 whose equation does not contain or contains exactly I neg-ative squares, respectively. In these spaces, a frame is formed by points ejthat are vertices of an autopolar simplex with respect to the absolute and nor-malized in such a way that Q(e1, ej) = e ii51j , where EQ = -1 , a < 1, eu =1, u > 1. The orthonormal frames in these spaces depend on the same num-ber, n (n + 1)/2, of parameters as the groups On+1 of orthogonal matricesand 0n+1 of pseudo-orthogonal matrices which are locally isomorphic to thegroups of motions of these spaces.

The conformal space Cn can be represented in the form of an oval quadricQ(x, x) = 0 in the projective space Pn+ 1 . Thus, the group of conformaltransformations of the space Cn coincides with the group of motions ofthe space S'1 and is locally isomorphic to the group of pseudo-orthogonalmatrices On+2 . Such a representation is determined by the Darboux transferwhich maps the points of the space Pn+ 1 that are outside of the quadricQ onto real hyperspheres of the space Cn , the points that are inside of thequadric Q onto imaginary hyperspheres of the space Cn

, and the points ofthe hyperquadric Q itself onto the points of the space Cn.


As a frame in the space S1 +1 , one can take a system of n + 2 pointsthat are vertices of an autopolar simplex with respect to the absolute, butin such a frame one point is always inside the absolute and the remainingpoints are outside of it; in the space Cn , to such a frame there corresponds aframe consisting of n + 2 mutually orthogonal hyperspheres one of which isimaginary. This kind of frame is inconvenient, and this was the reason whyCartan chose, in the space Cn , a conformal frame consisting of two pointse0 and en+1 and n mutually orthogonal real hyperspheres passing throughthese two points. In this frame, the equation of the absolute of the spaceS1 +1 has the form

(5.4) Q(x,x)=(x')2+2x°x"'=O, i=1,... n.i

The conformal transformations of the space Cn are represented by lineartransformations of coordinates preserving equation (5.4).

§5.4. The derivational formulas

The derivational formulas are the formulas that determine the transi-tion from a frame Ra of a given homogeneous space Xn to an infinites-imally close frame Ra+da . To find these formulas, we fix a frame R. anddenote by Sa the transformation mapping the frame Ro onto the frameRa , Ra = Sa Ro . Then, the transition from the frame Ra to the frameRa+da is defined by the transformation Sa+daSa 1 . Since SaSQ 1 = I, thistransformation is in a neighborhood of the identity I of the group of ad-missible transformations of frames. Thus, it can be written in the formSa+daSQ 1 = I + Sw + o (d a) . Cartan called the transformation Sw the in-finitesimal transformation of a frame of the homogeneous space under consid-eration. Using this transformation, the derivational formulas can be writtenas dRa = SwRa. Now we can say that the transformations S. belong to theLie algebra G of the group G of transformations of the homogeneous spaceX n . Denote by cvu , u = 1, ... , r, the coordinates of the transformationSw in the algebra G. These coordinates are invariant forms of the Lie groupG.

In the affine space En the frame Ra consists of a point x and vectorse! , and the frame Ra+da consists of a point x + d x and vectors ei + d e! .Thus, in this space, the derivational formulas can be written as

(5.5) dx= oei, de,=wjej ,

where co' and wj are differential forms, depending on parameters a (thatdetermine the position of the frame) and their differentials da. Since thegroup of transformations of frames of the space En is the (n2 + n) -parametergroup, the f o r m s w` and a , whose number is also equal to n2 + n, arelinearly independent.

§5.4. THE DERIVATIONAL FORMULAS 151

In the spaces Rn and R! , the derivational formulas have the same form(5.5), but now the forms a, are not linearly independent. By differentiat-ing the relations e! ej = alj , we find that in the space R" these forms areconnected by the relations

(5.6)

Similarly, by differentiating the relations elegy = eA , we find that in thespace Rn these forms are connected by the relations

(5.7) = ii

The derivational formulas for the frames Ra = {e!} of the space P' canbe written as

(5.8) de1=coyel, i, j=0, 1, ... , n.

Since now the vectors e . allow multiplication by a common factor, the fam-ily of frames can be reduced by imposing the condition of equality for thevolumes of the parallelepipeds [e0, el , ... , en] constructed on these vectors.From this condition we obtain the relation

(5.9) c00+c01 +...+con =0,

connecting the forms coJ'.. Relation (5.9) distinguishes the unimodular groupSLn+1 in the general linear group GLn+1 .

The derivational formulas in the spaces Sn and S have the form (5.8),areare connected by relations similar to relations (5.6)but now the forms coJ

and (5.7). These relations follow from the fact that the corresponding framesconsist of vertices of simplices that are autopolar with respect to the absoluteof the space.

The derivational formulas in the spaces Cn also have the form (5.8) wherei , j = 0 , 1 , ... , n + 1. However, since in the Cartan frame the equation ofthe absolute has the form (5.4), the forms coj are connected by the relations

(5.10)W0=-(D n+1a wt =-(0 n+1

c)j (D , i,j= 1,... n.o n+1 0 i i j

Suppose further that, in a homogeneous space X" with an r-parametergroup G of motions, there is given a smooth family I of frames dependingon p < r parameters. On this family, the forms co" defining the infinitesimaldisplacements of frames also depend on p parameters and their differentials.Cartan noted that if there are two families and f of frames such thatf = SY where S is a fixed transformation of the group G , the forms couand 'co" defining the infinitesimal displacements of frames in these familiescoincide. Conversely, if two families X and f of frames in a homogeneousspace X" depend on the same number, p < r , of parameters and under


an appropriate bi j ective correspondence between frames of these families wehave 'ctlu = ,u , then these families can be superposed by a transformationof the group G. This theorem is important when one studies submanifoldsof homogeneous spaces by means of the method of moving frames.

§ 5.5. The structure equations

Invariant forms co" of the group G of transformations of a homogeneousspace X" satisfy the structure equations

(5.11) dcvu=cvwc)tAo?, u,v,w= 1,... r,

which are equivalent to equations (2.12). For the groups of transformationsof the classical homogeneous spaces, these structure equations can be ob-tained from derivational formulas (5.5) and (5.8).

Taking exterior differentials of equations (5.5) and equating to zero thecoefficients of the linearly independent vectors e! , we obtain the structureequations of the spaces E", R" , and Rl :

(5.12) dcvco Acvki

`, dw'=w Aw ,! k

where in the space R" the forms cvl satisfy relations (5.6) and in the spacesRl they satisfy relations (5.7).

Similarly, exterior differentiation of equations (5.8) leads to the structureequations of the spaces P" , S", Sl , and C":

(5.13) d cvf = co A cvfi i k

where for the spaces P" , S" , Sl , i , j , k = 0 , 1, ... , n , and for the spaceC" , i , j , k =01 1, ... , n+ 1 , and, in addition, in the space P" the formscvj satisfy relations (5.9), in the spaces S" and Si" they satisfy relations(5.6) and (5.7), and in the space C" they satisfy relations (5.10).

The structure equations of a homogeneous space X" are the conditionsof complete integrability of its derivational formulas. From this follows theimportant theorem which Cartan noted in all his works devoted to the methodof moving frames: Let the forms cv , u = 1, ... , r, be given; supposethat they depend on p , p < r , parameters and their differentials and satisfythe structure equations of a homogeneous space X" ; then they define in thisspace a p-parameter family I of frames uniquely, up to a transformationS of the group G . This theorem is a generalization of the theorem ondetermination of a curve in the space R3 by its curvature and torsion andthe 0. Bonnet theorem on determination of a surface in the space R3 by itsfirst and second fundamental forms. As we will see, the Codazzi and Gaussequations, which the coefficients of these forms must satisfy, follow from the

3structure equations of the space R .

§5.6. APPLICATIONS OF THE METHOD OF MOVING FRAMES 153

§5.6. Applications of the method of moving frames

Cartan applied the method of moving frames to the study of submanifoldsin various homogeneous spaces. We will give the general scheme of investiga-tion of a submanifold VP in a homogeneous space X' indicated by Cartan.With every point x of a submanifold VP there is associated a family Yx offrames subject to only one condition: the point x belongs to all frames ofthe family,,. Such frames are called "frames of order zero". These framesdepend on p principal parameters u 1, ... , up, on which the point x of thesubmanifold VP depends, and on r - n secondary parameters whose num-ber is equal to the difference between the dimension r of the group G andthe dimension n of the space X n

. The whole family of frames of orderzero is a fiber bundle whose base is the submanifold VP and the fibers arethe families Y,,,. The number of secondary parameters can be reduced if onereplaces the frames of order zero by the frames of order one whose elementsare connected in a certain way with the first-order differential neighborhoodof the point x of the submanifold VP . The frames of order one form a fibersubbundle 1( 1) of the fiber bundle Y which has the same base VP . Further,families of frames of orders two, three, etc., are constructed whose elementsare chosen by means of the corresponding differential neighborhood of thepoint x of the submanifold VP . This procedure is called the specializationof frames. There are two possibilities when we follow this procedure.

In the process of specialization we exhaust all the secondary parametersand, for some number k , the family y(k) of frames will depend only onp principal parameters. Such a family of frames is called canonical. Inthis case, all differential forms in the derivational formulas are linear com-binations of the differentials of the principal parameters. The coefficientsof these combinations are invariants defining the submanifold VP up to atransformation of the fundamental group of the space.

The second possibility is that the process of specialization of frames stopsbefore reaching the end, i.e., on a certain step 2;(k+1) _ y(k) but not allsecondary parameters will be exhausted. Then, the submanifold Vp admitsa certain group of transformations into itself.

For instance, for a curve in the Euclidean plane R2 , the family Y offrames of order zero depends on one principal and one secondary parameter-the angle of rotation of the orthonormal pair of vectors e 1 and e2 relativeto a point x of the curve. When we construct the family 11 of frames oforder one, we take the vector el to coincide with the tangent to the curve.This family is canonical since it depends only on the unique principal param-eter. The frames constructed are the Frenet frames for a plane curve. Thefamily of canonical frames for a curve in the space R3 can be constructedin a similar manner. In this case, the canonical frame is determined by thetangent line and the principal normal to the curve, and this canonical frameis a frame of order two. For a curve in the space R" , the Frenet frame is


a frame of order n - 1. The nonvanishing differential forms in the deriva-tional formulas for the Frenet frames have the form co'+1 = - i+ 1 = -kids ,

and the quantities ki form a complete system of invariants defining a curvein the space Rn up to a motion.

The Darboux frames for a hypersurface in the space Rn are also canonicalframes. These frames are formed by the vector en parallel to the normal tothe hypersurface and the vectors e1, e2 , ... , en_ 1 parallel to its principaldirections. These frames are frames of order two.

§5.7. Some geometric examples

Cartan noted that simple geometric considerations do not always lead tothe construction of a canonical frame. In such cases the construction maybe conducted purely analytically by means of the structure equations of thespace. We show how this can be done for an isotropic curve of the spaceCR3. These curves were considered by E. Vessiot in 1905. Cartan consid-ered them in the book The theory of finite continuous groups and differentialgeometry considered by the method of moving frames [157] and in his lectureson The method of moving frames, the theory of finite continuous groups andgeneralized spaces [144] which he delivered in Moscow in 1930.

An isotropic curve x = x(t) in the space CR3 is said to be a curve eachtangent vector x of which is isotropic, i.e., (x')2 = 0. The latter equationimplies x'x = 0 . The arc length of such a curve is equal to zero, and thenormal and tangent planes coincide. Thus, it is impossible to construct theFrenet frame for such a curve. For studying an isotropic curve, Cartan usedthe cyclic frames in the space CR3 whose vectors satisfy the relations

(5.14) e2=e2ele2=e2e =0, e2=e1e = I.3=3 3

Only three out of the nine forms w determining the infinitesimal displace-ments of this frame are independent. Differentiating equations (5.14) andusing equations (5.5), we easily find that they are connected by the relations

(5.15) w1=w=o2=0,3 3 l 3

While constructing a canonical frame, we save one step by immediatelyassociating with the curve the frames of order one. For this, we place theorigin of a frame at the point x of the curve and take its isotropic tangentvector (x)' as the vector el . Since now we have d x = o 1 eI , on the curvethe following equations hold:

(t)2=(t)3=0.

The form cvl is called the basis form. It contains the differential of theparameter t defining the location of a point x on the curve. If we apply

§5.7. SOME GEOMETRIC EXAMPLES 155

exterior differentiation to equations (5.16) with the help of the structure equa-tions (5.11), then, by (5.15) and (5.16), we obtain only one exterior quadraticequation c01 A rvi = 0. This equation implies that

(5.17) w1 = pcOl.

The form cvi is principal since it vanishes when the point x is fixed. More-over, there will be only two nonvanishing independent forms on the curve,namely, the forms to i and to . They determine the admissible transfor-mations of frames of order one. Thus, the family of frames of order onedepends on one principal and two secondary parameters.

For further specialization of frames, we apply exterior differentiation toequation (5.17). This gives

Ldp - 2pw1)

from which it follows that

(5.18)

Aw'=0,

dp - 2prvi = -2gcv1.

If we fix a point x on the curve, then rv 1 = 0, and equation (5.18) takes theform

(5.19) op - 2pii = 0,

where 6 denotes differentiation with respect to secondary parameters and1 1

7r1 = (01(0 '=0

In equation (5.19) we distinguish two cases. If p = 0 for all points of thecurve x = x(t) , then further specialization is impossible, and the family offrames of order two coincides with the family of frames of order one. Sincein this case equation (5.16) implies that a = 0, it follows from equations(5.5) that

dx = coIel, de1 = cve1.

It follows from this that, in the case p = 0, a curve x = x(t) is an isotropicstraight line.

If p 0, equation (5.19) can be written in the form

6lnp -2n = 0.

It is easy to check that dirt = 0 if w1 = 0. Thus, the secondary form nl isa total differential: n i = a In cp . Substituting this value and integrating theprevious equation, we obtain p = C9 2 . Here (p is a secondary parameterwhich determines the magnitude of the vector e 1 . By an appropriate choiceof this parameter, we can reduce the quantity p to + 1 or -1 . Let us take


the first case. In this case, equations (5.17) and (5.18) take the form

(5.20)2 1 1 1w1=w, cv1=qcv.

These equations define the family of frames of order two associated with anisotropic curve.

To construct a family of frames of order three, we take exterior differentialsof the second of equations (5.20). As a result, we obtain the equation

(dq + OJI) A OJI = 0,

from which we find that

(5.21) dq+cv2 = kcv'.

If we fix the point x on the curve, we obtain

aq+irl=0.

Here again the form ire is a total differential: 12 = -a w . This implies

oq=5 , q= yi+C.It follows from this that by an appropriate choice of the secondary parameterw the quantity q can be reduced to 0. Now the second equation in (5.20)and equation (5.21) can be written in the form

(s.22) 0)1 =0,

These equations show that all secondary forms are already expressed in termsof the basis form cv 1 . Therefore, the frame of order three is canonical.

Note that, by previous formulas, d cv1 = 0. Thus, the form cv1 is atotal differential: cvl = d a . The parameter a is called the pseudoarc ofan isotropic curve x = x(t) . It was introduced by E. Vessiot in the paper[Ves] mentioned above. The quantity k in the second equation of (5.22) isan invariant which is called the pseudocurvature of an isotropic curve. Byprevious relations, the Frenet formulas for an isotropic curve have the form

A de1 de2 de3(5.23) = e1, = e2, = ke1 - e3, = ke2,d6 da da da

Two isotropic curves coincide up to a motion of the space CR3 if for bothcurves, the pseudocurvature k is the same function of the pseudoarc a .

The method of moving frames can be applied to the study of manifoldswith any generating element. As an example, in the book [ 157], Cartan con-sidered ruled surfaces of the space R3 . Starting with the family of orthonor-mal frames {x, e 1, e2 , e3} of order zero, where the point x belongs to agenerator 1 of the ruled surface and the vector e 1 is directed along thisgenerator, Cartan arrived at a canonical frame whose origin is located at thecentral point of the generator 1, the vector e3 is directed along the common

§5.7. SOME GEOMETRIC EXAMPLES 157

perpendicular of two infinitesimally close generators, and e2 = e3 x e1 . Thederivational formulas for the family of canonical frames have the form

(5.24)dx

= ae + kde1 de2 -

bde3

= -b, e3 ,da_= e3,dQ da

e2, -e2+ ee.dQ

The last three of formulas (5.24) are the Darboux formulas for a curve onthe sphere described by the terminal point of the vector e1 . This curve isthe spherical image of the ruled surface under consideration. The parametera coincides with the arc length of this spherical image since dQ = ide1 kand the invariant b represents its geodesic curvature. The invariant k is thedistribution parameter of the ruled surface which is equal to the limit of theratio of the shortest distance between its two rectilinear generators and theangle between them when one of these generators approaches another. Thisinvariant is determined by the first-order differential neighborhood of thegenerator l , and the invariants a and b are determined by its second-orderdifferential neighborhood. If three arbitrary functions k = k (a) , a = a(a)and b = b(o) are given, then there exists a unique ruled surface for whichthese functions are the corresponding invariants. If k = 0 , a ruled surfaceis developable.

Cartan also showed how the method of construction of a canonical movingframe which he developed can be applied to nonmetric geometries. In hisbook [ 144], he considered the theory of plane curves in affine geometry, andin the book [157], he considered the theory of plane curves in projectivegeometry. In both cases, starting with the family of frames of order zero, heconstructed the family of canonical frames, found the derivational formulasfor this family, and gave the geometric characterization to the invariantsin these formulas. In addition, he found some special cases for which theconstruction of the canonical frame is impossible.

Thus, the Cartan books [ 144] and [ 157] contain not only the general the-ory of the method of moving frames but also its applications to a series ofconcrete geometric problems.

Among similar problems which Cartan solved in his other papers, we notethe analogue of the Frenet formulas which Cartan obtained in his paper Ona degeneracy of Euclidean geometry [147a] (1935) for the isotropic planeI2 In this paper, with every point x of a curve in the plane I2 , Cartanassociated the frame consisting of the unit tangent vector e1 and the unitvector e2 which is parallel to the isotropic straight lines of this plane andwrote the analogue of the Frenet formulas in the form

(5.25) ds = e, , dkel+e2 dS2 = 0.

15 8 5. THE METHOD OF MOVING FRAMES AND DIFFERENTIAL GEOMETRY

§5.8. Multidimensional manifolds in Euclidean space

In the book Riemannian geometry in an orthonormal frame [108a), Cartanconsidered certain special topics of the theory of p-dimensional manifoldsV" in n-dimensional Riemannian manifolds Vn and, in particular, in theEuclidean space R". We consider in more detail how Cartan constructedthis theory.

With every point x of a p-dimensional manifold V' of the space Rn , weassociate a family of orthonormal frames whose vectors e! , i = 1, ... , p ,are located in the space Tx (V") tangent to the manifold V" , and the vectorsea , a = p + 1 , ... , n , belonging to its normal space Nx (V") . Then, themanifold V" is determined by the following system of Pfaffian equations:

(5.26) toa = 0,

and the forms co' are linearly independent on the manifold V' . The squareof the linear element of the manifold V' has the form ds2 = >11(a)')2. Thisform is called the first fundamental form of the manifold VP.

Exterior differentiation of equations (5.26) by means of the structure equa-tions (5.12) leads to the equations

(s.27) dcva=a)tA(= 0.

Applying Cartan's lemma to this equation, we find that

(5.28) a = b 1a) 30

bcc = baii )!.

The coefficients b are the coordinates of the vectors b . = b a ea .

From derivational formulas (5.5) it follows that on the manifold V" wehave

(5.29) dx = wrei , d2x = (dco + cv'cv`.)e! + cv`cvaea.t

Therefore, the quadratic forms

(5.30) a=cc)ta0 a=b1C01 Cc)J

define the deviation of the manifold V" from its tangent space TX( VP).

The vector-valued quadratic form V = (paea = bi .co'd is called the secondfundamental form of the manifold P.

Consider a curve x = x(s) on the manifold V" given by a vectorial func-tion of its arc length s. The vector = a = a`e! is its unit tangent vector.Since a` = rv` Ids, with the help of (5.30), we obtain for this curve

(5.31)d2x da da` `

ds2 ds dsa

ds) ei + tj

§5.8. MULTIDIMENSIONAL MANIFOLDS IN EUCLIDEAN SPACE 159

The vector d2x/ds2 is the vector of curvature of the curve x = x(s) . For-mula (5.31) gives its decomposition into the tangent and normal components.Because of this the vector of normal curvature of this curve has the form

(s.32) kn = bl.Ia of .

It follows from this that the vector kn depends only on the direction of thetangent line to the curve x = x(s) .

The linear span of the set of vectors kn coincides with the linear spanof the system of normal vectors bil and determines the principal normalNx (V") of the manifold VP . Its dimension is equal to p1 <min{n - p, p(p + 1)/2}. The direct sum of the principal normal and thetangent space Tx (V") is the first osculating space Tx (V") of the manifoldVP at the point x. Its dimension is dim TX (VP) = p + p 1 .

If we change the tangent vector a in the tangent space Tx (Vp) and p 1 > p,the terminal point of the vector kn describes a (p - 1)-dimensional algebraicsurface in the principal normal Nx (V"). This surface is called the indicatrixof curvature. If p1 < p, the terminal point of this vector describes a closeddomain in NN (V") which is called the domain of curvature.

As an example, following Cartan, we consider a two-dimensional surfaceV 2 in the space R n

. In this case, p 1 < 3, a = e1cos 0 + e2 sin 0, and

formula (5.32) takes the form

kn = b11 cost 0 + 2b12 cos 0 sin 0 + b22 sine 0(5.33)

=1(b +b22)+b12sin28+

2(b 11-b22)cos20.

We can see from this that if p1 = 2 or 3, the terminal point of the vectorkn describes an ellipse in the normal Nx (V 2) with the center determined bythe vector

2(b11 + b22) and the vectors b 12 and

2(b11 - b22) parallel to its

conjugate diameters. Cartan called this ellipse the ellipse of curvature of thesurface V2 . If p1 = 1, then the terminal point of the vector kn describes asegment in the one-dimensional normal NN (V 2) which is called the segmentof curvature. Its ends correspond to the principal directions of the surface

2V .

A p-dimensional manifold VP of the space Rn depends on n - p arbi-trary functions of p real variables. As these functions, in the general caseone can take the functions expressing the coordinates x`' of a point x ofthis manifold in terms of the coordinates x` taken as independent variables.In the general case the indicatrices of curvature of p-dimensional manifoldsare the Veronesians and the quasi- Veronesians.

We will now illustrate the application of the Cartan test by investigatingthe system of Pfaffian equations (5.26) which determines a manifold Vpin the space Rn . The character s1 of this system is equal to the numberof linearly independent exterior quadratic equations (5.27) obtained as a


result of exterior differentiation of system (5.26); i.e., it is equal to n - p :sl = n - p . The ranks r, of the system of linear equations determining the(i + 1)th integral elements are rl = i(n - p) . Thus, the remaining charactersSi = r! - r!_ 1 are also equal to n - p . The sum of the characters sl + s2 +

+ sp = q is equal to the number p(n - p) of independent forms co'. TheCartan number

Q +psp {= 1 +2+ +p)(n-p) =p(P +1)(n-p)2

coincides with the number N of independent coefficients b'. Since Q = N,by Cartan's test, system (5.26) is in involution, and its solution dependson sp = p(n - p) functions of p real variables. This corresponds to thearbitrariness of the existence of a manifold VP in the space Rn which weindicated above.

§5.9. Minimal manifolds

In the book Riemannian geometry in an orthonormal frame [108a) Car-tan also considered minimal surfaces V 2 in the Euclidean space R4 . Thecondition for V2 to be minimal is that the variation of surface area be equalto zero. This condition has the form J fw' A cv2 = 0. It follows from thiscondition that the vector

2(b11 + b22) = 0, i.e., the center of the ellipse of

curvature of a minimal surface coincides with its point x.In the case of a manifold VP in the space Rn , Enrico Bompiani (1889-

1975), one of the founders of multidimensional differential geometry, calledthe vector b = Ej b11 the vector of mean curvature. If p > 2, the equationb = 0 also characterizes the minimal manifolds.

However, the minimal surfaces V2 of the space R4 are also remarkableby the fact that they carry a complex structure (this is not true for minimalmanifolds VP for p > 2). Namely, each minimal surface V 2 in the spaceR4 can be viewed as a real interpretation of an analytical curve y in a two-dimensional complex plane endowed with the metric of a complex Hermitianplane CR2 where the length of the vector z = {z', z2) is equal to I z I _

Z 2.Generalizing this property of minimal surfaces V2 in the space R4 , we

can arrive at the notion of strongly minimal manifolds Vp in the Euclideanspace R2n . For this, we consider the complex Hermitian space CR11 andconstruct its real interpretation R2n . We take the vectors of a unitary or-thonormal frame of the space CR" as the vectors e2k_

1of a frame in the

space R2n , and as the vectors e2k we take the products of the vectors ofthe same frame in the space CR" by i. Then, in addition to the conditionscvk = -cv1 5 the differential forms cvk in the equations of the infinitesimal

§5.9. MINIMAL MANIFOLDS 161

displacements of the frame in the space R2" are also connected by the con-ditions:

(5.34) (0u-, u Zr-, u2k- I 2k 2k _(02k- I k,1=1,2,... , n.

Note that the number of relations (5.34 ), whose form remind us the Cauchy-Riemann conditions (4.4 ), is equal to the difference n (2n + 1) - n (n + 2) ofthe dimensions of the group of motions of the spaces R2" and CR".

Next, we consider a (2p )-dimensional manifold V 2p of the space R2"representing an analytic manifold C V" of complex dimension p in the spaceCR" . With every point of this manifold, we associate an orthonormal frameconsisting of vectors e2i_ 1 , e21, i = 1 , 2) ... , p , lying in the tangent spaceTX ( V

2p), and vectors e2,_ 1 , e2a , a = p + 1, ... , n , lying in the normal

space NX ( V 2p) . Then, the manifold V 2P is defined by the following systemof Pfaffian equations:

(5.35)2cx-1 = cv2a = 0.

Exterior differentiation of equations (5.3 5) leads to the following exteriorquadratic equations:

(5.36)21-1 2a-1 2i 2a-1 2i-1 2a 2i 2a

A w2i_ 1 + A w2, = 01 w A (02i_ 1 + a) A (021 = 0.

Since the forms c)21-1 and a 2' are linearly independent on the manifoldV2p , application of Cartan's lemma gives

cv2°`=11 = b2°` 112 1 c02j- i + b2 112 cv2'

2tr j_ j

2a-1 = b2a- 1(0

2j-1 + b2°`-1(0 2i

21 21,2j-1 21,2j2a 2a 2j-1 2a 2j

2i-1 = b2i-1 , 2j-1 + b2i-1, 2j2a _ b2cx 2 j - 1 b2cx 2j21

2i , 2j-1+

21, 2i

where bli = bit. By conditions (5.34 ), we also find the following relationsbetween the coefficients b :

(5.38)b2

12 = b2J i 21-1 = b21 2 _ 1 = b2 1 21 = -b2J _ 2 211j- 1 l- l l j> 2i lb2z 2 =b 2°` =

b2a

i- 1 = b2z 112 = -b2 1 21-1 = -b21 1 2j- 1j 2j, 21 2j, 2 j jOne consequence of these relations is that

(5.39) (bi,21_1 +b) = J>22cr(i,2i_i + b2cr21) ,

i.e., the vector of mean curvature of a strongly minimal manifold is equalto zero, and this manifold is minimal in the usual sense. A p-dimensional


analytic manifold C VP in the space CA" is defined by n - p analytic func-tions of p complex variables. But to define each of these functions, it issufficient to take two functions of p real variables. Thus, a strongly minimalmanifold Yep in the Euclidean space R2n depends on 2(n - p) functions ofp real variables. This result can also be obtained by applying the Cartan testto the system of Pfaffian equations (5.35). The character si of this systemis equal to the number of linearly independent exterior quadratic equations(5.36), i.e., s1 = 2(n - p) . The ranks ri of the systems of linear equationsthat determine the (i + 1)-dimensional integral elements are r, = 2 i (n - p) .

This implies that the remaining characters s, = r1 - r._ i are also equal to2(n - p) : s, = 2(n - p). The sum of characters si + s2 + + sp = q isequal to the number 2p (n - p) of independent forms 2a_ 11 and cv2a

1

The Cartan number

Q=s1+2s2+. .+psp=p(p+l)(n-p)

coincides with the number N of independent coefficients bl . Since Q = N,system (5.35) is in involution and its solution depends on sp = 2(n - p)functions of p real variables.

§5.10, "Isotropic surfaces"

The bulk of Cartan's unpublished paper Isotropic surfaces of a hyperquadricin seven-dimensional space [ 1771 is devoted to the differential geometry of"isotropic surfaces", i.e., two-dimensional surfaces on the absolute of the hy-perbolic space Sa which can be considered as the pseudoconformal spaceC3 C. Cartan assumed that all tangent two-dimensional planes of these sur-faces are plane generators of the absolute of the space SS , and hence througheach of its two-dimensional planes there passes one three-dimensional planegenerator of this absolute of the first and second family. While in the con-formal space C' Cartan used a frame consisting only of two points of theconformal space and a few mutually orthogonal hyperspheres, in the spaceC3 he used a frame consisting of points represented by such points of theabsolute that the straight lines joining pairs of these points are mutually polarwith respect to the absolute. If the equation of the absolute is

(5.40) (x) = x°x7 + xlxb + x2x5 + x3x4 = 0,

Cartan takes as the points of the frame analytic points A. , A i , ... , A7 forwhich the quadratic form (5.40) vanishes: n (A) = 0 , i = 0 , 1) ... , 7 , andthe bilinear form SZ(A1, A1) obtained by the polarization of the quadraticform (5.40) is equal to 1 if i + j = 7 and 0 in all other cases. This meansthat the straight lines AA_i are mutually polar with respect to the absolute.Note that in the manuscript of this paper, Cartan denoted analytic points notonly by capital Latin letters, as he did in most of his works, but also by smallLatin letters with arrows -b over them; moreover, he also called the forms

§5.10. "ISOTROPIC SURFACES" 163

fl(x) and (x, y) the "inner square" of an analytic point and the "innerproduct" of two analytic points; i.e., in fact, he considered analytic pointsas vectors of the pseudo-Euclidean space R8 . The derivational formulas ofthis frame have the form (5.8), and the structure equations have the form(5.13). Differentiating the equations AA = 1 , i + j = 7, Cartan obtainedthe relations

(5.41) 7'+(v7-t=0,(c)! i

which show that the matrix (o) is skew-symmetric with respect to the sec-ondary diagonal, and in particular, all the entries of this diagonal are equalto 0.

With every point of an isotropic surface, Cartan associated the frameswhose point A0 coincides with this point of the surface, the points A 1 andA2 belong to the tangent isotropic plane to the surface at this point, the pointA3 lies in that "generating space" (three-dimensional plane generator) of thefirst family which passes through the tangent isotropic plane, the point A4lies in the "generating space" of the second family, the points A5 and A6 lieoutside of the "generating spaces" mentioned above, in the hyperplane whichis tangent to the absolute at the point A0 , and the point A7 lies outside ofthe latter hyperplane. Thus, the Pfaffian equations of the isotropic surfacehave the form

(5.42)3=(04 5 6

(00= 0=(o0_ (000

(the analogous equation coo = 0 is a consequence of relations (5.41)). By ex-terior differentiation of equations (5.42) Cartan found the following exteriorquadratic equations:

3(5.43) da)o=cooncoi+cooA co2=0,

(5.44) dcoo=cooA 4Aco2=0,

(5.45) dwo=cooA coi=0,

(5.46) dwo=wonw6=0.

Since equations (5.41) imply that coi = co2 , it follows from (5.45) and (5.46)that

(5.47) w1=(v2=0.

Exterior differentiation of equations (5.47) leads to the relations:

d (v5 = w 3 A w5 + w A w5 = o , d (v2 = (v2 n cv3 + (v2 A (t) 04

I 1 I 3 1 I 4


which, by (5.41), are reduced to one relation:

Thus, the closed system of differential equations defining an isotropic sur-face V2 of the space C3 consists of Pfaffian equations (5.42) and (5.47) andexterior quadratic equations (5.43), (5.44), and (5.48). Investigation of thissystem by Cartan's test shows that since new forms cvi , cv2 , cv4 , and cv2enter into the quadratic equations, i.e., their number q = 4, and the numberof independent exterior quadratic equations is s, = 3, the second Cartancharacter is s2 = q - s, = 1 and the Cartan number is Q = s, + 2s2 = 5.

Applying Cartan's lemma to relations (5.42) and (5.43), Cartan obtainedthe equations:

(5.49)(v3 = acvl + b()2 , cvi = bcvl + ccv2

4O02 4 '1 0,

2cv1 =a(v 1 cv2=bcv0+ccv0.

Substituting expansions (5.49) into equation (5.48), he found the relation:

(5.50) acs + cap - 2bb' = 0.

Therefore, the degree of freedom of the most general integral element ofan isotropic surface V2 is equal to N = 6 - 1 = 5, and the system of Pfaffianequations (5.41) and (5.47) defining this surface is in involution, and, sinces2 = 1 , an isotropic surface V2 in the space C3 depends on one functionof two real variables.

Since the differential of the analytic point A0 tangent to the surface V2is equal to dA0 = (voA0 + (voA1 + (voA2 , the second differential of this pointmodulo the tangent plane to the surface V2 is equal to d 2A0 = (vo(O A , , i =1, 2, a = 3 , 4 , ... , 7. But by equations (5.42) and (5.47), modulo thesame tangent plane, we have

(5.51) d2 A0 _" cv0cvIA3 + U0U)1A4.

It follows from this that the osculating plane TT (V 2) of the isotropic surface

V2 is determined by the points A0 , Al , A2 , A3 , A4 , and thus it is four-dimensional. (The osculating plane of the general surface V2 in the space ofdimension exceeding five is five-dimensional.) This plane coincides with thefour-dimensional plane which is determined by those two "generating spaces"of the quadric that pass through the isotropic tangent plane to the isotropicsurface V2 . This plane is a polar plane of the tangent plane relative to thequadric. The coefficients of the points A3 and A4 in expression (5.51) are thesecond fundamental forms & = (00(v3 and (D4 = (vo(va of the surface V2 .

Substituting relations (5.49) into these expressions, we obtain the following

§S.IO. "ISOTROPIC SURFACES"

form for the second fundamental forms of the surface VZ

(5.52)

(D 3 =a(coo)2 + 2bcoocoo +

c(coo)2

165

(D4 = a (c0o)Z + 2b'coomo + c

Cartan called the nets (D3 = 0 and D4 = 0 the nets (I) and (II). Next, heconsidered the general case when, at any point of the surface, the equationsc 3 = 0 and (D4 = 0 do not have common roots (the pairs of points definedby these equations on the line A1A2 have no common point) and found onthe surface V2 a net (III) which is "harmonic for the nets (I) and (II)", i.e.,a net of lines on this surface such that if the points A 1 and A2 belong tothe tangents to the lines of this net then the forms 3 and (p4 can both besimultaneously reduced to algebraic sums of squares. In this case b = b' _

0, a = c = a _ -c = I. At present, the net (III) is called the conjugate netof the surface V2 . In this case

(5.53) w, = coo, W3 = wo, 4 1 (02 _(02W1

=oa 2

=0),

and the forms (5.52) become

(5.54) 03 = (1)2k(we)z, cp4 = (we) 2 - (co0)2

The tangents to each family of lines of the net (III) form a two-parameterfamily of straight lines. For the general two-parameter family of straight linesin the spaces PN , N > 5, the limiting positions of three-dimensional planes,passing through a straight line of the family and infinitesimally close straightlines tending to it, do not coincide but belong to a certain five-dimensionalplane (called the tangent plane of the family). Unlike the general case, in thecase of the two families of tangents considered above, these three-dimensionalplanes coincide. Therefore, the neighborhoods of lines of these families havestructure similar to that of straight lines of the congruences of straight linesof the space P3 , and each of these straight lines have two foci the pointsof intersection of this line with infinitesimally close straight lines. The latterinfinitesimally close lines define two developable ruled surfaces of this familypassing through each of its lines. Such two-parametric families of straightlines are called focal families. One of the foci of each of these straight linesis the point A. . We place the points A 1 and A2 into the second focus ofthe tangents from the first and second family, respectively, and choose thepoints A3 and A4 in such a way that the points AO) A1, A2 , A3 + A4 andAo, A, , A2 , A3-A4 define three-dimensional tangent planes of the straightlines of the first and second family, respectively. The foci A 1 and A2 gener-ate two-dimensional surfaces 'V 2 and " V 2 obtained from the surface V2 byLaplace transforms. These surfaces also belong to the absolute of the spaceS4 , but they are not isotropic. Thus, they can be considered as nonisotropic


surfaces of the space C3 . Cartan introduced a system of curvilinear coor-dinates u and v on the surface V2, whose coordinate lines are the curvesof the net (III) and whose differentials are the forms cvo and coo , and con-structed a canonical frame.

Cartan also considered special classes of isotropic surfaces. In the casewhen the foci A 1 and A2 generate not surfaces but lines, Cartan expressedall points Ai in terms of analytic points U and V depending on the variableu alone and the variable v alone, respectively. These expressions of thepoints of the canonical frame are:

(5.55)

A0 = U+V, Al = UA2 = VA3+A4 = U", A3-A4 = VA = 1U"

A6_1Vill A7 _ 1UIV=_1VIV.s_ 2 ^ 2 - 2

Completing the construction of the canonical frame, Cartan found thecomplete system of invariants defining an isotropic surface V2 on the abso-lute of the space S4 up to a motion of this space, i.e., a surface V2 of thespace C3 up to a conformal transformation of this space.

§ 5.11. Deformation and projective theory of multidimensional manifolds

While in his books Cartan considered problems of differential geometryusing mainly examples of curves and surfaces in spaces of lower dimensions,in his theoretical papers on differential geometry he studied problems of thetheory of multidimensional manifolds in the spaces R" , S" , P" , and C" .

The first paper of this type was his paper The deformation of hypersurfacesin the real Euclidean space of n dimensions [47] (1916). By the well-knowntheorem of Richard Beez (1827-1902), in general, the hypersurfaces in thespace R" , n > 3 , are not deformable; i.e., if n > 3, any pair of hypersurfacesof the general type can be superposed by means of a motion. In the paperindicated above, Cartan investigated such hypersurfaces in the space R" thatadmit a nontrivial deformation, i.e. such a deformation that leaves themisometric, but such that they cannot be superposed by means of a motion.After solving this problem in the space R" , in the paper The deformationof hypersurfaces in the real conformal space of n > 5 dimensions [48] (1917 ),Cartan considered a similar problem in the conformal space C" . Later,applying the notion of projective deformation of surfaces introduced in 1916by Fubini, in the paper On the projective deformation of surfaces [54] (1920),Cartan solved the problem of projective deformation. In the same year in hislecture On the general problem of deformation [55], Cartan also defined theprojective deformation of the congruences and complexes of straight lines inthe projective space P3 and noted that, with the help of Pliicker transfer,this problem can be reduced to the problem of conformal deformation ofsurfaces of dimensions two and three in the space C4 . In 1919-1920 Cartanreturned to the problem of deformation of manifolds in the space R" ; in the

§5.11. PROJECTIVE THEORY OF MULTIDIMENSIONAL MANIFOLDS 167

paper On the manifolds of constant curvature of Euclidean and non-Euclideanspace [51], [52], he solved similar problems in n-dimensional non-Euclideanspaces. In all these papers Cartan systematically used the method of movingframes in the spaces R" , P" , S" , S! , and C" .

During the study of deformation of manifolds in n-dimensional Euclideanand non-Euclidean spaces, it was detected that in this theory the projectiveproperties of manifolds, i.e., properties that are invariant under projectivetransformations of the space, play an esssential role. This was one reasonCartan studied the geometry of a manifold VP in the projective space P" inChapter 4 "Manifolds of p dimensions in the projective space of n dimen-sions. Osculating planes. Asymptotic linear systems" of his two-part paper[51], [52].

With a manifold VP he associated a moving point frame whose pointe0 coincides with a varying point x of the manifold VP , whose pointse! , i = 1, ... , p , are in its tangent space Tx (Vp) and whose points a , a =p + 1, ... , n, are outside of this space. If we denote the Pfaffian forms wk

in derivational formulas (5.8) by wk , then the Pfaffian equations definingthe manifold VP in P" will have the same form (5.26) as in the space R" ,their exterior differentials will have the form (5.27), and the application ofCartan's lemma will again give relations (5.28). Cartan called the quadraticforms (5.30) asymptotic forms of the manifold VP . The linear system ofthese forms is projectively invariant and does not depend on the metric prop-erties of the manifold VP even if the latter belongs to the space R" . Simi-larly the first osculating space Tx (V") defined for the manifolds Vp in theEuclidean space R" has also an invariant meaning. If one places the pointse! , i 1 = p + 1, ... , p + p 1 , into this osculating space, then the asymptotic

forms "+p' become linearly independent, and the forms 0A , A > p + p1 ,

are identically equal to zero. After this Cartan defined the asymptotic andconjugate directions on the manifold VP .

In the same manner, Cartan introduced the osculating spaces TX (V") , k >1, and linear systems of asymptotic differential forms of higher orders andestablished relations among them. Then he posed the problem of projectiveclassification of multidimensional manifolds according to the structure oftheir linear systems of asymptotic differential forms of certain order.

As the first example, Cartan considered the manifolds on which the asymp-totic forms of first order can be reduced to the form

cP+r = (a)1)2 , I 0 = 0, A > 2p.

At present, these manifolds are called Cartan manifolds. They carry a con-jugate net, and their osculating spaces T, ,(V-) have dimension 2p. Cartaninvestigated the system of Pfaffian equations defining such manifolds andproved that they exist and depend on s2 = p (p - 1) functions of two vari-ables.


Next, Cartan considered tangentially degenerate manifolds VP whose tan-gent spaces Tx (Vp) depend on q < p parameters. The number q iscalled the rank of the tangentially degenerate manifold VP. The asymp-totic quadratic forms of such a manifold can be expressed in terms of qlinearly independent forms cr)a and have the form

ab

Later on, it was proved that a tangentially degenerate manifold VP isfoliated into a q-parameter family of (p - q)-dimensional planes along eachof which the tangent space T, (V-) is the same.

Cartan considered two classes of tangentially degenerate manifolds. Forthe first class, the asymptotic forms can be reduced to the form

op+a

= (01")2, a=1,...,q; 4A=0, 2>p+q.

If q > 1, such manifolds depend on s2 = q (q - 1) functions of two variables.If q = 1, they are envelopes of a one-parametric family of p-dimensionalplanes and depend on sl = n - 1 functions of one variable. The second classis characterized by the fact that, among the forms 4)', there is the maximalpossible number q(q + 1)/2 of linearly independent forms. In this case, ifq > 1, the manifold VP is a cone with a (p - q - 1)-dimensional vertex and(p - q)-dimensional plane generators.

These results of Cartan were further developed by many geometers. Akivisin the paper on multidimensional surfaces carrying a net of conjugate lines[Ak3] (1961) and Vyacheslav T. Bazylev (1919-1989) in the paper on a classof multidimensional surfaces [Baz] (1961) considered the manifolds Vp inthe space pn for which all asymptotic quadratic forms can be reduced tosums of squares. As the Cartan manifolds discussed above, such manifoldscarry a conjugate net, but their osculating spaces Tx (V") have dimensionnot exceeding 2p. For these manifolds, conditions of holonomicity of theirconjugate net were found (for the Cartan manifolds it is always holonomic)as well as conditions under which the manifold Vp belongs to its osculatingspace TX (Vp) .

Manifolds Vp c pn that carry a conjugate system with multidimensionalcomponents were considered by Valery V. Ryzhkov (b. 1920) in the pa-per Conjugate systems on multidimensional surfaces [Ryl] (1958) and byAkivis in the paper on the structure of two-component conjugate systems [Ak6](1966).

Tangentially degenerate manifolds were studied in detail by Akivis in thepapers Focal images of a surface of rank r [Ak2] (1957), and on a classof tangentially degenerate manifolds [Ak5] (1962), by S. I Savelyev in thepaper A surface with plane generators along which the tangent plane is fixed[Say] (1957) and by Ryzhkov in the paper On tangentially degenerate surfaces[Ry2] (1960). In particular, Akivis studied the structure of focal images of

§5.11. PROJECTIVE THEORY OF MULTIDIMENSIONAL MANIFOLDS 169

a (p - q)-dimensional plane generator of a tangentially degenerate mani-fold of dimension p and rank q. Further, in the paper Multidimensionalstrongly parabolic surfaces [Ak9] (1987), he showed that the structure of reg-ular strongly parabolic manifolds in Euclidean and non-Euclidean spaces isconnected with focal properties of the tangentially degenerate manifolds.

In the paper on n-dimensional surfaces with asymptotic fields of p-direc-tions [Lu] (1959), Ulo G. Lumiste (b. 1929) showed that, in the general case,such surfaces possess an (n - p)-parameter family of p-dimensional planegenerators. In the same paper, he considered manifolds with a complete sys-tem of asymptotic directions. He evaluated the dimension of their osculatingspace and described their structure.

Phillip A. G rifths (b. 1938) and Joseph Harris (b. 1951) devoted theirpaper Algebraic geometry and local differential geometry [GrH] (1979) to thestudy of the projective structure of multidimensional manifolds. In this pa-per, they applied methods of algebraic geometry for studying linear systemsof asymptotic differential forms of the manifold Vp introduced by Cartan.The main goal of the paper wasthe study of manifolds whose projective struc-ture is not general. Griffiths and Harris again studied tangentially degeneratemanifolds (they called them manifolds with degenerate Gauss mappings), andnext they studied manifolds with degenerate dual varieties, manifolds withdegenerate Chern forms, manifolds with degenerate secant varieties, etc. Lin-ear systems of asymptotic differential forms of such manifolds have a specialstructure and their algebraic-geometric analysis allowed Griffiths and Harristo study not only local but also global structure of such manifolds.

Finally, in 1988 Akivis and Polovtseva created a new procedure in theproblem of projective classification of multidimensional manifolds (see theabstract of Akivis's lecture on projective differential geometry of submani-folds [Ak10] (1988)). Let PNX(V") = TX(V")/Tx(V") be a "projective nor-mal" of the manifold VP which is a projective space of dimension p 1 - I,and let PTx (Vp) be the projectivization of its tangent space. Consider themapping b : PTx (Vp) -+ P NX (Vp) defined by the formula y" = b a. x` xJ .

JThis mapping can be represented as a superposition b = flo v where vPTx (Vp) Pm , m = p(p + 1) /2 - 1 is the Veronese mapping (3.11), and

Pm PNX (VP ) is the linear mapping ya = ba.xlJ . Arbitrary projectiveJ

transformations of the space PTx (V") induce projective transformations ofthe space Pm that preserve the Veronesian Wl , defined by parametric equa-tions (3.11), and also the algebraic manifolds Wk defined by the equationsrank(xi) = k, k = 2, ... , p - 1 (if k = 1, we obtain again the manifoldW,). These manifolds form a filtration f : Wl C W2 C C Wp_ 1 C Pm .The kernel K = ker /3 of the mapping /3 which is a subspace of the spacePM of dimension m - p1 - I. To the points of intersection K n w, therecorrespond the asymptotic directions on Vp, and to the points of intersec-tion K n W2 there correspond the pair of conjugate directions on VP . The


projective structure of the linear systems of the asymptotic quadratic formsand, consequently, of the manifold VP , are determined by the location of thekernel K relative to the filtration f in Pm. A similar construction can becarried out for osculating spaces and linear systems of asymptotic differentialforms of higher orders.

Polovtseva in her dissertation Projective differential geometry of three-dimensional manifolds [Poll (1988) applied this method to the study andclassification of three-dimensional manifolds V3 in Pn . For V3 , the di-mension of the osculating space TX (VP) is equal to 3 + p1 where p1 < 6.Since the cases p 1 = 1, 2 were studied in detail earlier, Polovtseva consid-ered in her dissertation only the cases p1 = 3, 4, 5, 6. For p1 = 6, thelinear systems of the asymptotic quadratic forms of all manifolds V 3 be-long to one class. For this reason, in this case the projective classification ofmanifolds V 3 is determined by the structure of linear system of asymptoticcubic forms which arise in the third-order differential neighborhood of themanifold VP . If p1 = 5, 4, or 3, there are 3, 8, or 15 classes of the lin-ear systems of the asymptotic quadratic forms, respectively. Each of theseclasses determines a class of manifolds V3 which is projectively invariant.For each of these classes, Polovtseva investigated the geometric structure ofmanifolds V 3 , indicated the presence, if any, of conjugate pairs and asymp-totic directions in them, and evaluated the dimension of the osculating spacesof orders higher than one. In addition, for the cases where n = 3 + pt , sheproved the existence of manifolds from each of the classes and establishedtheir arbitrariness.

§5.12. Invariant normalization of manifolds

The problem of construction of the canonical moving frame, which Cartanconsidered in some of his papers and monographs, is directly connected withthe problem of invariant normalization of manifolds embedded in homoge-neous spaces.

Let VP be a manifold of an n-dimensional Euclidean or non-Euclideanspace. Its invariant normalization is a family of normals naturally determinedby the geometry of the ambient space. The normals Nx (Vp) of the manifoldVP are completely orthogonal to its tangent spaces Tx (Vp) and defined in thefirst-order differential neighborhood of a point x of the manifold VP . Theinvariant normalization induces in VP the inner geometry, i.e., the metric,the geodesic lines, the Gaussian curvature, etc. None of these properties ofthe manifold VP depend on the choice of a system of curvilinear coordinateson VP or on the choice of coordinates in the normals Nx(V") .

For manifolds in spaces with a wider group of transformations (the affine,projective, or conformal or some other spaces), it is impossible to determinean invariant normalization in the first-order differential neighborhood. Forthe first time the problem of construction of invariant normalization arose

§5.12. INVARIANT NORMALIZATION OF MANIFOLDS 171

in affine differential geometry. It turned out that for a surface V2 in thespace E3 , the invariant normal, i.e., a straight line passing through its pointx and not lying in the tangent space Tx (V") , can be determined only in thethird-order differential neighborhood. It was first constructed by Blaschke inhis paper on affine geometry. V. Characteristic properties of ellipsoids [Bla2](1917).

I n projective differential geometry the situation is even more complicated.In this case, to construct an invariant normalization of a surface V2 in thespace P3 , one must find its normals of the first and second kinds. Theformer is defined in the same way as in affine geometry, and the latter is astraight line in the plane T(V2) not passing through the point x {see thebook Spaces with of ne connection [Nor] (1950) by Alexander P. Norden (b.1904) ). The problem of construction of invariant projective normals of asurface V 2 in the space P3 was considered by Wilczynski in 1909, Fubiniand Eduard Cech (1893-1960) in 1927, and Finikov in 1937. However, intheir works the invariant normalization was connected with the choice of acertain coordinate net on the surface.

A normalization of a surface V2 in the conformal space C3 is determinedby a family of tangent spheres Cx and normal circles Sx passing throughthe point x E V2. In this case the construction of invariant normalization isconnected with the second-order differential neighborhood and was consid-ered in 1924 by Blaschke and in 1948 by Norden.

In 1953, Herman F. Laptev (1909-1972) in the paper Differential geome-try of imbedded manifolds [Lap3] developed a general method of differential-geometric investigations of manifolds embedded in homogeneous spaces orspaces with connections. This method is based on the theory of represen-tations of Lie groups and the Cartan method of moving frames. The ideaof the method is that during the differential prolongations of the system ofequations, defining the manifold V" under consideration in the space X n ,

one constructs a sequence of geometric objects connected with this manifold.This sequence contains complete information on the differential geometry ofthe manifold V" and is the basis for all geometric constructions related tothis manifold. Using this sequence, one can construct an invariant normal-ization of the manifold V" and also other geometric images connected withit. The construction indicated above does not require us to fix a coordinatesystem on the manifold V" . Because of this, this construction is invariant.

As an example, we consider the application of the Laptev method to thestudy of the geometry of a hypersurface V'_ 1 in the affine space EN

. Ifwe place the origin of a moving frame at a point x of the hypersurface andits vectors e! , i = 1 , ... , n - 1 , in the tangent space T (V'"- 1) , then theequation of the hypersurface can be written in the form a/ = 0. After tripleprolongation of this system (i.e., three exterior differentiations followed bythe application of Cartan's lemma), we obtain


wn = A.(.

(5.56) DA + A wn = A kk

ilk ilk n (ii k)1 n ij 1

where V is constructed according to the rule

VA = da. -A. wk -A wkii ii rk i ki i

and the quantities Aij , Ai jk , and a'ijkl are symmetric in all indices. Thesequantities form the fundamental sequence of geometrical objects indicatedabove. Here and in what follows parentheses mean a cycle of the indicesi, j, and k followed by division by 3.

Let us fix a point x on the hypersurface Vn-1. Then (vt = 0, wn = 0,

and the remaining equations of system (5.56) take the form:

(5.58) DA A 71=0.6 1Jk iik n (i k)1 n

Here, as earlier, the symbol a denotes differentiation relative to the sec-ondary parameters, and 7ru = wv (o) , u, v = 1, ... , n . The previous equa-tions show that the quantities Aij form a double covariant symmetric relative

tensor the asymptotic tensor of the hypersurface Vn -1, and the quantities

Aijk do not form a tensor since they depend on the choice of the vector enat the point x. The latter fact follows from equations (5.58) containing theforms 7r, defining the displacement of the vector en .

Suppose that det(Ai .) 0 0, i.e., the hypersurface Vn- is not tangentiallydegenerate. We construct a geometric object whose coordinates depend onlyon the displacement of the vector e. . For this, we first construct the tensorI%' which is the inverse tensor of the asymptotic tensor. By (5.57), this tensorsatisfies the equations

(5.59)

Next, we set

Ak=

D u 7tn = 0.n

1 .n-1 ijk'xi = Aik xk .

Differentiating the last equations relative to the secondary parameters andusing formulas (5.5 7) and (5.58), we find that

D 1 = 71nn 17rt

= 0.8 nn+ l n

§5.12. INVARIANT NORMALIZATION OF MANIFOLDS 173

It follows from this that the quantities A' form the desired object.It is easy to check that the vector n = en _n-1Aiei constructed with the

help of the object A' satisfies the equation on = inn , and therefore, its di-rection does not depend on the choice of the vector en at the point x. Thisdirection is internally connected with the geometry of the hypersurface andgives the affine normal of Blaschke. This normal is determined by the third-order differential neighborhood of a point x of the hypersurf ace V' 1 sincefor its construction we used the quantities 2iik connected with this neigh-borhood. It is possible to prove that this normal is parallel to the diameter ofthe paraboloid which has a tangency of second order with the hypersurfaceV1 at its point x .

The quantities 2jJk and 2i allow us to construct an important tensor thatis connected with the third-order differential neighborhood of a point x ofthe hypersurface Yn-1 . This tensor is defined by the formula

bIk=)"ijkn-1

- 3(l.in 1 'k)

and is called the Darboux tensor. It satisfies the condition bi JkA' = 0 , i.e.,it is apolar to the asymptotic tensor Ai . For a two-dimensional surface inthe three-dimensional space this tensor was constructed by Darboux. For ahypersurface this was done by Galina V. Bushmanova (b. 1919) and Nordenin their paper Projective invariants of a normalized surface [BN] (1948). Thevanishing of this tensor characterizes the hypersurfaces of second order.

The method developed by Laptev was widely used for solving concreteproblems in the theory of embedded manifolds. In the paper An invariantconstruction of the projective differential geometry of a surface [Lap l I (1949),Laptev himself used this method for study of the geometry of a surface V2 inthe space P3 , and later in the paper On fields of geometric objects on imbeddedmanifolds [Lap2] (1951), for study of the geometry of a hypersurface Vn-

in the space pn. For these manifolds, he found the fundamental sequence

of geometric objects, considered a family of osculating hyperquadrics, andconstructed a few invariant normalizations. Later Akivis solved similar prob-lems, in the paper Invariant construction of the geometry of a hypersurface of aconformal space [Ak 11 (1952 ), first for a hypersurface V'_ I

, and second, inthe paper On the conformal differential geometry of multidimensional surfaces[Ak4] (1961), for a manifold V" of arbitrary dimension p in the conformalspace Cn . In the paper Invariant constructions on an m-dimensional surfacein an n-dimensional afjine space [Shy] (1958), Petr I. Shveikin consideredthe problem of constructing invariant normalization for a manifold V" inthe affine space En . In the papers On the geometry of a multidimensionalsurface in a projective space [Os 1] (1966) and Distributions of m-dimensionallinear elements in a space with a projective connection II [Os2] (1971) Natalia


M. Ostianu (b. 1922) studied this problem for a manifold V' and a distri-bution of hyperplane and p-dimensional elements in the space P" and in aspace with a projective connection.

Note that if in the beginning of its development the method of movingframes was very often opposed to the tensorial methods which also wereof great importance for differential geometry, then after the creation of theLaptev method, it became clear that these two methods can easily be com-bined and can complement one another. The example considered above is agood evidence of this.

§5.13. "Pseudo-conformal geometry of hypersurfaces"

While in his works on differential geometry that we have considered so farCartan considered manifolds embedded in spaces with Lie groups as theirgroups of transformations, in his two-part paper On the pseudo-conformalgeometry of hypersurfaces of the space of two complex variables [136, 136a](1932), Cartan studied the geometry of three-dimensional surfaces of the two-dimensional complex space with analytic transformations of this space, whichform a Lie pseudogroup. The term "pseudoconformal geometry" which Car-tan used and which means a generalization of conformal geometry is presentlyused for the geometry of the pseudoconformal space C! . In Cartan's papersindicated above this term had another meaning: Cartan understood the term"conformal geometry" in the sense of geometry of the pseudogroup of an-alytic transformations of the plane of one complex variable and the term"pseudoconformal geometry" in the sense of geometry of the pseudogroupof analytic transformations of the space of several complex variables. Thestudy of real hypersurfaces of a two-dimensional complex space was initiatedby Poincare in the paper Analytic functions of two variables and conformalmapping [Poi5] (1907). In this paper Poincare proved that such a hyper-surface possesses an infinite set of invariants relative to transformations ofthis space. Using the analogy with analytic transformations of one complexvariable, Poincare himself called these transformations conformal mappings.The term "pseudoconformal mappings" was suggested by Severi. In 1931Beniamino Segre (1903-1973) in the papers On the Poincare problem onpseudo-conformal mappings [SeB 1 ] and Geometric questions associated withfunctions of two complex variables [SeB2] found new geometric properties ofthese hypersurfaces. Cartan, who became interested in the "Poincare prob-lem" under the influence of these papers by Segre, gave a classification ofreal hypersurfaces of the complex plane CE 2 (which can be considered asthe space E4 with complex structure) according to the groups of "pseudo-conformal mappings" admitted by these hypersurfaces.

These Cartan papers were substantially developed in the paper Real hyper-surfaces in complex manifolds [ChM] (1974) by Chern and JUrgen Kurt K.Moser (b. 1928).

§5.13. "PSEUDO-CONFORMAL GEOMETRY OF HYPERSURFACES" 175

The Laptev method generalizing the Cartan method of moving frameswas expanded by Anatoly M. Vasil'ev (1923-1987) to spaces where infinite-dimensional Lie groups act in his papers General invariant methods in dif-ferential geometry [Va 1 ] (1951) and Differential algebras and differential-geometric structures [Va2] (1973) and the book Theory of differential-geometric structures [Va3] (1987 ). The Vasil'ev method encompasses a widercircle of differential-geometric investigations.

CHAPTER 6

Riemannian Manifolds. Symmetric Spaces

§6.1. Riemannian manifolds

The Euclidean space Rn , the elliptic space Sn , and the Lobachevskianspace Si are particular cases of the Riemannian manifold V' introducedby B. Riemann in his famous lecture On the hypotheses which lie at the foun-dations of geometry [Rie2] (1854). Cartan contributed much to the geometryof Riemannian manifolds. His books Geometry of Riemannian manifolds[84] (1926), Lectures on the geometry ofRiemannian manifolds [ 114] (1928),[ 183] (1946), and Riemannian geometry in an orthogonal frame [ 108a] (1927)and many of his papers were devoted to this topic.

A Riemannian manifold Vn is a manifold whose points are defined byreal coordinates x1 , x2 , ... , xn , and the transition from these coordinatesto another coordinate system is performed with the help of differentiablefunctions. In addition, the distance ds between infinitesimally close pointswith coordinates x t and x t + d x' is given by the formula

(6.1) ds2 = gt .dx'dx3 ,

where gt . are differentiable functions of the coordinates xt of the pointsand the quadratic form (6.1) is positive definite.

Integrating the expression d s defined by formula (6.1) along a curve inthe space Vn , we find the arc length of this curve. Comparing differentcurves joining two points of the space Vn , we find geodesics: the curve isgeodesic if and only if, for any of its points sufficiently near, the arc of thiscurve between these two points is the shortest one. On the other hand, thecoefficients gt . allow us to find the angle (p between the differentials {dx'}

and

g. dxlox3(6.2)

gjjdxtdx3 Vgij6x'6dx

The volume element of the space Vn can be expressed in terms of the de-terminant of the matrix (g1) by the formula

(6.3) dV = vl-gdx I Adx2A

177

178 6. RIEMANNIAN MANIFOLDS. SYMMETRIC SPACES

and the volume of a particular domain of the space Vn is equal to theintegral of expression (6.3) over this domain. Volumes of the domains ofany m-dimensional surfaces of the space Vn and, in particular, the areas ofdomains of two-dimensional surfaces of this space can be defined in a similarmanner. Formula (6.2) for finding the angles between curves of the space Vn

and the formula for finding the area of domains of two-dimensional surfacesof this space allow us to define the most important notion of Riemanniangeometry, namely, the sectional (Riemannian) curvature of the space V nat a given point and a given two-dimensional direction. For calculation ofthe sectional curvature at a given point x(x') and a given two-dimensionaldirection defined by the differentials dx' and ax` of the coordinates of thispoint, one must take two geodesics through this point in the direction of thesedifferentials, join points y and z of these geodesics by the third geodesic,find the area S. of the geodesic triangle xyz obtained, the angles A, B, Cof this triangle, and their sum A + B + C. Then, the sectional curvature atthe given point in the given two-dimensional direction is the limit of the ratioof the difference A + B + C - ir to the area S, of this triangle as the triangleis shrunk to the given point in such a way that its sides remain tangent to thegiven two-dimensional direction:

(6.4) K = limA+B+C-7t

A--+O Sa

In particular, for the space Sn , by (3.4), we have S,=r 2 (A + B + C - n) ,

and the expression under the limit sign is equal to 1 /r2 . This implies thatthe sectional curvature K of the space Sn at all its points and in all two-dimensional directions is equal to the curvature 1 /r2 of this space. For thespace Si , by (3.5), we have Se = q2(n - A - B - C), and the expressionunder the limit sign is equal to -1 /q2 . This implies that the sectional cur-vature K of the space S1 at all its points and in all two-dimensional direc-tions is equal to the curvature -1 /q2 of this space. Thus, the elliptic spaceSn and the Lobachevskian space S1 are Riemannian manifolds of constantcurvature, positive and negative, respectively. Since, in his lecture of 1854Riemann paid special attention to the spaces of constant curvature, the spaceSn is often called the non-Euclidean Riemannian space. The Euclidean space

also a particular case of the space Vn . Since in the space Rn the sumof angles of any triangle is equal to 7r, the expression under the limit sign informula (6.4) is equal to zero. Hence, in this space the curvature K at all itspoints and all two-dimensional directions is equal to zero, and the Euclideanspace a Riemannian manifold of constant zero curvature.

With any point x of the Riemannian manifold V n there is associated theEuclidean space T( V n) tangent to Vn at this point, and the differentialsdx' of coordinates of points of the space V n can be considered as coordi-nates of vectors of the space Tx (V n) tangent to V n at this point. In the first

§6.1. RIEMANNIAN MANIFOLDS 179

edition of his Lectures on the geometry of Riemannian manifolds [114] witheach point of the space V" , Cartan associated the so-called natural framethe vectorial basis of the space TX (V") consisting of the vectors el that aretangent to the coordinate lines of a coordinate system {x'} up to infinitesi-mals of higher orders coinciding with the segments joining the points x (x` )and x' (x ` + dx ") . Therefore, up to infinitesimals of higher orders, the co-ordinates of vectors of the space TX (V") are equal to the differentials dx` .

The vectors ei of the natural frame are often denoted by 91,9x' .

Riemannian geometry in the natural frame usually is presented by meansof the tensor calculus developed by Levi-Civita in the paper Methods of theabsolute differential calculus and their applications [LeC] (1901 ).

Under the coordinate transformations x` =fit

(x', x2 , ... , x") , the vec-tors e, of the natural frame undergo a transformation according to the fol-lowing rule:

ax`(6.5) el,=ei a,.

x`

Under the same transformation, the coordinates of contravariant vectorsa= {a'} undergo the transformation

(6.6) a = a l ax`,

ax

where (Ox'/Ox') is the inverse of the matrix (Ox'/Ox'), and the coordi-nates of covariant vectors a = {a,} undergo the transformation

ax`(6.7) al. = al 1,

ax

with the same matrix as the vectors e.. The tensorsT','' ' 1° with p con-J 1 J2...Jq

travariant indices and q covariant indices undergo the transformation

11 i2...it it i2...ip axis ax`p axi' aXJq(6.8)

J ij?... j TjI j2... jq it ...

ipJ ... J

q ax ax ax t axq

The differentials d x ` of coordinates x' undergo the same transformationas coordinates of a contravariant vector, and the partial derivatives a (P/axlundergo the same transformation as coordinates of a covariant vector. Thequantities g,3 form a doubly covariant tensor which is called the metrictensor. In the natural frame, the equations of geodesics of the space V"have the form

d2x` , dxj dxk(6.9} 2 + r k ds ds = 0 ,

ds


where the functions I'`.k , which are called the Christoffel symbols and do notform a tensor, can be expressed in terms of the tensor gij by the followingformula:

(6.10) 1 agjh aghk agjk ihrjk k+ a xjax h g

ax

where g'-' is the inverse tensor of the tensor gi . (gthgJh . = 6). The sectionalcurvature in a two-dimensional direction defined by the differentials {dx'}and can be expressed by the formula

R.. kldx`dxkax'6xI'(6.11) K= '(gikgIjl - gIjkgil )dx dxkaxjax` '

where Rij , kl is the Riemann-Christoffel tensor and can be expressed in termsof the tensor gij and its derivatives by the formula

(0k(6.12) R

a r= - ik + 0 I,g - rig rgi kl g

49xt axi ig jk ik hl

It follows from formula (6.12) that the tensor R,3 , kl satisfies the relations:

(6.13) Rij,k1 = Rji,kl = -Rij,lk = Rji,lk,

(6.14) Ri j , k1 = Rkl , ij '

(6.15) Rij,kl + Rjk,il + Rki,jl = 0.

Relation (6.15) is called the Ricci identity, it is named after the founder ofthe tensor analysis Gregorio Ricci-Curbastro (1853-1925) who discoveredthis identity.

Cartan encountered the Riemannian manifolds in his works on the theoryof simple Lie groups. In his thesis he proved that the condition for a complexLie group to be semisimple is the nondegeneracy of quadratic form (2.20).In other words, he showed that semisimple complex Lie groups are complexRiemannian manifolds. Similarly, Cartan's result of 1914 that the conditionfor a real compact Lie group to be semisimple is the negative definiteness ofthe same quadratic form (2.20) (see [38]) means that semisimple compact realLie groups are real Riemannian manifolds if one takes the form -V(e) asds2 . Note that the Riemannian metric, defined in this manner in the complexand compact real simple Lie groups and at present called the Cartan metric,is a unique (up to the scale of the Riemannian metric) metric in these groupswhich is invariant relative to the group operations x -+ ax , x -' xa , andx -' x .

At the end of the introduction to his Lectures on the geometry of Rieman-nian manifolds Cartan wrote: "I was forced to leave aside many important

§6.3. PARALLEL DISPLACEMENT OF VECTORS 181

problems. They might compose the content of the next volume where themethod of moving frames and its numerous applications will be presented"[ 114, p. 6]. The main part of this material was included in the second edition[ 183] of this book.

§6.2. Pseudo-Riemannian manifolds

In 1914, in the same paper [38] discussed in the previous section, Cartanactually introduced the so-called pseudo-Riemannian manifolds which differfrom the Riemannian manifolds by the fact that the quadratic form (6.1),defining the metric of this space, is no longer positive definite; it can beany indefinite nondegenerate form. In this paper, Cartan showed that ifone takes the form - (e) as ds2 , then a noncompact real Lie group is apseudo-Riemannian manifold. At present, a pseudo-Riemannian manifold,whose metric is defined by a nondegenerate quadratic form (6.1) of index I,is called a pseudo-Riemannian manifold of index l and is denoted by Y" .

Thus, in the Cartan metric, a noncompact real group Lie of dimension r,whose character is equal to 6, is a pseudo-Riemannian manifold Y" wherethe index l is connected with the dimension r and the character 6 of thegroup by the relation a = r - 21.

The tangent pseudo-Euclidean spaces Tx (Y") for the space Y" play thesame role as the tangent Euclidean spaces 7x (V) for the Riemannian man-ifold Vn . The ideal domain of the hyperbolic space S1 and both domainsof the spaces Sn , l > l , are pseudo-Riemannian manifolds of constant pos-itive or negative curvature, and the pseudo-Euclidean space Rn is a pseudo-Riemannian manifold of constant zero curvature.

General relativity theory, created by Albert Einstein (1879-1955) in 1916,played the important role in attracting the interests of mathematicians to thegeometry of Riemannian and pseudo-Riemannian manifolds since, accordingto this theory, the space-time is a pseudo-Riemannian manifold V14 whosecurvature is greater in those places where the density of matter is larger.

§6.3. Parallel displacement of vectors

In 1917, shortly after the appearance of the general relativity of Einstein,Levi-Civita introduced one of the most important notions of the Riemannianand pseudo-Riemannian geometries-the parallel displacement of vectors. Atthe same time as Levi-Civita, Schouten discovered parallel displacement ofvectors in the Riemannian geometry. Schouten's colleague Dirk Jan Struik(b. 1894) recently recalled: "One day in 1918 Schouten came bursting intomy office waving a paper he had just received from Levi-Civita in Rome.`He also has my geodesically moving systems,' he said, `only he calls themparallel.' The paper had in fact already been published in 1917, but the warhad prevented it from arriving sooner." [Row, p. 16].


In the same year, 1917, when the paper of Levi-Civita was published, inthe paper On the curvature of surfaces and manifolds [Svr], Severi gave thegeometric definition of this notion which Cartan widely used. The essence ofSeveri's definition is that to each vector a of the tangent space T( V n) at apoint x of the Riemannian manifold V" , one can set in correspondence acertain vector 'a of the tangent space at a point x' infinitesimally close tothe point x of the same space. This vector 'a is defined by a mapping ofa neighborhood of the point x onto a neighborhood of the point x' that isa result of the sequential reflecting about the point x of the neighborhoodof the point x along geodesics emanating from this point and the similarreflection in the point x0 of a neighborhood of the point x0 located on thegeodesic xx' , half way between the points x and x' (provided that theneighborhood of the point x0 contains the neighborhoods of the points xand 4. If, up to infinitesimals of higher orders, the vector a coincides withthe geodesic segment xa, and the mapping indicated above sends the pointa of the neighborhood of the point x into the point a' of the neighborhoodof the point x', then up to infinitesimals of higher orders, the vector 'acoincides with the geodesic segment x'a'. If the vector 'a is given at thepoint x(x') of the space V" , then the result of its parallel displacement intothe infinitesimally close point x' (x t + d x t) is a vector 'a with coordinates

(6.16)i i i ka =a +I'J.ka dx

Assuming that the scalars are not changed under a parallel displacement,we find that the parallel displacement of covariant vectors a = {ak} is definedby the formula

(6.17) ak = ak -

and the parallel displacement of an arbitrary tensor Tt p is defined asJ1 Jq

(6.18)11...i 11...1 i k12...1 1T; ...;F= T;..;p..r;kT, ...;pdx +...1 q 14 4

ip 11...ik ; I 111z...1pJ I 11...1p ;

+r;kT;1... -,P-1 dx ;;l Ti;z...;4 dx >j47'jj...;Q_ 1dx

If in the space V" , a vector or tensor field is given, i.e., at any point of thisspace a vector or a tensor of a certain type is defined that is a function of thispoint, then we can define the covariant derivative of this vector or tensor bysubtracting from the value of this vector or tensor at the point x' the resultof its parallel displacement from the point x into the point x' , dividing thisdifference by the difference of the coordinates xl + dx' and xt of the pointsx' and x and taking the limit of the ratio obtained when the point x' tendsto the point x . The covariant derivative Via' of the vector a = {a,} has

§6.4. RIEMANNIAN GEOMETRY IN AN ORTHOGONAL FRAME 183

the coordinates

i a`(xk + A X -'a(xk} aa` i k(6.19) via = lim + Fjka

&X --+o Ax a x

and the covariant derivative V . T.' ' of the tensor T.' has the coordi-J Ji ...Jq JI Jq

nates1 1

ll...i° - aT...lg iJ ki2...i° ° 1...i°-tk(6.20) VJT. -- +I'jJl ... jq

axe Iq z ' q

-ri. Ti!i2...i° -..._r`Ti' .'°

JJ1 J2 "Jq JJq Ji ..JQ- f

The covariant derivative of any vector or tensor is again a tensor whichhas one more covariant index than the original vector or tensor. The resultof the contraction of the covariant derivative V T" `° with the differential

J J ... Jq

dxJ is called the absolute differential and is denoted by D TJI Jq

Comparing formulas (6.9) and (6.19), we see that formula (6.9) can bewritten in the form

VJ.

(d x /d s) (dxJ /d s) = 0 from which it follows that thegeodesic lines of the space Vn can be also defined as the lines along whichtheir tangent vectors dx'/d s undergo a parallel displacement. Note also thatrelation (6.10) is equivalent to the relation V

k giJ= 0.

In the spaces V'1 , parallel displacement and covariant differentiation canbe defined in a similar manner.

§6.4. Riemannian geometry in an orthogonal frame

In Cartan's book Riemannian geometry in an orthogonal frame [ 108a],which was composed of his lectures of 1926-1927 written by Finikov andtranslated by him into Russian, and in the second edition of the book Lectureson the geometry of Riemannian manifolds [ 183], Cartan introduced a newpresentation of Riemannian geometry associating with every point of thespace an orthonormal frame {e,}(e,egy = oil) instead of the natural frame.

The derivational formulas of an orthonormal frame {e1} have the sameform (5.5) as for an orthonormal frame in the space R`1 (the differential dxhere is the vector with the coordinates d x but in this case the structureequations are more complicated than in the case of the space Rn , namely,they are:

(6.21) d a' = cok A Co` ,k dco' = cvk n coJ + 1 RJ cok A wr k 2 i,kl

where RI) kl is the curvature tensor which in this case can no longer beexpressed in terms of the tensor gig and its derivatives by formulas (6.12).

In an orthonormal frame, the sectional curvature K in a two-dimensionaldirection can again be calculated by the same formula (6.11) as in the case


of the natural frame; but, since in this case we have gij = c5, , , this formulabecomes:

(6.22) K = Rij, kla`akbJbI ,

where a' and b' are the coordinates of two unit orthogonal vectors a andb defining this two-dimensional direction.

Comparing formulas (6.21) with formula (5.13) held in the space of con-stant curvature 1 /r2 , we find that in this space and in an orthonormal framethe curvature tensor Rij k1 has the form

(6.23) Rij $kl 2(C5

ik JI it Jk)rThe application of orthogonal frames enables one to solve many problems ofdifferential geometry in the space Vn in the same simple way as in the spaceRn.

Contracting the tensor R.j , k1 of the space V n in the indices i and 1, weobtain the Ricci tensor :

(6.24) Rjk = Rij,klgil.

Since at each point of the space Vn two tensors gij and R. j are alwaysgiven, in the general case with each point of the space V n

, n principaldirections in the sense of Ricci are associated, and these directions are thedirections of the eigenvectors of the matrices R` = Rk gkl .

J J

§6.5. The problem of embedding a Riemannian manifoldinto a Euclidean space

The problem of embedding a Riemannian manifold Vn as a surface intoa Euclidean space RN of sufficiently large dimension was posed by one ofthe founders of multidimensional geometry, Ludwig Schlafli (1814-1895), inhis Note on the memoir "On spaces of constant curvature" of Mr. Beltrami[Scll] (1871-1873). Schlafli's argument was as follows. If the space Vnwith the metric form d s2 = giJ. d u d uJ is embedded into the space RN inthe form of a surface x = x(u1 , u2 , ... , un) , then the coefficients giJ. are

connected with the partial derivatives xi = 8x/8ui by the relations giJ . =x, x j . Since the number of coefficients gi j is equal to n (n + 1) /2 , the numberof the equations obtained is the same. Schlafli concluded from this that it ispossible, at least locally, to embed the space Vn into the space RN whereN=n(n+1)/2.

Schlafli's statement on the possibility of local embedding was proved byMaurice Janet (1888-1984) in the paper On the possibility of imbedding agiven Riemannian manifold into a Euclidean space [Ja] (1926). Janet's re-sults were revised by Cartan in the paper [104] (1927) with the same title.

§6.6. RIEMANNIAN MANIFOLDS SATISFYING "THE AXIOM OF PLANE' 185

While Janet wrote the problem of embedding the space Vn into the spaceRA in the form of a system of partial differential equations which he in-vestigated using rather complicated methods, Cartan applied his own theoryof systems in involution. He wrote the system of equations describing theembedding of the space V'7 into the space RN in the form of a system ofPfaffian equations &i = co' , 6a = 05 i= 1, ... , n, a = n+1. ... , N ,

where co' are basis forms of the space vn and the forms Cam` are basis formsof a surface Vn of the space Ri" onto which the space Vn is mapped. Thedifferential prolongation of these systems leads to the equations 6 ?. = COO- .

Applying structure equations (6.21) of the space Vn and the structure equa-tions of the surface Vn of the space R , Cartan showed that this systemof Pfaffian equations is in involution and its general solution depends on narbitrary functions of n - 1 real variables. This solution of the problem ofembedding the space V'7 into the space Ri" in the paper [104]. which wasmuch simpler than Janet's solution in the paper [Ja], offended Janet. AfterCartan's death Janet tried to convince Pommaret that his (own) methodsof solution of partial differential equations were better than Cartan's meth-ods, complaining that his methods were undeservedly forgotten. Pommaretwrote in his book Lie pseudogroups and mechanics [Pom3] (1988): "Thesecomments were given privately to us by Maurice Janet, again mathematicianand mechanician (sic.), who died in January 1984 at the age of 96." [Pom3,p. 7].

§6.6. Riemannian manifolds satisfying "the axiom of plane"

In 1927, Cartan published the paper The axiom of plane and metric dif-ferential geometry [90]. This paper appeared in the collection of articles "InMemoriam N. I. Lobatschevsky" which was published in Kazan, U.S.S.R.,on the occasion of the 100th anniversary of the discovery of non-Euclideangeometry by Lobachevsky. Figure 6.1 (next page) reproduces the first pageof Cartan's manuscript of this paper, which is kept in the Department of Ge-ometry of the University of Kazan. Cartan called a surface of a Riemannianmanifold geodesic at a certain point if this surface coincides with the unionof geodesics of the Riemannian manifold, emanating from this point andtangent to a plane element of the space at this point, and he called totallygeodesic a surface which is geodesic at each of its points. The requirementthat any geodesic surface be totally geodesic at each of its points was called"the axiom of plane" by Cartan. The notion of a totally geodesic surface wasintroduced by Jacques Hadamard (1865-1963) in his paper on linear ele-ments of many dimensions [Had] (1901). Hadamard defined these surfacesas surfaces such that each geodesic of them is a geodesic of the space.

In the case of Euclidean and non-Euclidean spaces, geodesics are straightlines and planes are totally geodesic surfaces. This explains the name of thisCartan axiom. It is obvious that the Euclidean and non-Euclidean spaces


aca:+ o(+*.- AZVn_r./ lies-77

r. jw*aG"A&%v.

to t dw+t C.

4P. &&WV) 06 VVt 0&

RYA-rt wee-xe, woe

1 41

s ,lam 4" ` &Courtesy of Department of Geometry, Kazan University, Tatarstan, Russia

FIGURE 6.1

satisfy this axiom. In his paper Cartan proved that if a Riemannian manifoldsatisfies "the axiom of plane", then it can be geodesically and conformallymapped onto a Euclidean or non-Euclidean space.

§6.7. Symmetric Riemannian spaces

In the note On Riemannian manifolds in which parallel translation pre-serves the curvature [87] (1926) Cartan remarked on Harry Levy's paper Thecanonical form ds2 for which the five-index Riemann symbols are annihilated[Lev]. Since Levi called the coordinates Ril kl of the Riemann tensor thefour-index Riemann symbols and the coordinates OhR1l kl of the covariantderivative of the Riemann tensor the five-index Riemann symbols, the spacessingled out by Levi are Riemannian manifolds satisfying the condition:

(6.25) OhR;l.kt = 0.

Levi established that condition (6.25) holds for the spaces of constantcurvature, i.e., the spaces R" and Si' and the Cartesian products ofthese spaces, but he did not find other Riemannian manifolds satisfying thisproperty. Neither Levi nor Cartan knew that the same class of Riemannianmanifolds was introduced by Petr A. Shirokov (1895-1944) in the paperConstant fields of vectors and tensors ofsecond order in Riemannian manifolds[Shl] published in 1925 in Kazan. Shirokov also established that the spaces

§6.7. SYMMETRIC RIEMANNIAN SPACES 187

of constant curvature satisfy this condition, and, in addition, unlike Levi, hefound the general form of three-dimensional Riemannian manifolds of thistype.

In his note [87] Cartan indicated that irreducible spaces of this type "areseparated into 10 large classes each of which depends on one or two arbi-trary integers, and, in addition, there exist 12 special classes correspondingto the exceptional simple groups G". He wrote further: "Among the generalclasses, besides the spaces of simple groups which were discussed in the noteI, I will only indicate the class of the spaces of constant curvature found byMr. Levi and the class of Hermitian hyperbolic and elliptic spaces." [87, p.245]. In the text of the note instead of the word "spaces" at the end of thequotation written above, it is incorrectly written "groups". In this quotationthe "note I" means the joint note on the geometry of the group-manifold ofsimple and semisimple groups [91] by Cartan and Schouten also publishedin 1926 but a little later than the note [87]. In this note [91], the authorsdefined three types of parallel displacements in Lie groups and denoted themby (-), (+), and (a). The first two types of these parallel displacements areabsolute parallelisms, and the third type in the case of simple and semisimpleLie groups is the parallel displacement of the vectors of the Riemannian orpseudo-Riemannian manifold satisfying property (6.25). Briefly mentioningthis fact in the note [87], Cartan indicated that the totally geodesic subman-ifolds of the group-manifold of simple and semi-simple Lie groups possessthe same property. Since the number 10 of "large classes" is the mean of thenumbers of types of symmetric Riemannian spaces with the classical funda-mental groups 11 types Al - IV , BI - II , CI - II , DI - DIII and 9 typesAl - IV , BDI - II , CI - II , DIII which Cartan used later, and the number12 of "special classes" coincides with the number of types of symmetric Rie-mannian spaces with the special fundamental groups El - IX, FI - II , andGI , and since Cartan indicated that in simple and semisimple Lie groupsone can define the metric of symmetric Riemannian spaces which can takeplace also on totally geodesic submanifolds of these groups, we see that whenCartan was publishing his note [87J, he already knew most of the results ofhis theory of symmetric spaces.

Cartan gave a systematic exposition of this theory in his paper on a re-markable class of Riemannian manifolds published in two parts [93] (1926)and [94] (1927). Cartan defined these space as the spaces characterized by"the property that the Riemannian curvature of any face is preserved undera parallel displacement, or in more abstract terms, by the property that thecovariant derivative of their Riemann-Christoffel tensor is identically equalto zero" [93, p. 214], i.e., by identity (6.25). In this paper, Cartan calledthese spaces the "spaces W". However, in his late works he gave them thename "symmetric Riemannian spaces".

In the papers [93]-[94], Cartan proved that condition (6.25) is equivalentto the fact that the reflection in each point of the space along geodesics is an


isometric transformation (motion) of the space. This very property was thereason that in his subsequent papers Cartan called these spaces the symmetricRiemannian spaces. In addition, Cartan showed that all compact simple andsemisimple Lie groups are symmetric Riemannian spaces provided that oneintroduces in them the Cartan metric which will be in this case a Riemannianmetric (in this case the reflection in the identity element of the group has theform x - x-1 , and the reflection in an element a has the form x - ax-1 a).In the same paper, Cartan proved that any irreducible compact symmetricRiemannian space can be realized in the form of a totally geodesic surface inthe group of motions of this space which passes through the identity elementof this group (if a is the reflection in an arbitrary point of this space andco is the reflection in a certain fixed point, then this totally geodesic surfacein the group of motions consists of products ooc).

Cartan also considered the symmetric Riemannian spaces whose groupsof motions are noncompact simple and semisimple Lie groups. These spacescan also be realized in the form of a totally geodesic surface in their groupsof motions if we introduce in these groups the Cartan metric. However, sincefor noncompact simple and semisimple Lie groups this form is nondegenerateindefinite, the Cartan metric in the Lie group is a pseudo-Riemannian metric.In this case, the Lie group is a symmetric pseudo-Riemannian space, i.e.,a pseudo-Riemannian manifold satisfying condition (6.25). For symmetricpseudo-Riemannian spaces, the two properties which were indicated aboveand which were established by Cartan for symmetric Riemannian spaces alsohold.

For the case, when the group of motions of a symmetric Riemannian orpseudo-Riemannian space is a simple Lie group, the Lie algebra of this groupadmits the "Cartan decomposition" (2.43). Moreover, the algebra H is theLie algebra of the isotropy group of this space (the group of rotations aboutits point), and the subspace E can be considered as the tangent space to thetotally geodesic surface in the group in which the symmetric space is realizedor, equivalently, as the tangent space to the symmetric space. By formulas(2.36) the subspace E of the Lie algebra G, which can be considered as thetangent space to a symmetric Riemannian space, is closed under the operation[[X, Y], Z ] . Such spaces are called Lie triple systems.

Since in the Lie algebra of a Lie group the Cartan-Killing form has theform (2.20), the Riemannian or pseudo-Riemannian Cartan metric in a Liegroup is defined by the linear element

(6.26) ds2 = C1 ck o)iU)fik i1

and the Riemann tensor of this symmetric space has the form

(6.27) Ri = l cj chi, kl 4 hr kl `

§6.7. SYMMETRIC RIEMANNIAN SPACES 189

In the case when a symmetric space corresponds to the Cartan decompo-sition (2.43) and the structure equations of the group are written in the form(2.36), the metric of the symmetric space is defined by the linear element

(6.28) ds2 = CaWCvawu

W'0

and the Riemann tensor of this symmetric space has the form

=(6.29) u, wz 4 au wz

Comparing formulas (6.26) and (6.28) with formulas (6.27) and (6.29), wesee that in both cases the Ricci tensor of the symmetric space is proportionalto its metric tensor. It follows from this that in the symmetric Riemannianand pseudo-Riemannian spaces it is impossible to define the principal direc-tions in the sense of Ricci.

Next, Cartan considered the classification of involutive automorphisms ofLie algebras of compact simple Lie groups which we presented in Chapter 2for Lie groups in the classes An) Bn , Cn , and Dn and gave the classifica-tion of symmetric Riemannian spaces whose groups of motions are compactsimple Lie groups. These symmetric spaces are characterized by the samecharacters 6 as noncompact simple Lie groups corresponding to the sameinvolutive automorphisms of compact simple Lie groups. In the case of sym-metric spaces with compact simple groups of motions, these characters havea simple geometric meaning: if G is the group of motions of a symmetricspace and the isotropy group is a subgroup H of this group (in this casethe space is denoted by G/H), then the character 6 of the symmetric spaceis equal to the difference dimE - dim H between the dimension dimE ofthe symmetric space and the dimension dim H of its isotropy group. (Sincethe dimension dim G of the group G is equal to the sum of dimensionsdimE + dim H, the character a of the symmetric space is equal to the dif-ference dim G - 2 dim H.)

Cartan also found the isotropy groups of irreducible symmetric Rieman-nian spaces, i.e., spaces that cannot be represented in the form of Cartesianproducts of other symmetric spaces. In the case of the symmetric space V N ,

the isotropy group is the group of rotations of the Euclidean space RN tan-gent to the space V N , i.e., the group ON or a subgroup of it. Moreover,for the case when the isotropy group is a simple group or a direct product ofsimple groups, Cartan found the linear representations in which the isotropygroup or its direct factors are realized in the group ON (in addition to thenoncommutative direct factors, the isotropy group may also contain a rep-resentation r of the commutative simple group D1 = 02 the group ofrotations of a circle). Cartan proved that, for irreducible symmetric Rieman-nian spaces, the isotropy groups of these spaces coincide with their holonomygroups. The latter are the subgroups of the isotropy groups defined by paralleldisplacements of vectors of these spaces along closed contours. (The require-ment of irreducibility is essential since, for example, for the Euclidean space


An Al n n n+32 Sp 1(On+1 )

All -n -2 (n -1)(n+2)2 'P2(

C(n+1)12)

AIII 41(n-I+1)- n(n + 2) 21(n-I+1) Sp1(A,-1)®c1(An-1)®90

AIV 2n - n2 2n Sp1(An_ 1) ®90

Bn BI 21(2n - 1 + 1) - n(2n + 1 ) 1(2n - 1 + 1) -V 1 (O1) ®V 1(O2n+1-1)

BII 2n - n2 2n 91 (02n )

Cn Cl n n(n + 1) SpI(An_1) ®90

Cl 81(n - 1) - n(2n + 1) 41(n-1) sp1(C1)®g1(Cn_1)

Dn DI 21(2n - 1) - n(2n - 1) 1(2n-1) V1(01)®V1(O2n-1)

DII (2-n)(2n-1) 2n-I c1(02n-1)

DIII -n n(n - 1) 92(An_1)®90

G2 GI 2 8 q (A1)®(p 1(A1)

F4 Fl 4 2 93 W3) ®;p1(A1)

Fl l - 20 16 rlrl(09 )

E6 El 6 4 So4 (C4 )

Ell 2 40 93(A5) ®cp1(A1)

EIII -14 32 W,(010) (9 coo

EIV - 26 26 (F4 )

E7 EV 7 70 V4(A7)

EVI -5 64 11(012) ®(p1(A1)

EVIL -25 54 V 1(E6) ®90

E8 EVIII 8 128 yi1(016)

EIX -24 112 (p1(E7) ®(p 1(A1 )

TABLE 6.1

R" which is the Cartesian product of n straight lines R 1

, the isotropy groupis the group of rotations On while the holonomy group consists of the identitytransformation alone.)

In Table 6.1 we give Cartan's notation for different types of compact sym-metric Riemannian spaces.

§6.8. HERMITIAN SPACES AS SYMMETRIC SPACES 191

In the first column of Table 6.1 we indicate the class of the simple Liegroup, in the second the Cartan notation for the type of the symmetric space,in the third and the fourth the character and the dimension of the symmet-ric space, and in the fifth the isotropy group of the symmetric space withindication of the linear representation of each of the direct factors of thisgroup.

In the book Symmetric spaces [Loo] (1969) by Ottmar Loos, the symmet-ric Riemannian spaces were considered as spaces with a certain algebraicstructure in which to any pair of points x and y there corresponds a thirdpoint z of this space which is the reflection of the point x in the point y .

A symmetric space with this operation is a quasigroup, i.e., it differs froma group by the absence of the identity element and the associativity of themain operation.

While Cartan found all symmetric spaces whose fundamental groups arecompact simple Lie groups and those noncompact simple Lie groups forwhich the stationary groups are compact, Berger in the paper Classification ofirreducible homogeneous symmetric spaces [Begs] (1955) and Anatoly S. Fe-denko (b. 1929) in the paper Symmetric spaces with simple non-compact fun-damental groups [Fed] (19 56) found all symmetric spaces with noncompactsimple fundamental groups of infinite sequences. In another paper Structureand classification of symmetric spaces with semi-simple groups of isometries[Beg2] (1955) Berger solved the same problem for noncompact exceptionalsimple Lie groups. Note also that in the paper Non-compact symmetric spaces[Beg3] (1957), Berger found the isotropy groups of all irreducible symmetricspaces and representations of the direct factors of these groups. This enablesone to find the orbits of the isotropy of these spaces.

The current status of the geometry of Riemannian manifolds is describedin the books Einstein manifolds [Bes] (1985) by Arthur L. Besse, Differentialgeometry, Lie groups and symmetric spaces [Hell ] (1978) by Helgason, andSpaces of constant curvature [Wo2] (1984) by Wolf.

§6.8. Hermitian spaces as symmetric spaces

The symmetric Riemannian spaces BII and DII of dimension 2n and2n - 1 , respectively, whose groups of motions are compact simple Lie groupsin the classes Bn and Dn (the groups of orthogonal matrices °2n+

1and

02n ), are Riemannian manifolds of constant curvature the elliptic spacesStn and 52n -1

The symmetric Riemannian space AIV of dimension 2n whose group ofmotions is a compact simple Lie group in the class An (the group CS Un+

1

of complex unimodular matrices is of this type) is the complex Hermitianelliptic space CSn , i.e., the space CPn where the metric is defined by for-mula (3.6). We have already mentioned that, as was noted by Cartan in his

1expanded translation of the Fano paper, the complex straight line CS of


curvature 1 /r2 is isometric to a sphere of radius r/2 in the space R3. Onthe other hand, assuming that the coordinates x and y' in formula (16)are real, we obtain formula (3.1) for finding the distances in the real ellipticspace S' of curvature 1/r2 .

Applying formula (6.29) for computing the curvature tensor to the Rie-mannian manifold V 2n which is isometric to the space CSn , we find thatin the orthonormal frame of V 2n , whose vectors e2i_ 1 coincide with thevectors fi of a unitary-orthonormal frame of the space CSn ((fi , f .) _ 3i

1 J

and whose vectors e2i coincide with the products ifi, the tensor R. j , k1 hasthe form

6.30)t

R. =r

- rl jk + rkjl - it jk +( , kl kl)

where ei j is the skew-symmetric tensor whose nonvanishing components are

e2i-1 ) 2i = -E 2i , 2i-1 = 1 . Formula (6.30) is similar to the formula for thecoordinates of the tensor RiJkl of the space V2n which is isometric to_ ,

the space CSn in the natural frame. The latter formula was found by P. A.Shirokov in his posthumously published paper On a certain type of symmetricspace [Sh2] (1957).

Substituting expressions (6.30) of the coordinates of the tensor Rij,kl intoformula (6.22), we obtain the expression for the sectional curvature K ina two-dimensional area defined by orthogonal unit vectors a and b in theform

(6.31) K = 2 (1 + 3 cost a),r

where a is the "angle of inclination of the two-dimensional element" intro-duced by P. A. Shirokov in the above-mentioned paper which is defined bythe relation (a, b) = i cos a. It is easy to check that the values of a and Kdo not depend on the choice of the vectors a and b in the two-dimensionalelement and that the angle a is equal to the angle between two complexstraight lines of the space CSn tangent to the vectors a and b. If the areaelement lies in the complex straight line (in this case the two-dimensionaldirection is called holomorphic), we have a = 0 and K = 4/r2 ; this canalso be seen from the fact that the line CSI of curvature 1 /r2 is isometricto a sphere of radius r/2 in the space R3 . If the two-dimensional elementlies in the manifold xi = x` or in the manifold obtained from it by a mo-tion of the space CSn (Cartan called such manifolds "normal space chains"),we have a = 7r/2 and K = 1/r2; this can also be seen from the fact thatif xi = xi , yi = yi , formula (3.6) takes the form (3.1). Formula (6.31)shows that the sectional curvature of any area element of the space V2n iso-metric to the space CSn lies in the interval: 1/r2 < K < 4/r2 . Since thesectional curvature of a holomorphic two-dimensional element of the space

§6.9. ELEMENTS OF SYMMETRY 193

CS,, (which is called the holomorphic curvature of this space) is equal to theconstant value 4/r2 , at present, the space CS,, is often called the Hermitianspace of constant holomorphic sectional curvature.

If I = I, the symmetric Riemannian space CII of dimension 4(n - 1)whose group of motions is a compact simple Lie group in the class C,, is the

quaternion Hermitian elliptic space HSn-1 , i.e., the space HPn-1 where themetric is defined by the same formula (3.6) as for the space CSn .

The symmetric Riemannian space FII of dimension 16 whose group ofmotions is a compact simple Lie group in the class F4 is the octave Hermitian

elliptic plane OS2 defined by A. Borel and Freudental in 1950-1951. Thesectional curvatures of these spaces Yon and V 16 are calculated by the sameformula (6.31) as in the space Yen where a is the "angle of inclination ofthe two-dimensional element" defined in the same manner as in the spaceCSn . In the case of holomorphic two-dimensional elements, i.e., the two-dimensional elements situated in quaternion straight lines of the plane HSnor in octave straight lines of the space aS2, we have a = 0 and K=4/r2. This corresponds to the fact that the straight lines HS 1 and OS 1 areisometric to hyperspheres of radius r/2 of the spaces R5 and R9 . Next,if the two-dimensional element lies in the normal real plane chain, we havea = ir/2 and K= 1 /r2 .

Note that applying formula (5.2 9) for computation of the Riemann tensorof a symmetric Riemannian space to the spaces Yon and V 16 (which areisometric to the Hermitian elliptic space HSn and plane OS2) we obtain theformula:

(6.32)

R, _ , a --6 -

)1+ -E +2E Et}kl (ikj1 rl k (ajk6aj1 ail aJ'k aiJakdor

where the matrices of the tensors Eaik are the matrices of the operators ofthe complex structure defined by the units is of the algebras H and 0.

The symmetric Riemannian spaces EIII, EIV, and EVIII of dimension32, 64, and 128 whose groups of motions are a compact exceptional simpleLie group in the classes E6 , E7, and E8 respectively, admit similar inter-pretations in the form of Hermitian elliptic planes over the tensor productsO®C,O®H,and O®O.

§6.9. Elements of symmetry

In the introduction to his paper on a remarkable class of Riemannianmanifolds [931, Cartan wrote: "The new spaces immediately admit a directgeometric definition: they can be represented by geometric figures admittinga simple definition in the ordinary space (of dimension three or higher)" [93,p. 217). At present, these geometrical figures are called elements of symmetry.


To the geometric interpretation of symmetric Riemannian spaces in theform of manifolds of elements of symmetry, Cartan devoted his great paperon certain remarkable Ri emanni an forms of geometries with a simple funda-mental group [ 107] (1927), which we have already mentioned in Chapter 2 asthe paper where an interpretation of a compact simple Lie group in the classCn in the form of the group of motions of the space Hr- ! first appeared. InChapters 2 and 3 of this work entitled Classification of spaces of constant neg-ative curvature connected with real non-unitary groups and Spaces of positivecurvature connected with real unitary simple groups (in the title of Chapter 3instead of the word "unitary" the incorrect "nonunitary" was written), Cartangave geometric interpretations of noncompact and compact symmetric Rie-mannian spaces with compact isotropy groups, respectively. Cartan calledthese spaces the spaces of negative and positive curvature, respectively, byanalogy with the hyperbolic and elliptic spaces S1 and Sn which are theirparticular cases. The terms "unitary" and "nonunitary groups" are connectedwith the fact that these groups are represented by unitary and pseudo-unitarycomplex matrices, respectively. For compact symmetric spaces of types A IIIand BDI, Cartan wrote that these spaces are spaces or can be defined asspaces of "pairs of plane manifolds of q - l and p - l dimension mutuallypolar relative to the form F" [107, pp. 448 and 451]; the form F is theright-hand side of the equation of the absolute in the space CSn and thespace Sn , respectively. In other cases, Cartan indicated the type of sym-metry relative to the corresponding element of symmetry without giving thenames of these figures. For the compact symmetric spaces of types A I , All ,and AIII , these symmetries are the transformations:

(6.33)

(b.34)

(6.35)

1 1 -x =x ,

1 2i -i+1 ! i+1 2ix =x , x =x ,a a 1 u uX = x , x = -x

of the space Cr that are the reflection in the normal space chain, the shiftfor a half-line along the lines of a paratactic congruence, and the reflection in am-dimensional plane and its polar, respectively. For the compact symmetricspace of type B1, the symmetry is the transformation (6.35) of the spaceStn which is the reflection in an m-dimensional plane and its polar. Forthe compact symmetric spaces of types DI and DII, the symmetries are thetransformations (6.35) and

(6.36) !x2! _ -X2i+1 x2i+1 = x2i

which are the reflection in an m-dimensional plane and its polar and theshift for a half-line along the lines of a paratactic congruence of the spaceStn-1 . Finally, for the compact symmetric spaces of types CI and CII, the

§6.9. ELEMENTS OF SYMMETRY 195

symmetries are the transformations

(6.37)i . i .-1X =rxl

and (6.35) of the spaceHSn-1

, respectively, which are the reflections in thenormal space chain and in an rn-dimensional plane and its polar, respectively.

Note that symmetries (6.33), (6.34), (6.35), (6.36), and (6.37) correspondto involutive automorphisms (2.48), (2.49), (2.47) (2.50), and (2.51), respec-tively, in the Lie algebras of the groups of motions of symmetric spaces.

Thus, if m = 0, the compact symmetric Riemannian spaces of typesAIV, BIl, DII, and CII coincide with the elliptic spaces CS", S2n , S2n-1 ,

andHSn-1

; if m 54 0, the spaces of types AIII, BI, DI, and CII canbe interpreted as the Grassmann manifolds CGRn, m , GR2n, m , GR2n_ 1 mand HGRn_ 1,m of these spaces, the spaces of types Al and CI can beinterpreted as the manifolds of normal space chains of the spaces

CSnand

HSn_ 1

, and the spaces of types All and DII can be interpreted as themanifold of paratactic congruences of straight lines of the spaces Cr andStn-1

Cartan also noted that a compact exceptional simple group in the classG2 is the group of automorphisms of the alternative field 0 of octaves. Ifwe define a metric of the space R. in the alternative field 0 by setting thedistance between octaves a and /3 to be equal to the modulus 1/3 - al oftheir difference, then the group of automorphisms of the alternative field 0is a transitive group on a six-dimensional sphere which is the intersection ofthe hypersphere dal = 1 and the hyperplane a = -a. This sphere playedan important role in the history of mathematics: in the tangent space tothis sphere at its point representing an octave a, there is defined a complexstructure transforming the differential d a of the octave a into the productad a , and this structure is nonintegrable since it is impossible to define com-plex coordinates on a sphere. This sphere was the first example of a spacewith nonintegrable complex structure (at present this structure is called thealmost complex structure). The study of spaces with almost complex struc-ture started from the study of this sphere. Identifying antipodal points ofthis sphere, we get the space Sg6. Elements of symmetry of this space arethose planes of the space Sg6 which are cut in it by associative subfields ofthe field 0. The latter subfields are isomorphic to the alternative field H.These planes are holomorphic relative to the almost complex structure of thespace Sg6.

We have already indicated that compact exceptional simple Lie groupsin the classes F4, E6 , E7 , and E. are the groups of motions of the elliptic

_2 =2 -2 _2planes OS ,

(O®C)S , (0 ® H)S , and (0 ® O)S . The compact symmet-

ric Riemannian spaces of types FII, EIII , EVI, and EVIII coincide withthese planes. Most of the remaining compact symmetric Riemannian spaces


whose groups of motions are these groups can be interpreted as manifolds ofnormal plane chains of different kinds of these planes.

Cartan considered noncompact symmetric Riemannian spaces to be of thesame type as compact symmetric Riemannian spaces if the isotropy groupsof these spaces were isomorphic. If m = 0, the noncompact symmetricRiemannian spaces of types AIV, B11, DII, and CII coincide with properdomains of the spaces CSI ,

Stn , Stn-1 , and HSn-!

1 1 1 1;lf m 0, the non-

compact symmetric Riemannian spaces of types AIII, BI, D1, and CII canbe interpreted as the manifolds of m-dimensional elliptic planes of thesespaces, the noncompact symmetric Riemannian spaces of types Al and Allcan be interpreted as the manifolds of imaginary quadrics of the space PIand imaginary Hermitian quadrics of the space HP(I -1)12 , and the non-compact symmetric Riemannian spaces of types CI and DIII can be in-terpreted as the manifolds of imaginary quadrics of the spaces Sy2n-1 andHSyn-1

. The noncompact symmetric Riemannian space of type GI can beinterpreted as the manifold of holomorphic two-dimensional elliptic planesof the space Sy, ; the noncompact symmetric Riemannian spaces of typesFII, EIII, EVI, and EVIII coincide with proper domains of the hyperbolic

-2 t t tplanes OS 1, (0 ® QS 1, (0 ® H)S 1 , and (0 ® O) S 1 ; most of the remain-ing noncompact symmetric Riemannian spaces whose groups of motions arenoncompact simple Lie groups of these classes can be interpreted as man-ifolds of normal plane chains of different kinds of elliptic planes over the

algebras'O,O®'C,'O®C,O®'H,'O®H, and 'O®O.

§6.10. The isotropy groups and orbits

The isotropy groups of the symmetric Riemannian spaces VN which Car-tan found in the paper [94] are subgroups of the groups ON of rotations ofthe spaces RN tangent to the spaces VN . Because of this, they act in thehyperplanes at infinity of the spaces RN which themselves are the spacesSN_

1 even if these groups are not transitive in these spaces. They transformcertain surfaces of these spaces, the so-called local absolutes, into themselves.The isotropy groups are transitive in the spaces SN-1 only in the cases whenthe space VN is a space of constant curvature, i.e., in the symmetric spacesBDII.

In the case of the symmetric spaces VN of type BDI(N = (m + 1)(n - m)), whose models are the Grassmannians of rn-dimensional planes

of the spaces SI , the isotropy groups are isomorphic to the direct products

°m+l x ° _ m , and the local absolutes are the Segreans l,n, n -m-1 (3.12) in

the space 5(m+1)(n-m)-1

In the case of the symmetric space VN of type AIV, i.e., the space CSn ,the isotropy group is isomorphic to the direct product of the group of motionsof the space CSI -1 and the group D1 of motions of the line S 1 . In this

§6.10. THE ISOTROPY GROUPS AND ORBITS 197

case, the isotropy group transforms into itself a paratactic congruence ofstraight lines of the space Stn-1 which is isometric to the space CSn-1 (thegroup of motions of the line S 1 is the group of shifts along the lines ofthe congruence). In this case, the local absolute is the pair of imaginaryconjugate (n - 1)-dimensional plane generators of the absolute of the spaceStn-1 which is an imaginary focal surface of this congruence.

In the case of the symmetric space VN of type Al (N = 2 3) , whosemodel is the manifold of the normal space chains isometric to the space Sn ,the isotropy group is isomorphic to the group Q,+1 , and the local absoluteconsists of the Grassmannian Gn ,1 and n vertices of the autopolar simplex

of the space Sn-1 .

In the case of the symmetric spaces V4n-4 of type CII (1 = 1) , i.e., thespace HSn + 1 , the isotropy group is isomorphic to the direct product of thegroup of motions of the space HSn-1 and the group A = B = C, (thegroup of automorphisms of the field H). In this case, the isotropy grouptransforms into itself a paratactic congruence of three-dimensional planes ofthe space Son-5 which is isometric to the space HSn-1 . In this case, thelocal absolute is the imaginary Segrean 11,2n-3 (2.1 2) which is an imaginary

focal surface of this congruence and lies on the absolute of the space Son-s .

In the case of the symmetric spaces V15 , V32 , V64 , and V 128 of types=2 =2

FII , EIII , EVI , and EVIII, i.e., the planes OS , (0 0 QS 5, (0 ® H)S-2

and (0 ® O)S , the isotropy groups are isomorphic to the spinor group ofthe group 09 , to the direct product of the spinor group of the group 010and the group D, , to the direct product of the spinor group of the group0,2 and the group A, = B, = C, , and to the spinor group of the group 016 ,respectively. In these cases, the isotropy groups transform into themselvescongruences of planes of the spaces S 15 , S 31 )s 63 , and S127 which are iso-metric to the Hermitian lines over the same algebras. In these cases, the localabsolutes are the imaginary Lipshitzeans Q5 , Q6 , n7 , and Q8 (2.34) whichare imaginary focal surfaces of these congruences and lie on the absolutesof the spaces S 1 s , s31 S63 and S127. Note that in the case of symmetric- 64pseudo-Riemannian spaces n2n , TIZn 2 , Y816 v! 62) V32 , and Vb4 $ , isomet-ric to the Hermitian spaces and planes that can be obtained from the spacesand planes mentioned above by substitution of the algebras 'C, 'H, and 1O

for the fields C, H, and 0, the local absolutes are real pairs of planes, Seg-reans and Lipshitzeans (see the papers [RKoY] and [RB] of B. A. Rosenfeld,T. I. Yuchtina, T. A. Burtseva and others).

Cartan gave the unitary equation of the local absolute only for irreduciblesymmetric Riemannian spaces whose isotropy group coincides with theirholonomy group; he wrote this equation not in point but in line coordinates:

(6.38) Rrj, klpijpkl _ 0,


where Ri3, kt is the Riemann tensor of the symmetric space and p" are thePliicker coordinates of a straight line in the space SN-1 -the hyperplane atinfinity of the space RN tangent to the symmetric space V N (p'-' are alsothe coordinates of a bivector defining a two-dimensional element of the spaceYN).

For the simple Lie groups endowed with the Cartan metric, the role ofthe group of motions is played by the group of transformations g' = agb(a, b, and g are elements of the given group) which is isomorphic to thedirect product of the given group by itself, and the role of the isotropy groupis played by the adjoint group, i.e., the group of transformations g' = agawhich is locally isomorphic to the given group.

§6.11. Absolutes of symmetric spaces

In the paper [107], for the symmetric Riemannian space of negative cur-vature of type AI whose model is the manifold of nondegenerate quadricsof the space Fn, Cartan introduced the notion of the absolute of the sym-metric space. Considering this space as the manifold of positive definitequadratic forms, Cartan wrote: "The absolute is formed by degenerate def-inite quadratic forms" and further he wrote that, for n = 2, N = 5, "theabsolute is formed in a projective space (aid) by the part of a cubic manifoldobtained by equating to zero the discriminant of the form (aid) , i.e., that partwhich corresponds to the conics decomposing into a pair of imaginary conju-gate straight lines. It contains also the Veronese surface corresponding to thequadratic forms that are perfect squares" [107, p. 387). The term "absolute"is undoubtedly explained by the fact that, in the case n = 1, N = 2, i.e., inthe case of pairs of imaginary conjugate points of the projective line B 1 , thesymmetric space is isometric to the hyperbolic plane Sl and the manifoldof pairs of coinciding points of this line is represented by the absolute of theplane S1 . In the general case, we have N = n(n + 3)/2, and the space is alsorepresented by a convex domain of the space BA whose point coordinatesare the coefficients ail of equations of quadrics; in this case the absoluteis an algebraic hypersurface of order N - n which represents the imaginaryquadrics (the imaginary cones of second order with real point, line, and planevertices). This hypersurface contains the Veronesean (3.11) which representsthe quadrics decomposing into a pair of coincident hyperplanes.

Similar absolutes can be defined for all symmetric Riemannian spaces withnoncompact groups of motions. For the symmetric space of type All whosemodel is the manifold of nondegenerate Hermitian quadrics, the absoluterepresents the degenerate imaginary quadrics. For the symmetric spaces oftypes AIII, BDI, and CII representing the manifolds of m-dimensional

CSI nplanes of the hyperbolic spaces CSn Stn Stn-1 -n-1

1 1 , and HS1 , the absolutesrepresent the planes tangent to the absolute of the hyperbolic space and its

§6.12. GEOMETRY OF THE CARTAN SUBGROUPS 199

plane generators. Similar imaginary absolutes can be defined for symmetricRiemannian spaces with compact groups of motions.

Note that the Veronesean for the symmetric space AI and the similar sur-face z`' = x`z3 for the symmetric space All, as well as submanifolds ofthe absolutes of the symmetric spaces AIII, BDI, and CII (the latter sub-manifolds represent the parabolic elements which are the plane generators ofthe absolutes of the spaces CS

1n ,

1Stn 3,

S1tn-1 , and HS1n-1) possess the prop-erty that the groups of collineations of the corresponding projective spacespreserving these submanifolds are isomorphic to the groups of motions ofsymmetric spaces.

§6.12. Geometry of the Cartan subgroups

In the paper The geometry of simple groups [ 103] (1927), Cartan studiedgeometric properties of the most important class of symmetric Riemannianspaces the group spaces of compact Lie groups. The geometry of the Car-tan subgroups (which were called in [108] the "subgroups y") was studied inthis paper in utmost detail. As are all subgroups, these subgroups are totallygeodesic surfaces in Lie groups with their Cartan metric. However, unlikearbitrary subgroups, they have the property that every geodesic in a Lie groupwith its Cartan metric lies in one of those subgroups, and for a geodesic com-posed of regular elements such a Cartan subgroup is unique: it consists ofall elements of the group commuting with elements of this geodesic. Sincethe Cartan subgroups are commutative, all their structure constants are equalto zero. Since the Cartan metric in compact simple Lie groups is Rieman-nian, the metric of the Cartan subgroups is locally Euclidean. Thus, thesesubgroups are compact spaces with the Euclidean metric; these spaces are theso-called Clifford forms of Euclidean spaces. (The simplest of such forms isthe Clifford quadric in the space S3 , i.e., the locus of points of the spaceS3 which are equidistant from a straight line of this space and the polarof this line.) Cartan studied closed and nonclosed geodesics of the Cartansubgroups and showed that the shortest closed geodesics are determined bythe characteristic equations of these groups. In particular, for the simple Liegroups of rank two, these geodesics are situated as shown in Figure 6.2 (nextpage) (i.e., closed geodesics of these subgroups are directed along the vectorsrepresenting the roots indicated in Figures 2.1 and 2.4).

Further, Cartan considered the Weyl group of a given Lie group andpointed out that in his paper [82] he showed that in some cases (namely,for the groups An , Dn , and E6), there exist substitutions of roots that donot belong to this group. Next, Cartan defined the "fundamental domain"(D) of the Weyl group bounded by those hyperplanes the reflections in whichgenerate the Weyl group. He also showed that "each infinitesimal transforma-tion of the group y is homologic to one and only one transformation insidethis domain " [ 103, p. 215] and that every internal point of the fundamental


A2 B2

b) c)

D2

d)

G2

a)

FIGURE 6.2

e)

domain (D) is a set of "homologic infinitesimal transformations" dependingon r - n parameters (r is the dimension of the group and n is its rank). (Atpresent, this domain is called the Weyl chamber.) Passing from "infinitesimaltransformations" to finite elements, Cartan defined the "net R" consisting ofthe points with integer coordinates relative to the basis formed by the vectorsrepresenting the fundamental system of roots and the affine Weyl group asthe group generated by the transformations of the Weyl group and the trans-lations preserving the net R. He showed that this group is also generated byreflections in hyperplanes. Next, Cartan defined the "fundamental polytope(P)" of this group (at present, this polytope is called the Weyl alcove) andproved that every finite element of the "group y" is represented at least byone point of this polytope. It follows from this that, in its Cartan metric, theCartan subgroup of a compact simple Lie group is isometric to a polytopecomposed of a few polytopes (P) with the points of the boundary of thispolytope being identified.

§6.13. The Cartan submanifolds of symmetric spaces

In the paper [ 107], Cartan constructed a similar theory for symmetricRiemannian spaces. If such a space is represented by a totally geodesic sur-face in a Lie group with its Cartan metric, then the intersection of this totallygeodesic surface with the Cartan subgroup is called the Cartan submanifold ofthe symmetric space. Since the Cartan submanifold is the intersection of twototally geodesic surfaces, this submanifold itself is a totally geodesic surface.From the properties of the Cartan subgroup, it follows that every geodesicof the symmetric Riemannian space lies in one of these submanifolds, andin the general case, a geodesic lies in a unique submanifold, and that thegeometry of the Cartan submanifolds is locally Euclidean. The dimensionof the Cartan submanifold is called the rank of the symmetric Riemannianspace and is equal to the number of metric invariants of a pair of points ofthis space. Symmetric Riemannian spaces of rank one are the space S" andthe spaces V 2"

, V 4" , and V 16 which are isometric to the spaces CS" andHS" and the plane

4S2

. All geodesics of these spaces are closed and havethe same length in each of these spaces.

§6.14. ANTIPODAL MANIFOLDS OF SYMMETRIC SPACES 201

In the paper [107), Cartan found the ranks, the Cartan submanifolds, andthe form of geodesics for all irreducible symmetric Riemannian spaces. Inparticular, for the symmetric spaces BDI whose models are the Grassmanni-ans of m-dimensional planes of the space S' , the rank is equal to m + 1 ifm < n - m - l . The stationary distances (the lengths of the common perpen-diculars) between rn-dimensional planes can be taken as metric invariants ofthese spaces. In this case, the Cartan submanifolds are represented by thefamilies of m-dimensional planes intersecting with m + l mutually polarstraight lines, and geodesics are represented by one-parameter families ofplanes intersecting with the same m + l straight lines and cutting on themproportional segments. (If m = l , these families are the families of rulingsof ruled helicoids of the space S3 .) These families are called m-helicoids.Note that the ranks of symmetric spaces Ell, EVI, and EVIII are equalto 2, 4, and 8 respectively. This corresponds to the fact that the straight

N2 "'2 ~2lines of the planes (0 ® C)S , (0 ® H)S , and (0 ® 0)S are interpretedby the Grassmannians Gr9,1 , Gr11, 3, and Gr15 , 7 . The term "local abso-lute", which was not used by Cartan, is introduced by analogy with his term"absolute of a symmetric space".

§6.14. Antipodal manifolds of symmetric spaces

In the note on geodesics of spaces of simple groups [96] (1927), Cartanconsidered geodesics of simply connected compact simple Lie groups withthe Cartan metric, gave a classification of these geodesics, and defined theantipodal points the points that can be joined by an infinite set of closedgeodesics. He also defined the antipodal manifold of a point the manifoldof antipodal points of this point. (For the spinor group of the group 03 whichis isometric to a hypersphere of the space R4 , the antipodal manifold consistsof one point.)

Cartan developed the theory of antipodal manifolds in the paper The geom-etry of simple groups [ 103). He showed that, for a simply connected compactsimple Lie group of rank n, each of its points possesses n antipodal mani-folds. This can be explained by the fact that if one characterizes geodesics bythe "angular parameters" (i.e., the coordinates of the tangent vector relativeto that basis in the Cartan subgroup through the geodesic which is definedby the root system), then for geodesics directed towards antipodal points,all nonzero angular parameters are equal to one another. Cartan noted thatthese geodesics are directed along the edges or the facets of the polyhedron(B)

Antipodal manifolds can be also defined in symmetric spaces. The numberof such manifolds for a point of a symmetric space is equal to the rank ofthis space. In particular, for the symmetric spaces of rank one, there is onlyone antipodal manifold: in the case of a hypersphere of the space R12 it isthe antipodal point; in the case of the spaces Cr S" and HSn and the plane


OS2 , the antipodal manifolds are the polar hyperplanes and the polars of thepoints. In these cases the polar images are isometric to the sphere of two,four, and eight dimensions, respectively, and the geodesics joining antipodalpoints belong to these spheres.

Antipodal manifolds can also be defined in nonsimply connected groupsand symmetric spaces. However, in these cases, the geodesics joining thepoints with points of their antipodal manifolds can be unique, and the an-tipodal manifolds are defined as the manifolds consisting of the midpoints ofthe closed geodesics of a certain type. For the symmetric spaces representedby the Grassmannians Gr,, , of rn-dimensional planes of the space Sn ,

the antipodal manifolds are represented by the manifolds of rn-dimensionalplanes that lie in the polar plane of the given rn-dimensional plane or per-pendicular to this polar plane.

§6.15. Orthogonal systems of functions on symmetric spaces

We saw in Chapter 2 that Cartan's works on classification and theory oflinear representations of simple Lie groups were significantly developed inWeyl's paper [Wey3] of 1925. Continuing this work, Weyl, in the paperCompleteness of primitive representations of closed continuous groups [PeW](192 7) written jointly with F. Peter (the words "closed continuous groups" inthe title mean compact Lie groups), showed that all irreducible compact Liegroups (they can always be represented by real orthogonal or complex unitarymatrices) can be obtained by means of orthogonal systems of functions givenon the group. A representation of the space L2(G) of functions f(g) withintegrable square of their modulus given in a compact Lie group G (thisspace is an infinite-dimensional Hilbert space) as the direct sum of finite-dimensional spaces of orthogonal or unitary representations of this groupwas presented in the paper [PeW]. This representation is a generalization ofthe classical expansion of a periodic function in a Fourier series: assigningsuch a function with a period 2ir is equivalent to assigning a function f(t)in the compact group T = R/(2irZ) (R and Z are the additive groupsof the field R and of the ring Z of integers), and the expansion of such afunction in a Fourier series:

(6.39) f(t) =a°

+ akcos kt + 1: bksin kt2

k k

is equivalent to the representation of the Hilbert space L2 (T) as the directsum of a straight line and two-dimensional planes of "vector diagrams" whosevectors represent the harmonics ak cos kt + bk sin kt . Each of these planescan be considered as a complex plane of representation of the group T bythe complex numbers elk` . These representations are called the charactersof the group T. Similar characters of any commutative group form a groupthemselves whose identity element is the representation of all elements of thegroup by the number 1 (the "unit character").

§6.15. ORTHOGONAL SYSTEMS OF FUNCTIONS ON SYMMETRIC SPACES 203

For a simple commutative group, the group of characters is isomorphic tothe group itself. The group of characters of an infinite discrete commutativegroup is a compact commutative group, and, conversely, the group of char-acters of any locally compact commutative group (all noncompact Lie groupsare of this type) is a commutative group of the same type. In particular, thegroup of characters of the group T is isomorphic to the group Z , and con-versely. The group of characters of the group R is isomorphic to the groupR itself.

Departing from the paper [PeW] of Peter and Weyl, in the paper On thedetermination of a complete orthogonal system of junctions on a closed sym-metric Riemannian space [117] (1929), Cartan constructed a similar theoryfor functions defined on a compact symmetric space E that forms a Hilbertspace Lz(E) . The spaces Lz(G) and LZ(E) are infinite analogues of thecomplex Hermitian Euclidean space CRS where the role of vectors is playedby the complex-valued functions f (x) , the role of the inner product of vec-tors is played by the integral

(6.40) (f, g) = f(x) g (x) d V,E

where d V is the volume element of the space or the group, and the role ofthe square of the modulus of a vector is played by the integral

(6.41) jfj 2 = I

E

If we denote by ax the point of the symmetric space E which is obtainedas a result of application of an element a of the group of motions of thisspace to a point x of this space, the transformation

(6.42) TQ f (x) = f (ax)

is a linear transformation in the space LZ(E) . Since the group consideredby Cartan is compact, a linear transformation of this group arising in thismanner can be split into finite-dimensional transformations of type:

(6.43) f,(ax) = aijf (x).

The sequence of functions f1(x), f2(x), ... , fp(x) defining such trans-formations was called by Cartan the fundamental sequence of functions. Car-tan showed that the functions of these fundamental sequences can be chosento be orthogonal; i.e., they satisfy the conditions (f1, f.) = 8rj . Thus, onecan construct an orthogonal sytem of functions on a symmetric space. If weexpand an arbitrary function f(x) of the space LZ(E) with respect to thesefunctions:

0

(6.44) f(x) =°ifi(x)


then the coefficients a; of this expansion possess the property that the seriesof their squares is convergent to the integral:

D

(6.45) a? = f(X)IZdV = f12jplaying the role of the square of the magnitude of a vector of this functionalspace.

Thus, the functions f(x) of a fundamental sequence are analogues of thefunctions cos kt and sin kt determining the Fourier series and forming asystem of orthogonal functions on the group T.

Cartan also defined "zonal functions" on symmetric spaces, which are ana-logues of the spherical functions of the space R3 .

§6.16. Unitary representations of noncompact Lie groups

The consideration of systems of functions on homogeneous spaces whosefundamental groups are noncompact real Lie groups led to the theory ofunitary representations of noncompact simple and quasi-simple Lie groups,i.e., representations of these groups by unitary operators in Hilbert spacesL2 (E) of functions on homogeneous spaces whose fundamental groups aregiven groups. A unitary operator in the space L2(E) with inner product(f , g) is a linear operator U of this space which satisfies the condition(Uf, Ug) = (f, g). These operators are infinite analogues of matrices ofthe group C U, .

The first work on such representations was the paper on unitary represen-tations of the inhomogeneous Lorentz group [Wig] (1939) by the physicistWigner. In this paper, Wigner constructed unitary representations of a qua-sisimple group of motions of the pseudo-Euclidean space R4 , i.e., the space-time of special relativity. Such representations are characterized by one realand one integer parameters. Wigner connected this fact with the phenomenonthat the elementary particles in physics also characterized by one continuousparameter the mass and one discrete parameter the spin. Wigner con-cluded from this that these representations are important in physics.

In 1943 Gel'fand and Dmitry A. Raikov (1905-1980) published the paperIrreducible unitary representations of locally bicompact groups [GeR]. (Fromthe 1920s to the 1940s, in Soviet mathematical literature the word "bicom-pact" had the meaning "compact", and the word "compact" had a differentmeaning.)

In 1947, in the paper Irreducible unitary representations of the Lorentzgroup [Bag] by Valentine Bargmann (1908-1991) and in the similarly titledpapers [GeN] by Gel'fand and Naimark and [Harl] by Harish-Chandra, uni-tary representations of the noncompact simple Lorentz group the group ofrotations of the space R4 were found. Note that the paper [Har fl, the Ph.D.thesis of Harish-Chandra, was written by him when he was a young physicist

§6.16. UNITARY REPRESENTATIONS OF NONCOMPACT LIE GROUPS 205

studying particle physics under the supervision of Paul Adrian Maurice Dirac(1902-1984).

Since the Lorentz group O is locally isomorphic to the group CSL2(which is its spinor group), Gel'fand and Naimark posed the general prob-lem of studying infinite-dimensional unitary representations of all classicalcomplex simple Lie groups considered as noncompact real simple groups (theSatake graphs of such groups consist of two copies of the Dynkin graph ofthe corresponding complex group, provided that the corresponding dots ofthese graphs are joined by two-sided arrows). This theory was presentedby Gel'fand and Naimark in their monograph Unitary representations of theclassical groups [GeN2] (1950). Later Gel'fand and Graev solved the con-siderably more difficult similar problem for arbitrary noncompact groups ofinfinite series. These results were briefly presented in their note Unitary rep-resentations of the real simple groups [GeG] (1952) and, in more detail, inthe Graev paper [Grv] (1958) under the same title.

Harish-Chandra, who became a famous mathematician, independentlyconstructed the theory of these representations in a series of papers concludedby the paper Representations of semi-simple Lie groups [Hart] (1951-1956).In the papers of Gel'fand and his co-workers as well as in the papers ofHarish-Chandra, the Hilbert spaces L2(E) of functions in various parabolicspaces whose fundamental groups are the noncompact groups under consid-eration were investigated. For construction of the principal series of unitaryrepresentations of a noncompact group G, they considered the Hilbert spaceL2 (G/B) of functions given in the parabolic space G/B defined by the Borelsubgroup B . For construction of the "degenerate series", they considered theHilbert space L2 (G/B) of functions given in the parabolic space G/B de-fined by an arbitrary parabolic subgroup P. (The subgroup B is defined byall positive roots of the group G while an arbitrary parabolic subgroup P isdefined by a part of these roots or even by one of them.) For the principalseries of unitary representations of the Lorentz group, which is locally iso-morphic to the group of motions of the hyperbolic space S1 whose parabolicimages are the points of the absolute, the space G/B can be identified withan oval quadric in P 3 and with the extended complex plane. Thus, in thiscase, one can take the space of functions f (z) of a complex variable as thespace L2 (G/B) . (In this case, the group G is the group of linear-fractionaltransformations of the complex plane.)

The roots and the weights of finite-dimensional representations of semisim-ple Lie groups are characters of the Cartan subgroups of these groups. For theinfinite-dimensional representations of these groups, the role of characters isplayed by certain distributions which are linear functionals of the Hilbertspace. These distributions themselves, as characters of a locally compactcommutative group, compose locally compact commutative groups which arethe direct products of a certain number of infinite discrete groups Z and a


certain number of the groups R. The elements of the groups Z determinethe integer parameters of unitary representations, and the elements of thegroups R determine the real parameters of these representations.

For a complex simple Lie group of rank n , the maximal compact subgroupis a compact group of the same class and rank, and the Cartan subgroup isthe direct product of n groups T and n groups R. Thus, its group ofcharacters is the direct product of n groups Z and n groups R, and theunitary representations of this group are determined by n integers and nreal parameters.

For an arbitrary real noncompact group, there are a few nonisomorphicCartan subgroups each of which determines a series of unitary representationsof the group. If the Cartan subgroup is the direct product of r groups Tand m groups R, then its group of characters is the direct product of 1groups Z and m groups R, and the unitary representations of this groupare determined by r integers and m real parameters.

In the cases when the Cartan subalgebra is compact, i.e., it is the directproduct of n groups T, its group of characters is the direct product of ngroups Z, and there are only the discrete series of unitary representationsdepending on n integers (the group of characters of the Cartan subgroup ofa compact simple Lie group of rank n is isomorphic to the direct product ofn groups Z).

The theory of unitary representations of quasi-simple Lie groups is sim-ilar to the theory of unitary representations of noncompact semisimple Liegroups.

The analogues of the classical spherical functions in symmetric spaces,whose study was initiated by Cartan in the paper [1171, are closely connectedwith the unitary representations of compact and noncompact Lie groups.

We saw earlier that the functions cos kt and sin kt , where k is an integer,determine representations of the group T isomorphic to the group 02 ofrotations of the Euclidean plane R2 . Similarly, the hyperbolic functionscosh pt and sinh pt where p is a real parameter determine representations ofthe group 02 of rotations of the pseudo-Euclidean plane R i . The sphericalfunctions of the space R3 are the Legendre polynomials Bn (cos 0) where 6is the latitude of a point on a sphere and n is an integer. The sphericalfunctions of the space R 3 are the Legendre polynomials B (cos 0) where 6

Pis the analogue of the latitude and p is a real parameter. The analogues of thespherical functions in the plane R2 are the Bessel functions Jo(pr) where ris the first polar coordinate of a point in the plane and p is a real parameter.At present, similar theories are constructed for many symmetric spaces withcompact and noncompact semisimple and quasisimple fundamental groups(see the paper [Gel I) by Gel'fand).

In the book Special functions and the theory of group representations [V i l]{ 1965), Vilenkin showed that all classical special functions can be considered

§6.17. THE TOPOLOGY OF SYMMETRIC SPACES 207

as elements of matrices of infinite order determined by linear operators ofrepresentations of simple and quasisimple Lie groups.

Note that the hypergeometric functions are determined by a representationof the group O of rotations of the pseudo-Euclidean space R 3 , the Hankel-MacDonald functions are determined by a representation of the group ofmotions of the plane RZ , the Gegenbauer polynomials are determined by arepresentation of the group on of rotations of the space R' , and the Hermitepolynomials are determined by a representation of the group of motions ofthe space Rn .

§6.17. The topology of symmetric spaces

While in his thesis and in the papers of the 1890s and the early 1900s, fol-lowing Lie, Cartan restricted himself to considering only the neighborhoodsof the identity element of a Lie group, in his papers of the 1920s and the1930s, he was interested in the topological structure of Lie groups in the largeas well as in the topological structure of the compact symmetric spaces closelyconnected with the simple Lie groups.

In the paper The geometry of simple groups [103] (1927), Cartan investi-gated in detail simply connected and nonsimply connected compact simpleLie groups and, for the latter groups, he found the "connection groups" thehomotopy Poincare groups. Cartan showed that these "connection groups"are finite commutative groups isomorphic to the centers of simply connectedsimple groups of the same type and that these groups are isomorphic to thequotient groups of the weight lattice of the group which Cartan called the"net R" by the root lattice which is its subgroup. (We defined these lat-tices in Chapter 2.) Cartan called the orders of the "connection groups" theconnection indices of simple Lie groups. (In Chapter 2, we saw that theconnection indices, whose name is explained by their relation with the con-nection groups, are equal to the determinants of the Cartan matrices of thecorresponding Lie groups.)

In the paper The theory of finite continuous groups and Analysis situs [ 128](1930), Cartan showed that a noncompact simple Lie group, and, in particu-lar, a complex simple Lie group, is homeomorphic to the topological productof its maximal compact subgroup and a Euclidean space.

In Chapter 4, we already mentioned the paper On the integral invariants ofcertain closed homogeneous spaces and topological properties of these spaces[118] (1929). In this paper, Cartan showed that, if a compact homogeneousspace is a symmetric Riemannian space (in particular, the group manifold ofa compact simple Lie group with its Cartan metric), any integral invariant ofthis space is an integral of an exact differential form.

In the paper On the Betti numbers of spaces of closed groups [ 111 ] (1928),Cartan applied the theory of exterior differential forms for computing theBetti groups of simply connected compact ("closed") simple groups in theclasses An and Bn , i.e., the groups CSUn+1 and the spinor group of the


group 02n+1 . Here, for the first time, Cartan introduced the term Poincarepolynomial for the polynomial Ei pit` whose coefficients are the Betti num-bers pi and showed that the Poincare polynomials of simply connected com-pact groups A. and Bn are equal to

(6.4b)

and

(6.47)

n(t2t+1 + 1)

i=1

nlt4i-1 + 1)

i=1

respectively.The method used by Cartan in this paper was developed by Pontryagin

in the paper On the Betti numbers of Lie groups [Pon 1 ] (193 5), where hecomputed the Poincare polynomials of simply connected compact simplegroups of all four infinite series: for the groups An and Bn Pontryaginarrived at the same expressions (6.46) and (6.47) as Cartan; for the groupCn Pontryagin found the same expression (6.47); and for the group Dn , thespinor group of the group 02n , he got the polynomial

n-1(6.48) (t2' + 1) lt4i-1 + 1).

i=1

Later the Pontryagin method was also applied to compact exceptional sim-ple Lie groups. In the paper The Betti numbers of exceptional simple Liegroups [BoC] (1955), A. Borel and Chevalley calculated the Betti numbers forall simply connected compact simple Lie groups in the exceptional classes.The Poincare polynomials of simply connected compact simple Lie groupscan be expressed by one formula:

(b.49)n

11lt2a,+1 + 1)

i=1

where the integers a; are the exponents (2.34) and (2.35) of simple Lie groups{formulas (6.46), (6.47), and (6.48) are particular cases of formula (6.49)).The coincidence of the numbers a; in formula (6.49) with exponents (2.34)and (2.35) was explained by A. J. Coleman in the paper The Betti numbersof the simple Lie groups [Coll] (1958).

When Cartan found the Poincare polynomials (6.4b) and (6.47), he in factfound the exponents for the simple groups A. and B. .

In the paper [118], mentioned earlier, Cartan found the Poincare polyno-mial of the space CSR (the symmetric space AIV) in the form:

(Fi SO) t2n -I- t 2n-2 ... -I- t2A-

1 =t2n+2 _ 1

t _1

§6.18. HOMOLOGICAL ALGEBRA 209

He also found the "Clifford form" of the spaceCS2n+ 1

which is obtainedby identification of its points X(x2t , x2i+1) with the points 'X('x2` , ix2`+1)obtained from the points X by symmetry (6.34). This symmetry defines,in the space C+ 1 , a paratactic congruence, isometric to the space HSn .

In addition, the lines of this congruence are isometric to two-dimensionalspheres, which under the Cartan identification become elliptic planes S2 .These facts imply that the quotient of the Poincare polynomial found byCartan by the Poincare polynomial t2 in a plane S2 is equal to

(hS 1) ton t4n-4 ... !4 1

t4n+4 _ 1t4-1

This quotient is the Poincare polynomial of the space HSn (a particular caseof the symmetric space CII). Formula (6.51) was used by A. Borel in hispaper [Borl ]. In this paper, Borel also found the Poincare polynomial of theplane a (the symmetric space FII) in the form:

(6.52) tlb +t 8 + 1 =

t24

t 8

analogous to polynomials (6.50) and (6.51).Another generalization of the polynomial (6.50) was found by Ehresmann

in his thesis [Eh 11; it is the Poincare polynomial of the Grassmannian CGrn m(i.e., the symmetric space AIII) and has the form:

(t2' n+- llrt2n - 1`... (t2n-2m-1 - 1)(b.53) (t2m+1 _ 1)(t2m_1)...(t2_ 1)

In the paper on the topological properties of complex quadrics [137] (1932),Cartan considered the topological properties of another symmetric Rieman-nian space the (2n )-dimensional real space represented by the quadricEiW) 2 = 0 of the complex projective space CPn+ 1 . In the space CPn+1 ,

Cartan introduced the metric of the Hermitian elliptic space CSn+ 1 . As aresult, the quadric in the space CPn+1 becomes the Riemannian manifoldYen . Cartan noted that the space Yen is a symmetric space. It is not difficultto see that this space admits the representation in the form of the Grassman-nian Grn+1

1(the absolute of the space Sn+1 is an imaginary quadric in the

space CPn+1 , and to any straight line of the space Sn+1 there correspondsa pair of imaginary conjugate points of this quadric at which the straightline intersects the quadric). In this paper, Cartan also computed the Bettinumbers of this symmetric space.

§6.18. Homological algebra

Cartan's paper [ 118], where the topological problem of computing the Bettinumbers in a compact Lie group was reduced to the purely algebraic problem


in the corresponding Lie algebra, gave birth to a new algebraic disciplinewhose terminology was adopted from the homology and cohomology theories.For this reason, this discipline got the name "hmological algebra". Followingthe homology theory of Lie algebras constructed by Cartan, the cohomologytheory of associative algebras was developed by Gerhard Paul Hochschild(b. 1915) in the paper On the cohomology groups of an associative algebra[Hoc] (1945). The cohomology groups of arbitrary groups was constructedby Samuel Eilenberg (b. 1913) and Saunders MacLane (b. 1909) in thepaper Cohomology theory in abstract groups [EM] (1947), and the cohomologygroups of Lie algebras over an arbitrary commutative ring were defined byChevalley and Eilenberg in the paper Cohomology theory of Lie groups andLie algebras [ChE] (1948).

The cohomology groups of abstract groups can be defined as follows. Leta multiplicative group G and an additive group A (as the latter group theadditive group Z of integers or the finite cyclic group m often is taken) begiven, where the elements of the group G operate on the left of the elementsA of the group A and x(A1 + A2) = xA1 +xA2, x2(x1A) = (x2x1)A, 1

-A = A.

Define an n-dimensional cochain of G over A as a homogeneous functionF(x0 , xl , ... , xn)(F(xx0 , xxl , ... , xxn) = xF(x0 , xl , ... , xn)) with itsvalues in A. Since Fl + F2 is again an n-dimensional cochain, these cochainsform an additive group Cn (G, A) . The coboundary of this cochain is an(n + 1)-dimensional cochain

bF(x0,x1,. xn)= >(-l)'F(xo, x1,. .. , xi-l, xi+1, . .. , xn).

The coboundaries have the following properties: b (F1 + F2) = b Fl +oF2 , 66F = 0. The n-dimensional cochains satisfying the conditionOF = 0 are called the n-dimensional cocycles. The coboundaries are partic-ular cases of cocycles. The n-dimensional cocycles and the n-dimensionalcoboundaries form commutative groups which are denoted by Z n (G, A) andBn (G, A), respectively. The second of these groups is a subgroup of the firstone. The quotient group of the first group by the second one is called ther-dimensional cohomology group Hn (G, A) of G over A.

If, in this definition, we change the group G to an associative algebraA or to a Lie algebra G, we obtain the cohomology groups Hn (A, A) andHn (G, A) of these algebras. In both cases, since the functionsF(x0, x1, ... , xn) are homogeneous, they can be written in the form of

xo xi' ...X ipi where A . is a tensor. In themultilinear forms: A 1011 ... , loin ...i,

case of Lie algebras, since the operation of commutation in them is skew-symmetric, this tensor is skew-symmetric in all indices and defines an exteriorform considered by Cartan.

However, if the Cartan theory was related only to the complex Lie groups,this theory is related to the Lie algebras of arbitrary Lie groups.

All aspects of this theory were presented as a single theory in the bookHomological algebra [CaE] (1956) by Henri Cartan and S. Eilenberg.

CHAPTER 7

Generalized Spaces

§7.1. "Affine connections" and Weyl's "metric manifolds"

We have already pointed out the importance of Einstein's discovery of gen-eral relativity for the development of Riemannian and pseudo-Riemanniangeometry. According to the general theory of relativity, the space-time andthe gravitational field of matter are described by means of a four-dimensionalpseudo-Riemannian manifold Va whose curvature is connected with the den-sity of matter. The problem of construction of a unified field theory posedby Einstein has played an exceptional role for the creation of further gener-alizations of the notion of space. Einstein departed from the idea that allphysics could be reduced to mechanics and electrodynamics, that the gravi-tational field of matter is already taken into account in the geometry of thespace I Ia , and that for a description of a unified theory of a physical fieldit is necessary to construct a more general spatial scheme which would de-scribe not only the gravitational field but also the electromagnetic field. InEinstein's special and general relativity, the electromagnetic field, which wasdefined in classical electrodynamics by the tension vector E of the electricfield and the tension vector H of the magnetic field, is characterized by asingle skew-symmetric tensor F'3 , i , j = 1, 2, 3, 4, of the electromagneticfield whose coordinates are connected with the vectors E and H and thespeed of light by the relations

F41= F14=cEi, F21__F12=H3,

F13=_F31 _H2, F32=_F23_H1.

The first attempt to construct a geometry more general than the Rieman-nian or pseudo-Riemannian geometry, which would describe both the grav-itational field and the electromagnetic field, was made by Weyl in his paperPure infinitesimal geometry [Wey 1 ] (1918). In this paper, Weyl distinguishedthree types of manifolds: "manifold-place" (situs - Mannigfaltigkeit), whichhe identified with the "empty world", i.e., with the world without matter, the"affinely connected manifold" (affin zusammenhangende Mannigfaltigkeit),which he understood to be a manifold with a parallel displacement of vec-tors and which he called the "world with a gravitational field"; and the "metric

211

212 7. GENERALIZED SPACES

manifold", which he also called the "ether" and which he understood to bethe "world with the gravitational field and the electromagnetic fields". Ac-tually, Weyl's "affinely connected manifold" coincides with the space-timeof the Einstein theory of general relativity, i.e., with a pseudo-Riemannianmanifold. Weyl's "metric manifold" is a generalization of the Riemannianand pseudo-Riemannian manifolds. If in the spaces Y" and Y" a paralleldisplacement of vectors induces an isometric mapping of the tangent spacesTx (Y") and 7x

(Vn) onto tangent spaces in infinitesimally near points, in theWeyl "metric manifolds", the mapping of the tangent spaces (which as wasthe case for the spaces V" and Y" are the Euclidean and pseudo-Euclideanspaces Rn and R!) onto the same kind of tangent spaces in infinitesimallynear points takes place. However, in the Weyl "metric manifolds", thesemappings are not isometric mappings anymore they are similarity mappings(i.e., transformations preserving the angles between vectors and multiplyingthe linear dimensions by real numbers).

Weyl expressed the same ideas in his book Space-Time-Matter [Wey2j. Thefirst edition of this book was published in 1918. This edition was followedby a series of new editions (the fifth was in 1923). In 1922, the book wastranslated into French and later into English. The book became very popularthroughout the mathematical world. (In this book the well-known point-vector axiomatics of n-dimensional affine and Euclidean spaces En and R"was presented.)

Although Einstein was occupied with the construction of a unified fieldtheory for a few decades, neither he nor other physicists were able to con-struct such a theory. On the contrary, as physics was developing, new formsof interactions of matter (the "weak interaction" and the "strong interac-tion") were discovered, and they were not reduced to either mechanical orelectromagnetic interactions. Nevertheless, for multidimensional differentialgeometry, the impetus that it received from physicists who were trying toconstruct a unified field theory was very helpful.

§7.2. Spaces with af'ine connection

The term "affinely connected manifold" introduced by Weyl soon receiveda wider meaning than that given by Weyl. Namely, it was applied to suchspaces A" whose tangent spaces Tx (A") are affine spaces E and for whichthe mappings of tangent spaces in infinitesimally near points are defined andare affine mappings of these spaces. Such spaces were defined by Schoutenwho arrived at them while generalizing the parallel displacement of vectorsin a Riemannian manifold which he discovered simultaneously with Levi-Civita. This was the reason that he called the mapping of tangent spacesEn of the space A" a displacement (Ubertragung). Schouten defined thesespaces in his paper On different kinds of displacements which can be taken as

§7.2. SPACES WITH AFFINE CONNECTION 213

a basis of differential geometry [Sco I] (1922). The Schouten theory was pre-sented in detail in his book Ricci calculus [Sco2] (1924) (as was mentionedabove, Ricci calculus is one of the names of the tensor calculus). Schoutendefined the space An as a manifold with coordinates at each point of which

are given. Under coordinate transformations, thesefunctions ri k = rkiJ

functions are transformed according to the same rule as the Christoffel sym-bols of a Riemannian manifold; however, in general, these functions cannotbe expressed in terms of the metric tensor g,j by formulas (6.10). Thus, inthe general case, in the space A', it is impossible to define the lengths oflines and the angles between lines, but it is possible to define geodesics thatare integral curves of differential equations (6.9). Furthermore, the parame-ter s for which these differential equations preserve their form is defined upto an "affine transformation" s - as + b. This is the reason this parameteris called the affine parameter of geodesics.

With each point x of the space An there is associated a tangent spaceTX (A

n) which is an affine space En similar to the tangent spaces T( V n )and T, (Y") . As was the case for the spaces Tx (Vn) and T, (Y n) , in thespace A n

, the contravariant and covariant vectors a! and al and the tensorsT`-'...`-1 are defined. A parallel displacement in the space An is defined by

1i 14

means of the functions rJt.k according to the same formulas (6.19) and (6.20)as for th a spaces Tr (Vn) and TX (V") .

A somewhat more general definition of a space with an affine connectionwas introduced by Cartan in the paper on manifolds with an affine connec-tion and the general relativity theory consisting of three parts [66] (1923),[69] (1924), and [80] (1925). The book [209a] contains English translationsof the papers [66], [69], and [80], and [208] contains Russian translations ofCartan's papers on the spaces with affine, projective, and conformal connec-tions. Explaining the term "affine connection", Cartan wrote in the forewordto the paper [66] that "the expression `affine connection' is borrowed fromH. Weyl (here Cartan made reference to the Weyl book Space-Time-Matter),although it will be used here in a more general context" [209a, p. 25].

We will denote the Cartan spaces with an affine connection by the samesymbol An which was used for the Schouten space. As Schouten did, Cartandefined an affine transformation of a tangent space Tx (A n) onto a tangentspace T,1 (A n) at an infinitesimally near point, but the Cartan mapping wasmore general than the Schouten mapping. To define his mapping, Cartanconsidered a frame {x, e, } in the space Tx (An) and a frame {x', e) in thespace Tx, (An) and defined the principal part of the mapping by the relations:S# (x) = x + A, SP (e,) = e, + den where d x = w' e1 and de, = cc) a TheseJ Jformulas precisely coincide with derivational formulas (5.5) of a frame inthe affine space En ; however, here they are not completely integrable as theywere in the space En since the forms cv' and co in these formulas satisfy

J


the structure equations of a space with an affine connection:

dw' =fOk A wk + 'S' kW A kk j 1

Jk

1d( =Wi A (4 + A(0 1

which are more complicated than formulas (6.21). In formulas (7.1), S1

jk is

the torsion tensor and R.kl is the curvature tensor of the space An . Deriva-tional formulas (5.5) admit integration only along a curve x = x(t) belongingto the space An , and their one-dimensional integrals define "developments"of a space A n with an affine connection onto an affine space En .

If the vectors ei form the natural frame in An , i.e., if ei = (9/(9x ` andc0` = dx', the forms w' defining an affine connection on An are expressedin the form rte'. = r' kdxk . The coefficients 17'.k are called the coefficientsof affine connection. Unlike the similar Schouten coefficients, they are notassumed to be symmetric: ri.k = rk . Thus, the tensor

(7.2) rk rk1 1 1

arises. This tensor is called the torsion tensor. In addition, in the spacesunder consideration, there is also the curvature tensor:

(7.3)ark or11 i

R +r ijk1 oxj oxk 1k hl 11 hk

defined by a formula similar to formula (6.12) for calculating the Riemanntensor of the spaces Tx (Vn) and Tx { V") .

If SIk = 0, the space An is called a torsion free space, and if Rj.kl = 0)A

it is called a curvature-free space. Since in the spaces Vn and Vn , we have

ri.k = rk . , these spaces can be considered as torsion-free spaces with an1

affine connection, and the spaces En and E1 are both a torsion-free andcurvature-free space.

As we found for the spaces Vn and Vn, the result of a parallel displace-

ment of a vector a = {a'} along a closed contour defined in a neighborhoodof a point x of the space An by the differentials dx' and 5x` of the co-ordinates differs from the original vector a = {a`} by an increment whichis equal to the vector with coordinates Up to infinitesimals

of higher order, the vector with coordinates Sjk dx" axk in the tangent space

T,(An) is equal to the path between the end of the segment 5x', displacedin a parallel way from the point x along the segment dx` , and the end ofthe segment d x` , displaced in a parallel way from the point x along thesegment ax` . This gives a geometric meaning to the torsion and curvaturetensors of a space An with an affine connection.

§7.3. SPACES WITH A EUCLIDEAN, ISOTROPIC, AND METRIC CONNECTION 215

§7.3. Spaces with a Euclidean, isotropic, and metric connection

Cartan defined spaces with an affine connection in Chapter II of his paperOn manifolds with an affine connection and the generalized relativity the-ory [66). In Chapter I of this paper entitled Dynamics of continuous mediaand the notion of affine connection of the space-time, Cartan analyzed thespace-time of the theory of general relativity considering it as a space with a"pseudo-Euclidean connection" whose tangent spaces are pseudo-Euclideanspaces R4 . In the same chapter, he also considered the space-time of theclassical Galilei-Newton mechanics "from the point of view of the Einsteintheory", i.e., he considered this space-time as a space with an "isotropic con-nection" whose tangent spaces are isotropic spaces I4 . Cartan did not intro-duce this notion, but he wrote the transformations of spatial coordinates andtime under a passage from one inertial coordinate system of classical me-chanics to another, and these transformations coincide with the coordinatetransformations of the space I4

After he defined the spaces A' with an affine connection in Chapter IIof the paper (66), in Chapter III Cartan introduced the spaces with a metricconnection and the spaces with a Euclidean connection.

By analogy with Weyl's term "metric manifolds", Cartan used the termspaces with an affine connection for those spaces A n whose tangent spacesT(A) are Euclidean spaces Rn in which the group of similarities acts. Inthis case, the forms w are equal to each other (Cartan denoted these formsby cv ), and the forms co , i j , are connected by relation (6.31). The struc-ture equations of the spaces with a metric connection have the same form(7.1) as for the general spaces An . The mappings of the tangent spaces Rnof this space onto the tangent spaces in infinitesimally near points are simi-larity transformations. Cartan called a space with a Euclidean connection aparticular case of a space with a metric connection for which the form cois identically zero, i.e., the case when the mapping of the tangent spaces ofthis space onto the tangent spaces in infinitesimally near points are isome-tries. The Riemannian manifolds Vn are a particular case of spaces with aEuclidean connection for which the torsion tensor Sik is identically equalto zero. The "metric manifolds" of Weyl (at present called spaces with aWeyl connection) are distinguished by the same condition among the spaceswith a metric connection. Spaces with a pseudo-Euclidean connection whoseparticular cases are the pseudo-Riemannian spaces Y" can be defined in thesame way.

Cartan's memoir On manifolds with an affine connection and the generalrelativity theory [66], (69), and (80) was preceded by a series of notes de-voted to the attempts to construct a unified field theory. In the note On ageneralization of the notion of Riemannian curvature (58) (1922), Cartan de-fined a space with a Euclidean connection with torsion, and in the paper Ongeneralized spaces and relativity theory [59] (1922), he defined a space with


a metric connection and suggested characterizing the space-time as a spacewhose tangent spaces are pseudo-Euclidean spaces R4 .

In the paper Recent generalizations of the notion of space [711 (1924),Cartan gave a simple example of a space with a Euclidean connection: asphere on which the parallel displacement of tangent vectors is defined insuch a way that an initial vector and its parallel displacement compose equalangles with the meridians passing through their initial points. In this case,geodesics are loxodroms (rhumb lines). Since this parallel displacement doesnot depend on the path of displacement, it is an absolute parallelism. Cartangave the same example in his letter of May 8, 1929, to Einstein in connectionwith the fact that Einstein, who, independently of Cartan, arrived at thenotion of absolute parallelism in 1928 (he called it "Fernparallelismus"), triedto apply this notion in his unified field theory. This letter started an intensivecorrespondence between Cartan and Einstein concerning absolute parallelism.This correspondence was published with English translation in the book [210](1979).

§7.4. Afllne connections in Lie groupsand symmetric spaces with an af'ine connection

Although the notions of a space with an affine connection were initiallycreated by Schouten and Cartan independently, in 1926 two joint papersof both geometers were published: On the geometry of the group-manifold ofsimple and semi-simple groups [911 and on Riemannian geometries admittingan absolute parallelism [921. Both papers were concerned with Riemanniangeometry, but in both cases, one way or another, the geometry of a space withan affine connection was discussed. In the first of these notes, the authorsconsidered three affine connections associated with any Lie group. In thisnote these connections were called the (+)-connection, (-)-connection, and(o)-connection. The authors indicated that for simple and semisimple Liegroups, the latter connection is determined by the Riemannian or pseudo-Riemannian Cartan metric of this group. As we noted in Chapter 3, inthe second note the authors considered the absolute parallelisms (3.15) and(3.16) in the elliptic space S7 and similar absolute parallelisms in an arbitrarysimple compact Lie group with the Riemannian Cartan metric.

The theory of three affine connections was presented in more detail byCartan in his paper The geometry of transformation groups [10 1 ] (1927). Inthis paper Cartan called these connections the "absolute parallelisms of thefirst and the second type". The parallel displacement of vectors in the firsttwo connections is determined by mappings of neighborhoods of a point aonto a neighborhood of a point b by means of the translations x -' (ba' )xand x -* x(ba- 1) . Since these mappings do not depend on the path joiningthe elements a and b, the vector obtained as the result of a translationin both connections along a closed contour coincides with the initial vector.

§7.4. AFFINE CONNECTIONS, LIE GROUPS AND SYMMETRIC SPACES 217

This proves that these connections are curvature-free, i.e., they define anabsolute parallelism. At the same time, each of these connections possesses atorsion, and for the first of these connections the components of the torsiontensor Sr coincide with the structure constants c` of the Lie group andfor

.k

for the second one they differ from them in the factor -1 . Both of theseconnections are invariant under transformations of the group.

The third connection defined by Cartan on a Lie group, which is alsoinvariant under transformations of the group, is torsion-free. It is determinedby its geodesics and their affine parameter: the role of geodesics through theidentity element of the group is played by one-parameter subgroups, and therole of their affine parameter is played by their canonical parameter t. Forthe latter parameter the product of elements x(tl) and x(t2) of a subgroupcoincides with the element x (t1 + t2) , and this parameter is defined up to areal factor. The role of geodesics not passing through the identity elementof the group is played by cosets of one-parameter subgroups, and the affineparameter on these cosets is defined up to an affine transformation t -+ at +b. The curvature tensor of this space is expressed in terms of the structureconstants c`.k of the group by the same formula (6.27) which defines theRiemann tensors of the Riemannian or pseudo-Riemannian Cartan metricin the simple or semisimple Lie groups. This shows that in these groups theCartan torsion-free affine connection is defined by the invariant Riemannianor pseudo-Riemannian metric (6.26) of these groups.

The torsion-free affine connection in Lie groups defined by Cartan in thepaper The geometry of transformation groups is a particular case of a connec-tion of a symmetric space with an affine connection. In spaces with such anaffine connection, the mapping along geodesic lines preserves the affine con-nection, i.e., this mapping transfers geodesics into geodesics and preservestheir affine parameter. As he did for symmetric Riemannian spaces, Cartanshowed that condition VhRi-kl = 0 analogous to condition (6.25) is neces-sary and sufficient for a torsion-free space with an affine connection to bea symmetric space. These spaces can be realized in Lie groups in the formof totally geodesic surfaces oci passing through the identity element of thegroup and generated by the reflections in the points of these spaces. Herea is a reflection in an arbitrary point of a space with an affine connectionand a0 is a reflection in a certain fixed point of this space. As in the caseof symmetric Riemannian and pseudo-Riemannian spaces, the Lie algebra ofthe Lie group generated by reflections in points of a symmetric space with anaffine connection admits the "Cartan decomposition" (2.43) where the subal-gebra H is the Lie algebra of the stationary subgroup of a point of this space(the isotropy group), and the subspace E can be considered as the tangentspace to a totally geodesic surface in the group in which the symmetric spacewith an affine connection is realized or, equivalently, as the tangent space toa symmetric space with an affine connection.

The curvature tensor of a symmetric space with an affine connection is


expressed in terms of the structure constants ca and caw of the group gen-erated by reflection in points by the same formula (6.29) which defines theRiemann tensor in symmetric Riemannian and pseudo-Riemannian spaces.

As in the case of tangent spaces to symmetric Riemannian spaces, tangentspaces to symmetric spaces with an affine connection are closed with respectto the operation [[X, Y], Z], and therefore they are triple Lie systems.

As in the case of symmetric Riemannian spaces, in symmetric spaces withan affine connection, the Loos quasigroup (see [Loo]) is defined which as-sociates to any two points x and y of this space the point z that is thereflection of the point x in the point y along geodesics of this affine con-nection.

The Loos quasigroups, which were defined by Loos in symmetric Rie-mannian spaces and symmetric spaces with an affine connection, are smoothquasigroups. The idea of Loos was further developed by A. J. Leger in thepaper Generalized symmetric Riemannian spaces [Leg] (1957) and in the pa-per Affine and Riemannian s-spaces [LeO] (1968) written jointly with MorioObata (b. 1926). The Leger s-spaces which generalize symmetric Rieman-nian spaces and symmetric spaces with an affine connection were also studiedby Fedenko in the paper Regular spaces with symmetries [Fe2] (1973) and inthe book Spaces with symmetries [Fe3] (1977). Application of quasigroupsand loops to symmetric spaces and their generalizations was first suggestedby Mishiko Kikkawa in the paper On local loops on affine manifolds [Kik](1964) and was extended to generalizations of spaces with an affine con-nection which differ from the spaces with an affine connection in that theyhave fewer requirements on the differentiability of functions under consid-eration by Sabinin in the paper Methods of the non-associative algebra in thedifferential geometry [Sab] (1981). In this paper, which is a supplement toSabinin's translation of the book Foundations of differential geometry [KoN](1963-1969) by Shoshichi Kobayashi (b. 1932) and Nomizu, these general-izations are called "geoodular structures" (Sabinin used the word "odulus"for a nonassociative analogue of the modulus).

The theory of symmetric spaces with an affine connection was developedfurther by Rashevskii in the paper Symmetric spaces with an affine connectionwith torsion [Ra3J (1959). In this paper Rashevskii considered spaces with anaffine connection in which not only the curvature tensor is covariantly con-stant (VhRi.

kl = 0) but the nonvanishing torsion tensor is also covariantly

constant (VhSlk = 0) . Rashevskii showed that the fundamental group G ofthis space and its stationary subgroup H possess the property that the Liealgebra G of the group G admits decomposition (2.43) into the Lie algebraH of the subgroup H and the subspace E for which, as was the case fora symmetric Cartan space, the commutator of vectors h and e from thespaces H and E belongs to the subspace E, but the commutator of vectorse 1 and e2 from the subspace E does not belong to the subalgebra H. At

§7.5. SPACES WITH A PROJECTIVE CONNECTION 219

present, these spaces are called the reductive spaces. This term was suggestedby Nomizu in the earlier mentioned paper Invariant affine connections on ho-mogeneous spaces [Nom] (1954). In the same paper, applying formula (6.27),Nomizu calculated the curvature tensor of reductive spaces.

§7.5. Spaces with a projective connection

In the paper on manifolds with a projective connection [70] (1924), Cartan,by analogy with spaces An with an affine connection, defined spaces II n witha projective connection as n-dimensional manifolds in such a way that eachpoint x of the manifold is associated with a "tangent" space Tx (IIn) , and thelatter space is the space Pn. Moreover, to each pair of infinitesimally closepoints x and x' of the space IIn there corresponds a projective mapping ofthe space Tx (IIn) , and this mapping is an analogue of a parallel displacementof vectors of the space An . The derivational formulas of the projectiveframes in the spaces IIn have the same form (5.8) as for the space Pn , butthe structure equations of the spaces IIn differ from equations (5.13) andhave a more complicated form:

(7.4)h k I

dw1 'Wt AWh+IAi ktwoA(Do,

h,i, j=0, 1, ... ,n, k,1=1,... ,n,where the tensor A j kl (an analogue of the curvature tensor of the space A" )is called the tensor of projective curvature. For the case in which the tensor

Ai kt vanishes, Cartan called a space IIn a holonomic space. At present,such spaces are called projectively fiat spaces. Cartan denoted the exteriorquadratic forms Ai kI cvo A cv1 by S2 . As in the case of the space pnan infinitesimal displacement of frames in the space IIn with a projectiveconnection is determined by the forms cvJ . Cartan denoted the forms 100and S20 by cv' and Q', respectively. By analogy with the forms s2' of thespace An , Cartan called the forms S2` the torsion forms, and if S2` = 0 , hecalled a space with a projective connection a torsion free space.

As in the case of the spaces An , in the spaces IIn , geodesics can be definedas curves that preserve their direction under an infinitesimal displacementalong the line. However, in contrast to the spaces An , it is impossible todefine an affine parameter for geodesics in the spaces IIn .

The role which Riemannian and pseudo-Riemannian manifolds play forthe spaces An is played by normal spaces with a projective connection forthe spaces In

. These are torsion-free spaces with a projective connectionfor which Ak ij :A 0 and which are completely determined by the system oftheir geodesics. The normal spaces In are connected with the problem ofgeodesic mapping of Riemannian manifolds, i.e., a mapping of a Riemannianmanifold Vn onto another Riemannian manifold under which geodesics are


transferred into geodesics. For the solution of this problem, normal pro-jective connections in Riemannian manifolds defined by their geodesics areconstructed. The equivalence of these connections is equivalent to the exis-tence of a mapping of one of these spaces onto the other.

§7.6. Spaces with a conformal connection

In the paper Spaces with a conformal connection [68] (1923), also by anal-ogy with the spaces An , Cartan defined the spaces Kn with a conformalconnection, i.e., n-dimensional manifolds each point x of which is associ-ated with a "tangent" space Tx (Kn) , and the latter space is the space Cn .

Moreover, to each pair of infinitesimally near points x and x' of the spaceKn there corresponds a conformal mapping of the space T (Kn) , and thismapping is an analogue of a parallel displacement of vectors in the space An .

The derivational formulas of the conformal frames in the spaces Kn havethe same form (5.8) as in the space Cn , and the forms coy are connectedby the same equations (5.1 0), but the structure equations of the spaces Cndiffer from equations (5.13) and have a more complicated form:

(7.5) d co . = co A co ,I i h 2 rkl 0 0

h,i, j=0, 1,... ,n+1, k,1= 1,... , n.

The tensor Ai ki (an analogue of the curvature tensor of the space An) iscalled the tensor of conformal curvature. In case the tensor A' ki vanishes,Cartan called a space Cn a curvature-free space or a holonomic space. Atpresent, such spaces are called conformally fat spaces.

As was the case for the space Cn , an infinitesimal displacement of framesin the space Kn with a conformal connection is determined by the forms a .

Cartan denoted the exterior quadratic forms Aj lcok A cot by e . Cartani,k Q

denoted the forms coo and fo by co` and respectively. He also calledthe forms fl' the torsion forms, and if fl' = 0, he called the space Cn atorsion free space. Depending on coincidence of the forms of and coy oftwo spaces Kn , there are four types of isomorphisms of these spaces.

Among spaces Kn , the normal spaces are also defined: they are torsion-free spaces with a conformal connection for which Ak = 0. They playa role similar to that of Riemannian manifolds among spaces with a metricconnection. The theory of normal spaces Kn can be applied to the theory ofconformal mappings of Riemannian manifolds, i.e., mappings that preservethe angles between curves in these spaces. The linear elements ds at thecorresponding points of such spaces differ by a factor. Cartan consideredthree-dimensional normal spaces with a conformal connection in detail andalso constructed the theory of manifolds embedded into the spaces Kn .

The spaces with a conformal connection appeared under the name "gener-alized conformal spaces" as far back as 1922 in Cartan's note On generalized

§7.7. SPACES WITH A SYMPLECTIC CONNECTION 221

conformal spaces and the optical Universe [60]. It is well known that theconformal transformations of the space-time also play an important role inthe theory of special relativity since the Maxwell equations are invariant notonly with respect to the Lorentz transformations (the rotations of the spaceR4) and the Poincare transformations but also with respect to the confor-mal transformations of the space Ci ; the latter space is obtained from thespace R4 by adding the point at infinity and the ideal points. In the note[60], Cartan tried to construct a similar conformal theory for the theory ofgeneral relativity. Probably, this attempt was the principal stimulus for theconstruction of the theory of spaces with a conformal connection by Car-tan. Later on, by analogy with spaces with a conformal connection, Cartanconstructed the theory of spaces with a projective connection (his paper [68]on spaces with a conformal connection was written one year earlier than thepaper [70] on spaces with a projective connection). Cartan called the four-dimensional space with a conformal connection (actually it was the spaceKl with a pseudo-conformal connection but not K4) the "optical Universe"since the rays of light are propagated along isotropic lines of this space (inKi they are real while in K4 they are imaginary).

§7.7. Spaces with a symplectic connection

The geometry of spaces yv n with a symplectic connection, which is of-ten called simply "symplectic geometry", has important applications in thetheory of differential equations and theoretical mechanics. With each pointx of such a space there is associated the tangent space Tx (yvn) at whosehyperplane at infinity the geometry of the space Sy2n-1 is defined. This isequivalent to assigning a skew-symmetric tensor g.3 = - gji or an exterior

differential form co = gi .dx` A dx' at each point of the manifold tv' . Themost important of these spaces are those in which the form co is closed, i.e.,the exterior differential d cv of this form is equal to zero. The usage of thisspace in mechanics is based on the fact that a mechanical system given bygeneralized coordinates q' and generalized momenta pi can be consideredas a space with the closed exterior differential form a = d q` A dpi ; n-dimensional submanifolds of this space whose tangent n -planes cut, on thehyperplanes at infinity of the tangent spaces T (Yv'1) , (n - 1)-dimensionalnull-planes of the space Sy2n-1 are called Lagrangian submanifolds of thesespaces. This name is explained by the fact that a six-dimensional manifoldof this type was studied by Lagrange in his Memoir on the theory of varia-tions of elements of planets [Lag2] (1809). In this paper Lagrange consideredperturbations of motions of planets around the sun under the influence ofexterior forces. Lagrange took as his point of departure the fact that planetstravel around the sun along ellipses with the sun at one of the foci of these


ellipses, and a perturbed motion of a planet possesses the same property. Be-cause of this, Lagrange described a possible motion of a planet by means ofa plane passing through the sun, the major axis of an ellipse and a locationof a planet on this ellipse. He considered six "elements of planets" definedin this way as coordinates of a six-dimensional space in which he transferredto the coordinates q 1 , q 2 , q3 , p 1, p2 , and p3 . In his book Geometric the-ory of partial differential equations [Ra2] (1947), Rashevskii applied spaceswith a symplectic connection to the investigation of a wider class of differ-ential equations than the equations of mechanics. He called the spaces witha symplectic connection the "spaces of a linear form of even class". VictorP. Maslov (b. 1930) widely used the geometry of spaces with a symplecticconnection in his book Theory of perturbations and asymptotic methods [Mas](1965) where the term "Lagrangian submanifolds" was introduced.

§7.8. The relativity theory and the unified field theory

We have already indicated the exceptional role of Einstein's general rel-ativity in the development of the geometry of Riemannian and pseudo-Riemannian manifolds in the attempts to construct the unified field theoryand in the development of the theory of spaces with an affine connectionand other generalized Cartan's spaces. Thus, it is natural that a series ofCartan's works was devoted to the problems of relativity theory and unifiedfield theory.

Cartan became interested in the problems of general relativity even beforehe started to study the theory of generalized spaces. As far back as 1922,he wrote the paper On the equations of gravitation of Einstein [56], in whichhe investigated the equations of general relativity by means of his theory ofPfaffian equations in involution. Cartan found a system of Pfaffian equa-tions which is equivalent to Einstein's system of equations, calculated thecharacters of this system, proved that the system is in involution and its gen-eral solution depends on n(n - 1)/2 functions of n real variables (i.e., inthe case of four-dimensional space-time, it depends on six functions of fourvariables). In the paper on manifolds with an affine connection and generalrelativity theory [66], [69], and [80] (1923-1925), Cartan first considered thespace-time of general relativity and classical Galilei-Newton mechanics. Af-ter an exposition of the geometry of spaces An, spaces with a metric andEuclidean connection, and the theory of curves and surfaces in these spaces,in Chapter V, "The gravitational Universe of Newton and the gravitationalUniverse of Einstein", Cartan studied different spaces with an affine con-nection consistent with properties of "the Universe of Newton" and "theUniverse of Einstein". He studied these not only from the point of view ofdescription of mechanics of continuous media in these two "Universes", butalso from the point of view of description of electromagnetic fields in them.

§7.9. FINSLER SPACES 223

In the paper A historic note on the notion of absolute parallelism [124](1930), Cartan noted that the idea of absolute parallelism, which he intro-duced in one of his papers in 1922, was rediscovered by Einstein in 1928who decided to use it in the foundation of a unitary theory of gravitationaland electromagnetic fields. Einstein also defined the tensor F'3 of an elec-tromagnetic field in terms of the torsion tensor of this space. The Cartanbook Absolute parallelism and unitary field theory [ 130] (193 1) was devotedto the unitary theory of gravitational and electromagnetic fields based on thenotion of absolute parallelism.

In the paper The unitary (field) theory ofEinstein-Mayer [l 43a] (which waswritten in 1934 but published only in Cartan's Euvres Completes [207] afterhis death), Cartan gave a "geometrically intuitive" presentation of unitaryfield theory constructed by Einstein and Mayer in 1931. In this presentation,space-time is a totally geodesic surface in a five-dimensional space with aEuclidean connection.

§7.9. Finsler spaces

Another generalization of the Riemannian manifold is the Finsler spacein which a linear element or, using Cartan's words, the distance betweentwo infinitesimally close points x (x') and x' (x' + d x') on a manifold X isdefined by the formula

(7.6) ds=F(xI, ... , x"; dxI, ... , dx"),

where F is a positive function which is first degree homogeneous with re-spect to dxl , ... , dx" . This notion arose in connection with a geometricinterpretation of the variational calculus problem for the integral:

r2

J= F(x!,x)dt1

and was first considered by Paul Finsler (1894-1970) in his thesis on curvesand surfaces in generalized spaces [Fis] (1918). The extremals of this integralare geodesics of the Finsler space.

The simplest space of this kind was defined by Hermann Minkowski(1864-1909) in his (posthumously published) work Theory of convex bod-ies, especially the foundation of the notion of a surface [Min] (1911). Thespace considered by Minkowski is an affine space F" in which a metric isintroduced not with the help of a hypersphere (as in the Euclidean space R" )but with the help of a closed centrally symmetric convex "gauge surface".Minkowski showed that if one defines the distance between the points Xand Y of this space as the ratio of the segment X Y to the parallel segmentOP enclosed between the center 0 of the gauge surface and the point P ofthis surface, then the triangle inequality X Y + YZ > X Z holds in this space.The Finsler space is locally the Minkowski space since in each of its tangentspaces Tx (X ") , by means of the function F entering under the integral sign


in equation (7.7 ), a gauge surface is defined by the formula F (x' , fit) = 1 ,

where t are the coordinates of the tangent vector of the space TX (Xn).

The Finsler geometry was further developed in the paper Generalization ofRiemannian line-element [Sy] (1925) by John Lighton Synge (b. 1897) andthe papers On parallel displacement in spaces with commonly defined distances[Bew l ] (1926), On two-dimensional generalized metric spaces [Bew2] (1925)by Ludwig Berwald (1883-?), and other works. In 1934, in his lecture Onthe Finsler and related spaces [Ber3] at the Congress of Mathematicians ofSlavic Countries, Berwald replaced the vague term "general metric spaces"by the term "Finsler spaces" commonly used at present. In the paper On theaf, ne foundation of the metric of one variational problem [Win] (1930), ArturWinternitz (1893-?) gave the definition of the Finsler space as a space witha connection whose tangent spaces are Minkowski spaces.

In the book Finsler spaces [ 142] (1934) and in his lecture [ 152] under thesame title at the International Conference on Tensor Differential Geometryin Moscow, U.S.S.R., also in 1934, Cartan developed a new approach forstudying Finsler spaces. He indicated that the theory of these spaces can beconnected with general problems of equivalence. Such problems arise dur-ing the study of many objects in differential geometry. For example, if weconstruct the Riemannian geometry, we encounter the problem of findingout whether two differential forms with the same number of variables can betransformed into one another by a change of variables. Each of these differ-ential forms is the metric form of a point space with a Riemannian metric,and the equivalence of two differential forms is reduced to the geometricapplicability of these two Riemannian manifolds. As Cartan noted, for theFinsler space, the notion of a point space was insufficient. For this space,we are forced to consider spaces of linear elements with a Euclidean connec-tion. A linear element of Xn consists of a point x (x') of this manifold anda vector Sic (ic!) of the tangent space TX (Xn) of this manifold. This spacewill be defined if one assigns an expression glJ.d x' d x3 to the square of alinear element in it, where now glJ . = giJ. (xk , xk) , and the expression of the

absolute differential Dot of the vector = {c` (xk, ±k) } has the form:

(7.8) d i+ k(rkhll'Xh + CkyllXh).

The condition Dc' = 0 defines the parallel displacement o f a vector inthe Finsler space. The problem is: among all spaces of linear elements witha Euclidean connection, determine the space that is uniquely defined by thefunction F(xt , Sic`) assigning the distance between two infinitesimally closepoints in the Finsler space. To do this, we set

119 2F2 (xk , x k _ 1 agli(7.9) g13= 2 , C`IJk 2 kax ax ax

§ 7. 10. METRIC SPACES BASED ON THE NOTION OF AREA 225

C k Then crack Cjk = 0 and CiJJk = C Jik Under thiswhere CiJJk = gih J Jcondition, the parallel displacement preserves the length of a displacedvector , and the square of this length is II2 =

Further, Cartan extended the tensor calculus to the Finsler geometry andstudied submanifolds embedded in a Finsler space.

§7.10. Metric spaces based on the notion of area

In the book Metric spaces based on the notion of area [ 140] (1933), Cartanintroduced another generalization of the notion of a Riemannian manifold;particularly, the basic notion is that the area of a surface given by the equationz = f(x, y) in a three-dimensional space is expressed by the equation:

du=F x z az az dxd(7.10) ax' a y'' y' ' y

where the function F depends on coordinates x, y, z of a point and onthe quantities p = az and q = ay These quantities p and q deter-mine the position of the tangent plane to the surface z = f(x, y) at thepoint P(x, y, z). The surfaces giving the extremum to the integral ff dQ =ff F dx dy play the role of geodesics in this geometry.

For construction of such a geometry in an n-dimensional manifold X" ,

Cartan considered the set of "support elements" consisting of a point x ofthe manifold X" and an (n - 1)-dimensional subspace u of the tangentspace Tx(X") . If a coordinate system is chosen in the manifold Xn, thenthe point x is defined by coordinates x' and the subspace u is definedby homogeneous coordinates u,

.Cartan defined the square of the distance

between two infinitesimally close points x and x' by means of the quadraticform d s2 = giJ. d x' d x' as in the case of the Riemannian geometry, butnow the coefficients gi.I of this form depend not only on the coordinatesx' of the point x but also on the coordinates ui of the subspace u ofthe tangent space. Next, Cartan defined the absolute differential d of thesupport element = (x, u) by the formula:

(7.11) Dp' = ck(Ck'du, + r` du').k

The quantities Ck' and rk . determine a Euclidean connection in the spacekJof support elements.

After this, Cartan showed how to construct a Euclidean connection inthe space of support elements in such a way that this connection would beinvariantly related to the surface element d a indicated above and to a moregeneral hypersurface element on an n-dimensional manifold.

The construction of these Euclidean connections was further applied tothe solution of the equivalence problem for multiple integrals of the form


ff F(x, y, z, p, q) dx d y and similar integrals on an n-dimensional mani-fold. A necessary and sufficient condition for two such integrals to be equiv-alent is that the spaces with a Euclidean connection associated with theseintegrals must be geometrically equivalent; i.e., there exists a correspondencebetween these spaces (and therefore a correspondence between their supportelements) such that the metric and the Euclidean connection of the first spaceis transformed by this correspondence into the metric and the Euclidean con-nection of the second space.

Cartan's book [ 140] inspired many works devoted to the geometry of mul-tiple integrals. In particular, we mention here the paper Metric spaces of ndimensions based on the notion of area of m-dimensional surfaces [Au] (1951)of Maya V. Aussem (Vasil'eva) (b. 1926), in which the author studied thegeometry of an rn-dimensional integral

(7.12) f...fF(xk,p)dx1Adx2...Adxm

M

over an rn-dimensional surface xa = f(x') where pa = r? xa /r? x ` and thepaper The geometry of the integral f F (xa , x" , xa , xa , .. .) d x i A dx2..

A d xn-1 [Ev] (1958) by Leonid E. Evtushik (b. 1931), in which the authoralso considered the geometry of an integral of type (7.12) but with the func-tion F depending not only on the coordinates x` of the point x and thefirst order derivatives (9xa/r?x` but also on the derivatives of higher orderup to some order p . This forced the author to reconstruct the analytic appa-ratus that had been used previously for studying similar problems; instead ofthe classical tensorial methods he applied the invariant apparatus of exteriordifferential calculus, also originated by Cartan.

§7.11. Generalized spaces over algebras

Analogues of Riemannian manifolds and other generalized spaces havealso been constructed over commutative algebras-first, over the field C ofcomplex numbers and over the algebras IC and °C of split complex and dualnumbers. The most important among these spaces is the Hermitian space firstdefined by P. A. Shirokov in 1925 in the same paper [Sh I] in which he definedthe symmetric spaces. This space was also defined by Schouten in the paperon unitary geometry [Sco3] (1929).

The points of Hermitian spaces are defined by complex coordinates x` .The distance ds is defined between the points x(xand x'(x' + dx') ofthis space, and the square of this distance is given by the formula:

ds2 = gt1 ..dx'dx3 , gt.1 = g1.t..

Thus, an n-dimensional Hermitian space is isometric to a real Rieman-nian manifold V21 in which an operator J is given, and this operator has

§7.11. GENERALIZED SPACES OVER ALGEBRAS 227

the property J2 = - l and is covariantly constant with respect to an affineconnection determined by the metric of the space. With the Hermitian form(7.13) there is associated the exterior quadratic form

(7.14) icv = Im(gjjdx`dy'), Re(gr`dx`dyj) = 0

defining a symplectic connection in the space under consideration.If this form is closed, i.e., d cv = 0, the Hermitian space is called Kahlerian.

It was named after Kahler who first considered such spaces in his paper Ona remarkable Hermitian metric [Kah I) (1932). Hermitian spaces are a par-ticular case of spaces with an affine connection.

If at each point of the Riemannian manifold Yen or a differentiable man-ifold X 2n the operator J with the property J2 = -I is given, but it isimpossible to introduce complex coordinates x' in the space, we say that thespace is endowed with an almost complex structure (or a nonintegrable com-plex structure). As we discussed earlier, historically the first example of analmost complex structure was the six-dimensional sphere which is the inter-section of the hypersphere jal = l and the hyperplane a = -a of the algebra0 of octaves (where the geometry of the space R8 is realized). The opera-tor J of complex structure considered at each point of this sphere transfersthe differential d a into the product ad a . (We saw in Chapter 3 that onthis sphere a transitive subgroup of the group of its rotations is acting andthat this subgroup is isomorphic to a compact simple Lie group in the classG2 .) The almost complex structure on this sphere was first discovered byA. Frohlicher in the paper On the differential geometry of complex structures[Fro] (1955).

In complex and almost complex spaces it is possible to separate the holo-morphic manifolds whose tangent spaces are invariant under the operator ofa complex structure, the antiholomorphic (or "completely real") manifoldswhose tangent spaces are transformed into the planes totally orthogonal tothem under the operator of a complex structure, and the CR-submanifoldswhose tangent spaces are the direct sums of the tangent spaces to holomorphicand antiholomorphic submanifolds. In particular, in the spaces Can andCSn , the holomorphic submanifolds are complex straight lines and planes ofthese spaces, and the antiholomorphic submanifolds are their normal spacechains. Note that the spaces Yen which are isometric to the spaces CSnare Riemannian manifolds of variable sectional curvature given by formula(6.31) and taking on values from 1/r2 to 4/r2. But the sectional curva-ture of this space in holomorphic two-dimensional directions is equal to theconstant value 4/r2 , which explains the name Hermitian spaces of constantholomorphic sectional curvature for the spaces CSn .

If in the definition of complex and almost complex spaces we substitutesplit complex numbers or dual numbers for complex numbers, we obtain splitcomplex and almost split complex spaces or dual and almost dual spaces,


respectively. Since the algebra 'C of split complex numbers is isomorphicto the direct sum R ® R of two fields R, an n-dimensional space overthe algebra 'C can be represented in the form of the Cartesian product oftwo real spaces X". The split complex and almost split complex spaces areoften called the space-products. The dual and almost dual spaces are alsocalled the contact spaces and the almost contact spaces. Note that the six-dimensional sphere, which is the intersection of the hypersphere jal = land the hyperplane a = -a in the algebra '0 of split octaves (with thegeometry of the space R8 ), also forms an almost complex space, and thesix-dimensional sphere of imaginary radius, which is the intersection of thehypersphere I al 2 = -1 and the hyperplane a = -a in the same algebra withthe same geometry, forms an almost split complex space.

The most important results on the geometry of generalized complex spacesare given in the second volume of the earlier mentioned monograph Founda-tions of differential geometry [KoN] (1969) by Kobayashi and Nomizu, andthe results on the geometry of generalized spaces over more general algebrasare given in the book Spaces over algebras [VSS] (1985) by Vladimir V. Vish-nevskii (b. 1929), Alexander P. Shirokov (b. 1926), and V. V. Shurygin.

§7.12. The equivalence problem and G-structures

We have already mentioned the equivalence problem while discussing Car-tan's papers on the theory of Finsler spaces. In fact, this problem is con-nected with all generalized Cartan spaces, and Cartan became interested inthis problem as far back as 1902, long before he started to develop the theoryof generalized spaces.

In the general case, the equivalence problem is formulated as follows:let, on the one hand, a system of n linearly independent Pfaffian forms(01)(02, . *6 , w, with respect to independent variables x 1, x2 , ... , x" andm independent functions y1, y2 , ... , ym of these variables be given, and,on the other hand, let a system of n linearly independent Pfaffian forms921, Q2 , . . . , 9Z" with respect to independent variables X 1 , X 2 , ... , X"

and m independent functions Y 1 , Y2 , ... , Ym of these variables be given.It is required to find out whether there exists a change of variables that sendsthe functions y 1, y2 , ... , ym into the functions Y 1, Y2 , ... ,

Ym and al-lows the forms 01, Q2 , ... , S to be obtained from the forms cvl , cv2 , 440 ,

(o" by means of a linear substitution from some linear group IF, where thecoefficients of finite transformations of this group can depend on the func-

1 2 mtlons y , y , ... , yIn 1902 Cartan devoted his note on the equivalence of differential systems

[ 19] to this problem. He also considered this problem in his note On theintegration of certain systems of differential equations [40] (1914), and in thepapers on the absolute equivalence of certain systems of differential equationsand on certain families of curves [42] (1914), on an equivalence problem and

§7.12. THE EQUIVALENCE PROBLEM AND G -STRUCTURES 229

the theory of generalized metric spaces [ 126] (1930), and The problems ofequivalence [161a] (1937). The papers The subgroups of continuous groupsof transformations [26] (1908), The Pfaffian systems with five variables andpartial differential equations of second order [30] (1910), and on the pseudo-conformal geometry of hypersurfaces of the space of two complex variables[136, 136a] (1932) were also mostly devoted to the equivalence problem. Inthese papers Cartan solved this problem for various groups F.

The equivalence problem is closely connected not only with generalizedspaces but also with more general fiber spaces and with G-structures onsmooth manifolds.

The spaces that are closest to generalized spaces are fiber spaces whosebases are differentiable manifolds Xn

, and whose fibers are the sets of allframes {x, e! } in the tangent spaces Tx (X n) of the manifold Xn whichare transformed to one another by transformations of a subgroup G of thegroup GLn . The subgroup G is called the "structural group" of a fiberspace. At present, the fiber spaces are also called G-structures of first order.If we substitute the spaces Txk (Xn) of the differentials of kth order for thetangent spaces Tx (X n) in the above definition, we get the definition of aG-structure of kth order.

An example of G-structures of first order is the Riemannian manifold V n.

It is defined on a manifold Xn by means of a positive definite quadratic formds2 = g!J.d x` dx' . This form separates the subset of orthonormal frames inthe frame manifold of the tangent space Tx (X n) , and in this subset thisform is reduced to the form ds2 = >,(oi)2 and defines the group G = onof orthogonal transformations mapping the set of orthonormal frames intoitself. Thus, the Riemannian manifold Vn is a G-structure of first order withthe structural group G = on . Similarly, a pseudo-Riemannian manifold Ynis a G-structure of first order with the structural group G = a' of pseudo-orthonormal transformations of index I.

If the fibers can be identified with a certain group G, or more precisely,if the group G operates (on the right) on the space in such a way that G issimply transitive on these fibers, the fibration is called principal. A connectionin the principal fiber spaces plays an important role in the theory of G-structures. Let X n be the base of a fibration with fibers F = G of dimensionr forming the principal fibration Xn+r of dimension n + r. Further, let apoint y belong to the fiber F , and let, in the tangent space TY (X n+r) , ann-dimensional subspace Hy be chosen in such a way that it has only onecommon point y with the tangent space V, to the fiber F at the pointy and the space Ty (Xn+r) is the direct sum of the subspaces H,, and Vy V.These subspaces considered at all points y of the fibration Xn+r form thehorizontal and vertical distributions, respectively. If the distribution H,, isdifferentiable and invariant under the action of the group G on the fibrationXn+r , it is called a connection in the principal fibration.


Note that the spaces with affine, projective, and conformal connectionsconsidered above are particular cases of a connection in the principal fiberspace. In these cases, the fibers can be identified with the groups of affine,projective, and conformal transformations in the spaces En , Pn , and Cn ,

respectively. In particular, for the space An with an affine connection, thehorizontal distribution defines the parallel displacement of frames along acurve in the base.

The affine, projective, and conformal connections are G-structures of firstorder in the corresponding fiber space Xn+r. However, usually they areconsidered as G-structures of higher order on the manifold Xn ; for theaffine and conformal connections the order of this G-structure is two, andfor the projective connection the order is three.

The notion of a G-structure was first formulated by Cartan's student Ehres-mann in his note Fiber spaces of comparable structure [Eh2] (1 942), andthe term " G-structure" first appeared in Chern's paper Infinite continuouspseudo-groups [Chr3] (1954) in which these structures were connected withLie pseudogroups studied by Cartan under the name "infinite continuousgroups". Note also another of Chern's papers, The geometry of G-structures[Chr4] (1966), and the paper On the equivalence problem of certain infinites-imal structures [Lib] (1954) by Paulette Libermann (b. 1919).

The complex and almost complex structures played an important role inthe construction of the theory of G-structures. The first of these structures isdefined on an n-dimensional complex manifold CXn . Its structural groupis the group CGLn . If we take the real interpretation of the space CXn, weobtain a real manifold X 2n . In its tangent space Tx (X 2n) , the group G ofdimension 2n2 (which is the real interpretation of the group CGLn) acts.The elements g of this group commute with the operator J of the almostcomplex structure. This operator satisfies the condition J2 = - I and cor-responds to the scalar operator iI in the group CGLn . An almost complexstructure is a G-structure on the real manifold X2n whose structural groupis the same as the structural group of the complex manifold CXn . However,in contrast to a complex manifold, in general this structure cannot be ob-tained as a realization of the manifold CXn . The following problem arisesin this connection: to find under what condition an almost complex struc-ture becomes a complex structure. The solution of this problem is reduced tofinding conditions of complete integrability of two systems of Pfaffian equa-tions that define imaginary conjugate eigenspaces of the operator J in thetangent space Tx (X 2n) of the manifold X 2n

. This required condition is thevanishing of a certain operator, llTik , of the third valence (which is calledthe Nijenhuis operator) on the manifold X2n .

In a similar manner, the structure of an almost-product can be defined ona manifold X n+m . In this case, the operator J satisfies the condition J2 = Iand has real eigenspaces of dimensions m and 2n - m. The elements of the

§7.13. MULTIDIMENSIONAL WEBS 231

structural group G commute with this operator J. In particular, if m = n,the operator J defines an almost split complex structure on the manifoldX2n

There is the following problem in the theory of G-structures: given a man-ifold Xn carrying a G-structure, is it possible to define an affine connectionon Xn whose parallel displacements preserve the G-structure ? If the answeris positive, the G-structure is called a G-structure of finite type. Otherwise,it is called a G-structure of infinite type. Since the parallel displacements inRiemannian and pseudo-Riemannian manifolds generate a single affine con-nection, the G-structures associated with these spaces are of finite type. Onthe contrary, in a space with an almost complex structure, it is impossible tofind a single affine connection in which the operator J of the almost complexstructure will be invariant under a parallel displacement. Moreover, it canbe proved that in these spaces it is impossible to find a single affine connec-tion defined even by means of differential prolongations of this G-structure.Thus, this G-structure is a G-structure of infinite type.

There is an extensive bibliography on differential geometry of G-structures.We note here only the book Transformations groups in differential geometry[Ko] (1972) by Kobayashi.

§7.13. Multidimensional webs

Another interesting example of G-structures of first order is connectedwith webs on smooth manifolds formed by a certain number of smooth fo-liations. Web theory was founded by Blaschke at the end of the 1920s andduring the 1930s. In 66 papers composing the series Topological questionsof differential geometry [BlaT] (1928-1936) and many other publications,Blaschke and his co-workers considered webs formed by families of curvesin the plane and the families of curves and surfaces in three-dimensionalspace. This explains the title of the series of papers on webs indicated above.In these papers, it was established that web theory is connected with manybranches of geometry as well as with some other branches of mathematicsand, in particular, with some parts of algebra. These investigations in theweb theory were summarized by Blaschke and Bol in their book Geometry ofwebs [BlaB] (1936), and later in Blaschke's book Introduction to the geometryof webs [Blab] (1955).

However, as far back as 1908, in the paper The subgroups of continuousgroups [26], Cartan considered an example in which he posed the problem ofthe equivalence of two differential equations:

f(= x and g=F(X,Y)(7.15)dX , y)

with respect to transformations of the form

(7.16) z = X(x) , y = Y(y).


The latter transformations leave invariant the coordinate lines x = a, y = bin the plane xOy as well as the integral lines of equations (7.15). Thesethree families of lines form a three-web in the plane. Thus, the problem con-sidered by Cartan is equivalent to the problem of classification of curvilinearthree-webs in the plane. Cartan distinguished three classes of differentialequations of type (7.15): the equations admitting a three-parameter groupof transformations of type (7.16), the equations admitting a one-parametergroup of transformations of type (7.16), and the equations not admittingsuch transformations. To these three classes of differential equations therecorrespond three classes of curvilinear three-webs in the plane.

In the 1930s, along with webs in the plane and in three-dimensional space,webs on manifolds of dimension higher than three were studied. First, in1935 Bol published the paper on a three-web in a four-dimensional space[Boll in which he considered a three-web formed on a four-dimensional man-ifold by three two-dimensional foliations. Next, in 1936, Chern's paper Aninvariant theory of the three-web of r-dimensional manifolds in R2r [Chr 1 lappeared, in which Chern studied three-webs formed on a manifold R2r bythree r-dimensional foliations. During the last 20 years these studies werecontinued by Akivis, Vasil'ev, Goldberg, and their students and co-workers.

Let us consider, for example, a web W formed on a manifold X2n bythree foliations A. , a = 1, 2, 3 , of dimension n. Through any point x ofthe manifold X 2n there pass three leaves F belonging to these foliations A..Denote by Tx (F) the n-dimensional subspaces of the space Tx (X 2n) whichare tangent to the leaves F passing through the point x. The subgroup ofthe group of linear transformations of the space Tx (X 2n) preserving thesubspaces Tx (F) is the structural group of the G-structure induced on theamanifold X2n by the web W. It is not difficult to show that in this caseG = GL,. This G-structure is a structure of finite type since it definesan affine connection on X2n in which the web leaves are totally geodesicsubmanifolds of the manifold X2n .

The subspaces Tx (F) define in the space Tx (X 2n) an algebraic conewhich cuts the Segrean (3.12) in the hyperplane at infinity of this space.This cone is called the Segre cone. Since the Segrean defined by this cone hasrectilinear generators and (n - 1)-dimensional generators, the cone itself hastwo-dimensional generators and n-dimensional generators. Linear transfor-mations preserving this Segre cone form a group G which is the direct prod-uct of the groups GL, and SL2. This group G defines a new G-structurein the space X2n

, and this G-structure is called the almost Grassmann struc-ture AG n+ 1,1 . This name is explained by the fact that in the simplest case,

when this G-structure is integrable, the manifold X2n admits a mapping onthe Grassmannian Grn+1

1of straight lines of the projective space Pn+1 . In

this case, a web W is called Grassmannizable and is defined by a triple ofhypersurfaces in the space Pn+ 1 .

§7.13. MULTIDIMENSIONAL WEBS 233

The almost Grassmann structure AGr,+1,1 and the G-structure definedby the web W itself are of finite type. However, an affine connection onthe structure AGr, + 1,1 is defined by the differential neighborhood of thirdorder while an affine connection of the G-structure induced by the web Wis defined by the differential neighborhood of second order.

Multidimensional three-webs are connected with differentiable quasi-groups: if we map the n-dimensional bases of the foliations ).a forminga three-web W on a manifold X2n onto the same n-dimensional mani-fold Q, an algebraic operation arises in Q which defines a smooth localquasigroup. Moreover, to smooth quasigroups and loops there corresponddifferent classes of three-webs which are characterized by some closure con-ditions which are satisfied in these three-webs. In particular, the importantclasses of three-webs correspond to the Lie groups and the Moufang, Bol, andmonoassociative loops.

The theory of multidimensional three-webs is presented in the book Ge-ometry and algebra of multidimensional three-webs [AS] (1991) by Akivis andShelekhov, and the theory of multicodimensional (n + 1)-webs is presentedin the book Theory of mul ti cod imensional (n + I)-webs [Glb 1 ] (1988) byGoldberg.

Conclusion

As a rule, Cartan built his scientific research on works of his predecessors,developing their ideas so well that other mathematicians often forgot theoriginal works. This was the case in the theory of simple Lie groups withthe Killing paper, in the method of moving frames with the Cotton paper,and in the theory of symmetric Riemannian spaces with the Levy paper.It was somewhat different in the case of the theory of generalized spaces,since Cartan continued to work fruitfully with the founders of this theory,Weyl and Schouten. In some of the works of Cartan's predecessors (e.g.,the papers of Cotton and Levy), only the initial definitions were given forthe future theories which were later constructed by Cartan. In other cases,for example in the case of Killing's paper, the important notions of the newtheory were introduced and the main results of this theory were formulated,but the rigorous proofs of these results were given only in the famous Cartanthesis [5]; as a result, after the appearance of this thesis, Killing's paper [Ki12]on the structure of groups of continuous transformations was read by almostno one. This explains the enthusiasm that A. J. Coleman had while readingthe Killing paper mentioned above and that he expressed in his own paperentitled The greatest mathematical paper of all times [Co12] (1989). In thispaper, Coleman wrote:

"Cartan did give a remarkably elegant and clear exposition of Killing'sresults. He also made an essential contribution to the logic of the argumentby proving that the `Cartan subalgebra' of a simple Lie algebra is abelian.This property was announced by Killing but his proof was invalid ... In thelast third of Cartan's thesis, many new and important results are based uponand go beyond Killing's work. Personally, following the value scheme ofmy teacher Claude Chevalley, I rank Cartan and Weyl as the two greatestmathematicians of the first half of the twentieth century. Cartan's work oninfinite dimensional Lie algebras, exterior differential calculus, differentialgeometry, and above all, the representation theory of semisimple Lie algebraswas of supreme value. But because one's Ph. D. thesis seems to predetermineone's mathematical life work, perhaps if Cartan had not hit upon the ideaof basing his thesis on Killing's epoch-making work he might have ended his

235

236 CONCLUSION

days as a teacher in a provincial lycee and the mathematical world wouldhave never heard of him!" [Co12, p. 30].

A similar situation occurred with Cartan's works on the theory of Pfaffianequations, which was considered in the books of J. F. Pommaret [Pom 1-3].

Coleman's "prediction" was completely justified by the fate of Cartan'spredecessor in the correction of inaccuracies in Killing's results - C. A. Um-lauf, the author of the thesis [Um]; his further life and activities are unknown.However, this was not the case with Cartan. After Cartan gave the classifi-cation of complex simple Lie groups, he created a similar classification ofcomplex and real associative algebras and complex simple Lie pseudogroups.The latter led him to the theory of Pfaffian equations, whose application todifferential geometry implied a complete transformation of this disciplineand helped Cartan and his followers to solve numerous problems in the dif-ferential geometry of various spaces. Cartan's work on simple Lie groups wasfollowed by his remarkable theory of representations of these groups. Subse-quently Cartan solved the problem of classification of real simple Lie groups.The latter problem was posed by Killing, but the author of "the greatest math-ematical paper of all times" could not solve it himself. Following this, Cartancreated the geometries of "generalized spaces" and the theory of symmetricspaces by means of which the problem of classification of real simple Liegroups unexpectedly obtained a new and much more elegant solution.

Cartan's papers eclipsed the papers of many of his predecessors: afterthe publication of Cartan's papers, practically no one, except the historiansof science, read either Killing's or Janet's papers (Janet bitterly complainedabout this to Pommaret).

The most spectacular confirmation of the enormous influence that Cartanhas had on the development of contemporary mathematics was the creation ofthe encyclopaedia of mathematical sciences, Elements of mathematics [Bou]of Nicolas Bourbaki. This pseudonym was used by a group of mathemati-cians, among whom leading roles were played by Cartan's son Henri, AndreWeil, Jean Dieudonne, Claude Chevalley, and Jean Frederic Delsarte. Thetitle of this encyclopedia was supposed to indicate that, according to the ideaof its authors, this work would play the same role for mathematics of the20th century as Euclid's Elements played for ancient mathematics. Whilethe first part of this work contained a concise survey of the principal "math-ematical structures" on which algebra, topology, and analysis are based, itssecond part gave a systematic explanation of the theory of Lie groups and Liealgebras, the bulk of which was Cartan's creation. The authors of Elementsof mathematics, who belonged to another generation, often put in the fore-front what Cartan had not. While Cartan considered himself first of all as ageometer and headed the Department of Higher Geometry at the Sorbonne,in the Bourbaki work, geometry was dissolved in algebraical, topological, andanalytical "structures". Such pure geometrical structures as the affine, projec-tive, and conformal geometries, considered as sets of points in which some

CONCLUSION 237

specifically geometric subsets (straight lines and planes, circles and spheres)are distinguished, were not included in Elements of mathematics, neitherwere the spaces with affine, projective, and conformal connections whichare based on these geometries and which played so important a role in Car-tan's research. But nevertheless the influence of Cartan's papers penetratedElements of mathematics. Cartan's works also influenced those mathemati-cians whose research was out of the scheme of Elements of mathematics andwho continued to develop different directions of Cartan's research. In ourdescription of Cartan's scientific results, we often mentioned works of math-ematicians from different countries who developed those or other of Cartan'sideas: Weyl, Blaschke, Chern, Freudenthal, Ehresmann, Lichnerowitz, Serre,Tits, Finikov, Rashevsky, Norden, Wagner, Laptev, Vasilyev, and many oth-ers (including the authors and the translator of this book).

Another spectacular confirmation of Cartan's influence on many branchesof contemporary mathematics was the conference, "The Mathematical Her-itage of Elie Cartan", which was held in Lyons, France, on June 25-29, 1984,on the occasion of the l l 5th anniversary of Cartan's birth. The conferencetook place at the University of Lyons, and Henri Cartan and S. S. Chernwere the co-chairmen of its Organizing Committee. The participants in theconference made a trip to Dolomieu.

The program of the conference contains the following lectures:

(1) S. S. Chern: Moving frames.(2) J. M. Souriau: On differential forms.(3) J. Tits: Analogues of great classification theorems of Elie Cartan.(4) M. Gromov: Isometric immersions of Riemannian manifolds.(5) V. Kac: Computing homology of compact Lie groups and their in-

finite-dimensional analogues.(6) V. Guillemin: Some microlocal aspects of integral geometry.(7) B. Kostant: Simple Lie algebras, finite subgroups of SU2 , and the

MacKay correspondence.(8) A. Trautman: Optical structures in relativity theories.(9) M. Berger: The Riemannian manifolds as metric spaces.

(10) R. Bryant: The characteristic variety and modern differential geom-etry.

(11) Y. Choquet-Bruhat: Causality of supergravities theories.(12) J. L. Koszul: Schouten-Nijenhuis brackets and cohomology.(13) C. Feffermann: Conformal geometry.(14) M. Kuranishi: Cartan connection and CR-structures with nondegen-

erate Levy form.(15) M. Duflo: Noncommutative harmonic analysis and generalized Car-

tan subgroups.(16) S. Helgason: Fourier analysis on symmetric spaces.

2 38 CONCLUSION

(17) W. Schmid: Boundary value problems for group invariant differentialequations.

(18) G. D. Mostow: Discrete subgroups of Lie groups.(19) I. Piatetskii-Shapiro: L-functions for automorphic forms.(20) A. Weinstein: Poisson manifolds.(21) I. M. Singer: Families of Dirac operators with applications to physics.(22) I. M. Gel'fand: New models for representations of reductive groups

and their hidden symmetries.Following Gel'fand's lecture was the ceremony of his inauguration in thedegree of Doctor Honoris Causa of Lyons University.

The lectures of this conference were published in the book [ECM].

Dates of Cartan's Life and Activities

1869 Born in Dolomieu, France, April 91880-1885 Student at the College of Vienne1885-1887 Student at the Lycee of Grenoble1887-1888 Student at the Lycee Janson-de-Sailly in Paris1888-1891 Student at the Superior Normal School in Paris1891-1892 Drafted into the French army; achieved the rank of sergeant1892-1894 Boursier of the Pecaut Foundation1892-1894 Acquaintance with Sophus Lie and discussions with him in

Paris1894 In the Sorbonne defended the doctoral thesis The structure of

the finite continuous groups of transformations, in which heconstructed the theory of simple complex Lie groups

1894-1896 Lecturer of mathematics at the University of Montpellier1896-1903 Lecturer of mathematics at the University of Lyons1898 Constructed the theory of complex and real simple algebras in

the paper Bilinear groups and systems of complex numbers1899 Published his first paper on the Pfaff problem1903 Married Marie-Louise Bianconi in Lyons1903-1909 Professor of mathematics at the University of Nancy and the

Institute of Electrical Engineering and Applied Mechanics1904-1905 Published first papers in the theory of "infinite continuous

groups of transformations" (Lie pseudogroups) and the theoryof systems of Pfaffian equations in involution

1908 Published the paper Complex numbers for the French editionof Encyclopaedia of Mathematical Sciences, which contains asurvey and further development of the theory of algebras

1909-1912 Lecturer of mathematics in the Sorbonne, Paris1910 Published first papers on the method of moving frames1912-1940 Professor of Differential and Integral Calculus, and, from

1924, Professor of Higher Geometry in the Sorbonne, Paris;Professor of the Municipal School of Industrial Physics andChemistry, Paris

239

240 DATES OF CARTAN'S LIFE AND ACTIVITIES

1913 Constructed the theory of linear representations of complexsimple groups in the paper Projective groups under whichno plane manifold is invariant

1914 Constructed the theory of real simple Lie groups in the paperReal simple finite continuous groups and constructed linearrepresentations of these groups

1915-1918 Drafted into the Army and served at the rank of sergeantin the military hospital of the Superior Normal School

1915 Published the paper Theory of continuous groups and geom-etry for the French edition of Encyclopaedia of MathematicalSciences

1916-1920 Published papers on the theory of deformation of surfacesin the Euclidean, conformal, and projective spaces

1922 Published papers on the theory of gravitation and the bookLectures on integral i nvaria nts

1923-1925 Published papers on geometry of spaces with affine,projective, and conformal connections

1925 Published the book Geometry of Riemannian manifolds1926 Created the theory of symmetric Riemannian spaces in the

paper On a remarkable class of Riemannian manifolds1926-1927 Presented lectures in the Sorbonne afterward published

under the title Riemannian geometry in an orthonormalframe

1927 Created the theory of symmetric spaces with an affineconnection in the paper The geometry of transformationsgroups

1928 Published the book Lectures on the geometry of Riemannianmanifolds

1930 Presented lectures in the Moscow University laterpublished under the title The method of moving frames, thetheory of ftni to continuous groups, and generalized spaces

1931 Published the book Lectures on complex projective geometry1931 Elected to the Paris Academy of Sciences1934 Published the book Finsler spaces1937 Published the books Lectures on the theory of spaces with a

projective connection and The theory of finite andcontinuous groups and differential geometry

1938 Published the book Lectures on the theory of spinors1938 Awarded the Lobachevskian prize for geometric works1945 Published the book Exterior differential systems and their

geometric applications1946 Published the second largely augmented edition of the book

Lectures on the geometry of Riemannian manifolds1945 Member of Bureaux of Longitudes1951 Died in Paris, May 6

List of Publications of 1lie Cartan

In the List of Publications of Elie Cartan, we begin by listing chronolog-ically his mathematical works, and then his works in the history of science,his reminiscences, complete collections of his works and collections of hisselected papers, and publications of his scientific correspondence.

List of Cartan's mathematical works

The list of Cartan's mathematical works reproduces the lists published inthe editions [204] (before 1939), [207], and [209]. To our list, we added thetranslations of Cartan's books, as well as his works which were omitted inthe two lists mentioned above. If, in the lists published in the editions [207]and [209], a paper was given under the number with the suffix bis or ter, welist this paper under the same number followed by the letter a, b, or c.

1893

1. Sur la structure des groupes simplesfinis et continus, C. R. Acad. Sci.Paris 116, 784-786; cEuvres completes: Partie I, Groupes de Lie, vols.1-2, Gauthier-Villars, Paris, 19 52, pp. 99-101.

2. Sur la structure des groupes finis et continus, C. R. Acad. Sci. Paris116, 962-964; cEuvres completes: Partie I, Groupes de Lie, vols. 1-2,Gauthier-Villars, Paris, 1952, pp. 103-105.

3. Uber die einfachen Transformationsgruppen, Sitzungsber. Sachs. Ges.Wiss. Leipzig, Mat.-Phys. K1. 45, 395-420; Euvres completes: PartieI, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 107-132.

1894

4. Sur la reduction de la structure dun groupe a sa forme canonique, C. R.Acad. Sci. Paris 119, 639-641; cEuvres completes: Partie I, Groupesde Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 133-135.

5. Sur la structure des groupes de transformations finis et continus, These,Nony, Paris; 2nd ed., Vuibert, Paris, 1933; cEuvres completes: Partie I,Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 137-287.

241

242 LIST OF PUBLICATIONS OF ELIE CARTAN

6. Sur un theoreme de M. Bertrand, C. R. Acad. Sci. Paris 119, 902;TEuvres completes: Partie III, Divers, geometrie differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, p. 1.

7. Sur un theoreme de M. Bertrand, Bull. Soc. Math. France 22, 230-234; Euvres completes: Partie III, Divers, geometrie differentielle, vols.1-2, Gauthier-Villars, Paris, 1955, pp. 3-7.

1895

8. Sur certains groupes algebriques, C. R. Acad. Sci. Paris 120, 544-548; Euvres completes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 289-292.

1896

9. Sur la reduction a sa forme canonique de la structure dun groupe detransformations fini et continu, Amer. J. Math. 18, 1-61; Euvrescompletes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris,1952, pp. 293-353.

10. Le principe de dualite et certaines integrales multiples de l'espace tan-gentiel et de l'espace regle, Bull. Soc. Math. France 24, 140-177;Euvres completes: Partie II, Algebre. Formes differentielles, systemesdifferentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 265-302.

1897

11. Sur les systemes de nombres complexes, C. R. Acad. Sci. Paris 124,1217-1220; Euvres completes: Partie II, Algebre. Formes differen-tielles, systemes differentiels, vols. 1-2, Gauthier-Villars, Paris, 1953,pp. 1-4.

12. Sur les systemes reels de nombres complexes, C. R. Acad. Sci. Paris124, 1296-1297; Euvres completes: Partie II, Algebre. Formes differen-tielles, systemes differentiels, vols. 1-2, Gauthier-Villars, Paris, 1953,pp. 5-6.

1898

13. Les groupes bil ineai res et les systemes de nombres complexes, Ann. Fac.Sci. Toulouse 12B, 1-99; cEuvres completes: Partie II, Algebre. Formesdifferentielles, systemes differentiels, vols. 1-2, Gauthier-Villars, Paris,1953, pp. 7-105.

1899

14. Sur certaines expressions differentielles et le probleme de Pfaf,Ann. Sci. Ecole Norm. Sup. 16, 239-332; cEuvres completes: PartieII, Algebre. Formes differentielles, systemes differentiels, vols. 1-2,Gauthier-Villars, Paris, 1953, pp. 303-396.

1901

15. Sur quelques quadratures dont l'el ement differenti el contient des fonc-tions arbitraires, Bull. Soc. Math. France 29, 118-130; cEuvres com-

LIST OF PUBLICATIONS OF ELIE CARTAN 243

p/etes: Partie II, Algebre. Formes differentielles, systemes differentiels,vols. 1-2, Gauthier-Villars, Paris, 1953, 397-409.

16. Sur /'integration des systemes d'equations aux difjerentielles totales,Ann. Sci. Ecole Norm. Sup. 18, 241-311; (Euvres completes: PartieII, Algebre. Formes difjerenti elles, systemes d iffErenti el s, vols. 1-2,Gauthier-Villars, Paris, 19 53, pp. 411-481.

17. Sur 1 'integration de certain systemes de Pfaff de caractere deux, Bull.Soc. Math. France 29, 233-303; euvres completes: Partie II, Algebre.Formes difjerentielles, systemes diffErentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 483-553.

1902

18. Sur /'integration des systemes differentiels completement integrables. I,C. R. Acad. Sci. Paris 134, 1415-1418; euvres completes: PartieII, Algebre. Formes differentielles, systemes difjerentiels, vols. 1-2,Gauthier-Villars, Paris, 1953, pp. 555-558.

18a. Sur /'integration des systemes diff erentiels completement integrables.II, C. R. Acad. Sci. Paris 134, 1564-1566; euvres completes: PartieII, Algebre. Formes differentielles, systemes difj rentiels, vols. 1-2,Gauthier-Villars, Paris, 1953, pp. 559-561.

19 Sur /'equivalence des systemes difjerentiels, C. R. Acad. Sci. Paris 135,781-783; cEuvres completes: Partie II, Algebre. Formes difjerentielles,systemes difj rentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 563-565.

20. Sur I a structure des groupes infinis, C. R. Acad. Sci. Paris 135, 8 51-853; euvres completes: Partie II, Algebre. Formes difj rentielles, systemes difjerentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 567-569.

1904

21. Sur la structure des groupes infinis de transformations. I, Ann. Sci.Ecole Norm. Sup. 21, 153-206; Euvres completes: Partie II, Algebre.Formes difjerentielles, systemes difjerentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 571-624.

1905

22. Sur la structure des groupes infinis de transformations. II, Ann. Sci.cole Norm. Sup. 22, 219-308; Euvres completes: Partie H. Algebre.Formes difjerentielles, systemes differentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 625-714.

1907

23. Les groupes de transformations continus, infinis, simples, C. R. Acad.Sci. Paris 144, 1094-1097; CEuvres completes: Partie II, Algebre.Formes difjerentielles, systemes difjerentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 715-718.


24. Sur la definition de 1 afire dune portion de surface courbe. I, C. R.Acad. Sci. Paris 145, 1403-1406; Euvres completes: Partie III, Divers,geometrie diff'erentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 9-12.

1908

25. Sur la definition de 1 afire dune portion de surface courbe. II, C. R.Acad. Sci. Paris 146, 168; Euvres completes: Partie III, Divers,geometrie differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, p. 12.

26. Les sous-groupes des groupes continus de transformations, Ann. Sci.Ecole Norm. Sup. 25, 57-194; £uvres completes: Partie II, Algebre.Formes differentielles, systemes differentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 719-856.

27. Nombres complexes, Encyclopedia Math. Sci., edition francaise I 5, pp.329-468; Euvres completes: Partie II, Algebre. Formes difj'erentielles,systemes diff rentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 107-246.

1909

28. Les groupes de transformations continus, infinis, simples, Ann. Sci.cole Norm. Sup. 26, 93-161; Euvres completes: Partie II, Algebre.Formes differentielles, systemes diff'erentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 857-925.

1910

29. Sur les developpables isotropes et la methode du triedre mobile, C. R.Acad. Sci. Paris 151, 919-921; cEuvres completes: Partie III, Divers,geometrie differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955,pp. 141-143.

30. Les systemes de Pfaff a cinq variables et les equations aux deriveespartielles du second ordre, Ann. Sci. cole Norm. Sup. 27, 109-192;iuvres completes: Partie II, Algebre. Formes difjerentielles, systemesdifjerentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 927-1010.

31. La structure des groupes de transformations continus et la theorie dutriedre mobile, Bull. Sci. Math. 34, 250-284; Euvres completes:Partie III, Divers, geometrie diferentielle, vols. 1-2, Gauthier-Villars,Paris, 1955, pp. 145-178.

1911

32. Le calcul des variations et certaines families de courbes, Bull. Soc.Math. France 39, 29-52; cEuvres completes: Partie II, Algebre. Formesdifjerentielles, systemes vols. 1-2, Gauthier-Villars, Paris,1953, pp. 1011-1034

33. Sur les systemes en involution d equations aux derivees partielles dusecond ordre a une fonction inconnue de trois variables independantes,Bull. Soc. Math. France 39, 352-443; (Euvres completes: Partie


II, Algebre. Formes difjerentielles, systemes difjerentiels, vols. 1-2,Gauthier-Villars, Paris, 1953, pp. 1035-1125.

1912

34. Sur les caracteristiques de certains systemes d 'equations aux deriveespartielles, Soc. Math. France 40, C. R. des seances, p. 18; Euvrescompletes: Partie II, Algebre. Formes differentielles, systemes difjeren-tiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 1127.

35. Sur les groupes de transformations de contact et la Cinematique nou-velle, Soc. Math. France 40, C. R. des seances, p. 23; cEuvres completes:Partie III, Divers, geometrie differentielle, vols. 1-2, Gauthier-Villars,Paris, 1955, pp. 179.

1913

36. Remarques sur la composition des forces, Soc. Math. France 41, C. R.des seances, 58-60; Euvres completes: Partie II, Algebre. Formesdifferentielles, systemes differentiels, vols. 1-2, Gauthier-Villars, Paris,1953, pp. 247-248.

37. Les groupes projectifs qui ne laissent invariante aucune multipliciteplane, Bull. Soc. Math. France 41, 53-96; Selecta. Jubile scientifiquede M. Elie Cartan, Gauthier-Villars, Paris, 1939, pp. 137-151; Euvrescompletes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris,1952, pp. 355-398.

1914

38. Les groups reels simples ftnis et continus Ann. Sci. cole Norm. Sup.31, 263-355; Euvres completes: Partie I, Groupes de Lie, vols. 1-2,Gauthier-Villars, Paris, 1952, pp. 399-491.

39. Les groups projectifs continus reels qui ne laissent invariante aucunemultiplicite plane, J. Math. Pures Appl. 10, 149-186; cEuvres com-pletes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris,1952, pp. 493-530.

40. Sur i'integration de certains systemes d'equations diferentielles, C. R.Acad. Sci. Paris 158, 326-328; cEuvres completes: Partie II, Algebre.Formes diferentielles, systemes differentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp.1129-1131.

41. Sur certaines families naturelles de courbes, Soc. Math. France 42,C. R. des seances, 15-17; cEuvres completes: Partie III, Divers, geo-metrie vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 181-183.

42. Sur 1 equivalence absolue de certains systemes d equa ti ons diferentielleset sur certaines families de courbes, Bull. Soc. Math. France 42, 12-48;cEuvres completes: Partie II, Algebre. Formes diferentielles, systemesdifferentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 1133-1168.

43. La theorie des groupes, Revue du Mois 17, 438-468.


1915

44. Sur 1'integration de certains systemes indetermines d'equations differen-tielles, J. Reine Angew. Math. 145, 86-91; Euvres completes: PartieII, Algebre. Formes differentielles, systemes differentiels, vols. 1-2,Gauthier-Villars, Paris, 1953, pp. 1169-1174.

45. Sur les transformations de Backlund, Bull. Soc. Math. France 43,6-24; Euvres completes: Partie II, Algebre. Formes differentielles,systemes differenti el s, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 1175-1193.

46. La theorie des groupes continus et geometrie (the extended translationfrom German of Fano's article [Fa]), Encyclopedia Math. Sci. III 5,332-352; cEuvres completes: Partie III, Divers, geometrie differentielle,vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1727-1861.

1916

47. La deformation des hypersurfaces dans 1'espace euclidien reel a n di-mensions, Bull. Soc. Math. France 44, 65-99; £uvres completes:Partie III, Divers, geometrie differentiell e, vols. 1-2, Gauthier-Villars,Paris, 1955, pp. 185-219.

1917

48. La deformation des hyperfurfaces dans 1'espace conforme reel a n > 5dimensions, Bull. Soc. Math. France 45, 57-121; Euvres completes:Partie III, Divers, geometrie differentielle, vols. 1-2, Gauthier-Villars,Paris, 1955, pp. 221-285.

1918

49. Sur certaines hypersurfaces de 1 'espace conforme reel a cinq dimen-sions, Bull. Soc. Math. France 46, 84-105; Euvres completes: PartieIII, Divers, geometrie differentielle, vols. 1-2, Gauthier-Villars, Paris,1955, pp. 287-308.

50. Sur les varietes a 3 dimensions, C. R. Acad. Sci. Paris 167, 357-359;iuvres completes: Partie III, Divers, geometrie differentielle, vols. 1-2,Gauthier-Villars, Paris, 1955, pp. 309-311.

50a. Sur les varietes developpables a trois dimensions, C. R. Acad. Sci.Paris 167, 42 6- 42 9; cEu vres completes: Partie III, Divers, geometri edifferentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 312-314.

SOb. Sur les varietes de Beltrami a trois dimensions, C. R. Acad. Sci. Paris,167, 482-484; cEuvres completes: Partie III, Divers, geometrie differen-tielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 315-317.

SOc. Sur les varietes de Riemann a trois dimensions, C. R. Acad. Sci. Paris167, 550-55 1; Euvres completes: Partie III, Divers, geometrie differen-tielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 318-319.


1919

51. Sur les varietes de courbure constante dun espace eucl ud ien ou noneuclidien, Bull. Soc. Math. France 47, 125-160; (Euvres completes:Partie III, Divers, geometrie difjerentielle, vols. 1-2, Gauthier-Villars,Paris, 1955, pp. 321-359.

1920

52. Sur les varietes de courbure constants dun espace eucludien ou noneuclidien, Bull. Soc. Math. France 48, 132-208; tEuvres completes:Partie III, Divers, geometrie diferentielle, vols. 1-2, Gauthier-Villars,Paris, 1955, pp. 360-432.

53. Sur la deformation projective des surfaces, C. R. Acad. Sci. Paris 170,1439-1441; Euvres completes: Partie III, Divers, geometrie diferen-tiell e, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 433-435.

53a. Sur 1 applicabilite projective des surfaces, C. R. Acad. Sci. Paris 171,27-29; cEuvres completes: Partie III, Divers, geometrie difjerentielle,vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 437-439.

54. Sur la deformation projective des surfaces, Ann. Sci. cole Norm.Sup. 37, 259-356; £uures completes: Partie III, Divers, geometriedifjerentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 441-538.

55. Sur le probleme general de la deformation, C. R. Congres Math. In-ternat. (Strasbourg, 1920), pp. 397-406; Euvres completes: PartieIII, Divers, geometrie diferentielle, vols. 1-2, Gauthier-Villars, Paris,1955, pp. 539-548.

1922

56. Sur les equations de la gravitation d'Einstein, J. Math. Pures Appl. 1,141-203; tEuvres completes: Partie III, Divers, geometri e di ferenti elle,vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 549-611.

57. Sur une definition geometrique du tenseur d'energie d'Einstein, C. R.Acad. Sci. Paris 174, 437-439; Euvres completes: Partie III, Divers,geometrie difjerentielle, vols. 1-2, Gauthier-Vi1 sirs, Paris, 1955,pp. 613-615.

58. Sur une generalisation de la notion de courbure de Riemann et lesespaces a torsion, C. R. Acad. Sci. Paris 174, 593-595; Euvrescompletes: Partie III, Divers, geometrie difj'erentielle, vols. 1-2,Gauthier-Villars, Paris, 1955, pp. 616-618; English transl., Cosmol-ogy and Gravitation (Bologna, 1979), NATO Adv. Study Inst. Ser.B. Phys., vol. 58, Plenum Press, New York and London, 1980, pp.493-496.

59. Sur les espaces generalises et la theorie de la relativite, C. R. Acad. Sci.Paris 174, 734-736; cEuvres completes: Partie III, Divers, geometriedifferentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 619-621.

60. Sur les espaces conformes generalises et 1'Univers optique, C. R. Acad.Sci. Paris 174, 857-859; Euvres completes: Partie III, Divers, geo-


metrie differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 622-624; English transl., on manifolds with an affine connection and thetheory of general relativity, Bibliopolis, Naples, pp. 195-199.

61. Sur les equations de structure des espaces generalises et 1 expressionanalytique du tenseur d'Einstein, C. R. Acad. Sci. Paris 174, 1104-1106; Euvres completes: Partie III, Divers, geometrie differentielle,vols. 1-2, Gauthier-Villars, Paris, 1955, Partie III, pp. 625-627.

62. Sur un theoreme fondamental de M. H. Weyl dans la theorie de 1'espacemetri que, C. R. Acad. Sci. Paris 175, 82-85; (Euvres completes: PartieIII, Divers, geometrie difjerentielle, vols. 1-2, Gauthier-Villars, Paris,1955, pp. 629-632.

63. Sur les petites oscillations dune masse guide, Bull. Sci. Math. 46,317-352, 356-369; Euvres completes: Partie III, Divers, geometriedifjerentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 13-61.

64. Lecons sur les invariants integraux, Paris, Hermann, 2nd ed., 1958,3rd ed., 1971.

1923

65. Sur un theoreme fondamental de M. H. Weyl, J. Math. Pures Appl. 2,167-192; Euvres completes: Partie III, Divers, geometrie diferentielle,vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 633-658.

66. Sur les varietes a connexion affine et la theorie de la relativity generali-see. I, Ann. Sci. Ecole Norm. Sup. 40, 325-412; Euvres completes:Partie III, Divers, geometrie differentielle, vols. 1-2, Gauthier-Villars,Paris, 1955, pp. 659-746; English transl., On manifolds with an affineconnection and the theory of general relativity, Bibliopolis, Naples,pp. 29-105.

67. Les fonctions reelles non analytiques et les solutions singulieres desequations differentielles du premier ordre, Ann. Polon. Math. 2, 1-8; cEuvres completes: Partie III, Divers, geometrie differentielle, vols.1-2, Gauthier-Villars, Paris, 1955, pp. 63-70.

68. Les espaces a connexion conforme, Ann. Polon. Math. 2, 171- 221;cEuvres completes: Partie III, Divers, geometrie diff eren ti ell e, vols. 1-2,Gauthier-Villars, Paris, 1955, pp. 747-797.

1924

69. Sur les varietes a connexion affine et la theorie de la relativity generali-see. II, Ann. Sci. Ecole Norm. Sup. 41, 1-25; cEuvres completes:Partie III, Divers, geometrie differentielle, vols. 1-2, Gauthier-Villars,Paris, 1955, pp. 799-823; English transl., On manifolds with an affineconnection and the theory of general relativity, Bibliopolis, Naples,pp. 107-127.

70. Sur les varietes a connexion projective, Bull. Soc. Math. France 52,205-241; Selecta. Jubile scientifique de M. Die Cartan, Gauthier-Villars, Paris, 1939, pp. 165-201; Euvres completes: Partie III, Divers,


geometrie diff erentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp.825-861.

71. Les recentes generalisations de la notion d espace, Bull. Sci. Math. 48,294-320; Euvres completes: Partie III, Divers, geometrie differentielle,vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 863-889.

72. La theorie de la relativity et les espaces generalises, Atti V. Cong. In-ternat., Filosofia, pp. 427-436.

73. L a theorie des groupes e t les recherches recentes de geometrie diff eren-tielle, Enseign. Math. 24 (1925),1-18; Proc. Internat. Math. CongressToronto 1 (1928 ), 8 5-94; Euvres completes: Partie III, Divers,geometrie diff erentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp.891-904.

74. Sur lesformes diff erentielles en geometrie, C. R. Acad. Sci. Paris 178,182-184; Euvres completes: Partie III, Divers, geometrie diferentielle,vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 905-907.

75. Sur la connexion afne des surfaces, C. R. Acad. Sci. Paris 178, 292-295; Euvres completes: Partie III, Divers, geometrie differentielle, vols.1-2, Gauthier-Villars, Paris, 1955, pp. 209-212.

76. Sur la connexion affine des surfaces devel oppabl es, C. R. Acad. Sci.Paris 178, 449-451; Euvres completes: Partie III, Divers, geometriediferentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 912-914.

77. Sur la connexion projective des surfaces, C. R. Acad. Sci. Paris 178,750-752; Euvres completes: Partie III, Divers, geometrie diff erenti ell e,vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 215-217.

1925

78. Note sur la generation des oscillations entretenues (with Henri Cartan),Ann. Postes, Tel. et Tel. 14, 1196-1207; cEuvres completes: PartieIII, Divers, geometrie diff erentielle, vols. 1-2, Gauthier-Villars, Paris,1955, pp. 71-82.

79. Les groupes d 'hol on omi e des espaces generalises et l'A nal ysissitus, Assoc. Avanc. Sciences, 49e session, Grenoble, pp. 47-49;cEuvres completes: Partie III, Divers, geometrie differentielle, vols.1-2, Gauthier-Villars, Paris, 1955, pp. 919-920.

80. Sur les varietes a connexion affine et la theorie de la relativity generali-see, Ann. Sci. Ecole Norm. Sup. 42, 17-88; cEuvres completes: PartieIII, Divers, geometrie diff erentielle, vols. 1-2, Gauthier-Villars, Paris,1955, pp. 921-992; English transl., on manifolds with an affine connec-tion and the theory of general relativity, Bibliopolis, Naples, pp. 129-193.

81. Les tenseurs irred ucti bles et les groupes l ineai res simples et semi-simples,Bull. Sci. Math. 49, 130-152; cEuvres completes: Partie I, Groupes deLie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 531-553.


82. Le principe de d ual ite e t la theorie des groupes simples et semi-simples,Bull. Sci. Math. 49, 361-374; £uures completes: Partie I, Groupes deLie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 555-568.

83. Sur le mouvement a deux parametres, Nouvelles Ann. 1, 33-37;cEuvres completes: Partie III, Divers, geometrie differentielle, vols. 1-2,Gauthier-Villars, Paris, 1955, pp. 83-87.

84. La geometri e des espaces de R ieman n, Memorial Sci. Math. IX,Gauthier-Villars, Paris.

1926

85. L'application des espaces de Riemann et l'Anal ysis situs, Assoc. Avanc.Sciences, 50' session, Lyon, pp. 53-55; Euvres completes: PartieIII, Divers, geometrie differentielle, vols. 1-2, Gauthier-Villars, Paris,1955, pp. 993-995.

86. Sur certains systemes differentiels dont les inconnues sont des former dePfaff; C. R. Acad. Sci. Paris 182, 956-958; cEuvres completes: PartieII, Algebre. Formes differentielles, systemes differentiels, vols. 1-2,Gauthier-Villars, Paris, 1953, pp. 1195-1197.

87. Sur les espaces de Riemann dans lesquels le transport par parallelismeconserve la courbure, Rend. Accad. Lincei 31, 544-547; Euvrescompletes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris,1952, pp. 569-572.

88. Les groupes d'holonomie des espaces generalises, Acta Math. 48, 1-42; Euvres completes: Partie III, Divers, geometrie diferentielle, vols.1-2, Gauthier-Villars, Paris, 1955, pp. 997-1038.

89. Sur les spheres des espaces de Riemann a trois dimensions, J. Math.Pures Appl. 5, 1-18; tuvres completes: Partie III, Divers, geometriedifferentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1039-1056.

90. L'axiome du plan et la geometrie differentielle metrique, in Memoriamof N. I. Lobatschevskii, vol. 2, "Glavnauka", Kazan, 1927, pp. 4-12; Euvres completes: Partie III, Divers, geometrie diferentielle, vols.1-2, Gauthier-Villars, Paris, 1955, pp. 1057-1065.

91. On the geometry of the group-manifold of simple and semi-simple groups(with J. A. Schouten), Proc. Akad. Wet. Amsterdam 29, 803-815;Euvres completes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 573-585.

92. On Ri emanni an Geometries admitting an absolute parallelism(with J. A. Schouten), Proc. Akad. Wet. Amsterdam 29, 933-946;cEuvres completes: Partie III, Divers, geometrie diferentielle, vols. 1-2,Gauthier-Villars, Paris, 1955, pp. 1067-1080.

93. Sur une classe remarquable d'espaces de Riemann, Bull. Soc. Math.France 54, 214-264; Euvres completes: Partie I, Groupes de Lie, vols.1-2, Gauthier-Villars, Paris, 1952, pp. 587-637.


1927

94. Sur une classe remarquable d espaces de Riemann, Bull. Soc. Math.France 55 , 114-134; Euvres completes: Partie I, Groupes de Lie, vols.1-2, Gauthier-Villars, Paris, 1952, pp. 639-659.

95. Sur les courbes de torsion nulle et les surfaces developpables dansles espaces de Riemann, C. R. Acad. Sci. Paris 184, 138-140;Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1-2,Gauthier-Villars, Paris, 1955, pp. 1081-1083.

96. Sur les geodesiques des espaces de groupes simples, C. R. Acad. Sci.Paris 184, 862-864; Euvres completes: Partie I, Groupes de Lie, vols.1-2, Gauthier-Villars, Paris, 1952, pp. 661-663.

97. Sur la topologic des groupes continus simples reels, C. R. Acad. Sci.Paris 184, 1036-1038; Euvres completes: Partie I, Groupes de Lie,vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 664-666.

98. Sur 1 ecart geod esique et quelques questions connexes, Rend. Accad.Lincei 51, 609-613; cEuvres completes: Partie III, Divers, geometriedifferentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1085-1089.

99. Sur certaines formes riemanniennes remarquables des geometries agroupe fondamental simple, C. R. Acad. Sci. Paris 184, 1628-1630;cEuvres completes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 667-669.

100. Sur lesformes riemanniennes des geometries a groupefondamental sim-ple, C. R. Acad. Sci. Paris 185, 96-98; cEuvres completes: Partie I,Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 670-672.

101. La geometrie des groupes de transformations, J. Math. Pures Appl.6, 1-119; cEuvres completes: Partie I, Groupes de Lie, vols. 1-2,Gauthier-Villars, Paris, 1952, pp. 673-791.

102. Sur certains cycles arithmetiques, Nouvelles Ann. 2, 33-45; Euvrescompletes: Partie III, Divers, geometrie differentielle, vols. 1-2,Gauthier-Villars, Paris, 1955, pp. 89-101.

103. La geometrie des groupes simples, Ann. Mat. 4, 209-256; Euvrescompletes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris,1952, pp. 793-840.

104. Sur la possibilite de plonger un espace riemannien donne dans un es-pace euclidien, Ann. Polon. Math. 6, 1-7; cEuvres completes: PartieIII, Divers, geometrie differentielle, vols. 1-2, Gauthier-Villars, Paris,1955, pp. 1091-1097.

105. La theorie des groupes et la geometrie, Enseign. Math. 26, 200-225;cEuvres completes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 841-866.

106. Rapport sur le memoire de J. A. Schouten intitule "Erlanger programmand Ubertragungslehre. Neue Gesichtspunkte zur Grundlegung der Ge-ometrie", Izv. Kazan Fiz.-Mat. Obshch. 2, 71-76; CEuvres completes:


Partie III, Divers, geometrie differentielle, vols. 1-2, Gauthier-Villars,Paris, 1955, pp. 1099-1104.

107. Sur certaines formes riemanniennes remarquables des geometries agroupe fondamental simple, Ann. Sci. Ecole Norm. Sup. 44, 345-467; (Euvres completes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 867-989.

108. Sur un probl eme d u calcul des variations en geometrie projective plane,Mat. Sb. 34, 349-364; Euvres completes: Partie III, Divers, geometriedifferentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1105-1119.

108a. Riemannian geometry in an orthogonal frame (Cartan 's 1926-1927lectures in Sorbonne), Izdat. Moskov. Univ., Moscow, 1960. (Rus-sian)

1928

109. Sur les systemes orthogonaux complets de fonctions dans certains es-paces de Riemann clos, C. R. Acad. Sci. Paris 186, 1594-1596; Euvrescompletes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris,1952, pp. 991-993.

110. Sur les espaces de Riemann clos admettant un groupe continu transitifde deplacements, C. R. Acad. Sci. Paris 186, 1817-1819; *Euurescompletes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris,1952, pp. 995-997.

111. Sur les nombres de Betti des espaces de groupes clos, C. R. Acad. Sci.Paris 187, 196-198; cEuvres completes: Partie I, Groupes de Lie, vols.1-2, Gauthier-Villars, Paris, 1952, pp. 999-1001.

112. Sur la stabilite ordinaire des ellipsoides de Jacobi, Proc. Internat.Math. Congress Toronto 1, 9-17; Euvres completes: Partie III, Divers,geometrie differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955,pp. 103-111.

113. Complement au memoire "Sur la geometrie des groupes simples", Ann.Mat. Pura Appl. (4) 5, 253-260; Euvres completes: Partie I, Groupesde Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 1003-1010.

114. Lecons sur la geometrie des espaces de Riemann, Gauthier-Villars,Paris.

115. Sur les substitutions orthogonales imaginaires, Assoc. Avanc. Sci-ences, Congres de La Rochelle, pp. 3 8-40; cEuvres completes: Par-tie II, Algebre. Formes differentielles, systemes differentiels, vols. 1-2,Gauthier-Villars, Paris, 1953, pp. 249-250.

1929

116. Groupes simples clos et ouverts et geometrie riemannienne, J. Math.Pures Appl. 8, 1-3 3; cEuvres completes: Partie I, Groupes de Lie, vols.1-2, Gauthier-Villars, Paris, 1952, pp. 1011-1043.

117. Sur la determination d'un systeme orthogonal complet dans un espacede Riemann symetrique clos, Rend. Circ. Mat. Palermo 53, 217-


252; Euvres completes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 1045-1080.

118. Sur les invariants integraux de certains espaces homogenes clos et lesproprietes topologiques de ces espaces, Ann. Polon. Math. 8, 181-225;Selecta. Jubile scientiftque de M. Elie Cartan, Gauthier-Villars, Paris,1939, pp. 203-233;uvres completes: Partie I, Groupes de Lie, vols.1-2, Gauthier-Villars, Paris, 1952, pp. 1081-1125.

119. Sur la representation geometrique des systemes materiels non holo-nomes, Atti Cong. Internat. Mat. (Bologna, 1928), vol. 4, pp. 253-261; Euvres completes: Partie III, Divers, geometrie differentielle, vols.1-2, Gauthier-Villars, Paris, 1955, pp. 113-121.

120. Sur les espaces clos admettant un groupe transitif clos ftni et continu,Atti Cong. Internat. Mat. (Bologna, 1928), vol. 4, pp. 243-252;Euvres completes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 1127-1136.

1930

121. Les representations lineaires du groupe des rotations de la sphere, C. R.Acad. Sci. Paris 190, 610-612;uvres completes: Partie I, Groupesde Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 1137-1139.

122. Les representations lineaires des groupes simples et semi-simples clos,C. R. Acad. Sci. Paris 190, 723-725; cEuvres completes: Partie I,Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 1140-1142.

123. Le troisieme theoreme fondamental de Lie. I, C. R. Acad. Sci. Paris190, 914-916; cEuvres completes: Partie I, Groupes de Lie, vols. 1-2,Gauthier-Villars, Paris, 1952, pp. 1143-1145.

123a. Le troisieme theoreme fundamental de Lie. II, C. R. Acad. Sci. Paris190, 1005-1007; cEuvres completes: Partie I, Groupes de Lie, vols.1-2, Gauthier-Villars, Paris, 1952, pp. 1146-1148.

124. Notice historique sur la notion de parallelisme absolu, Math. Ann. 102,698-706; cEuvres completes: Partie III, Divers, geometri e diff erenti ell e,vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1121-1129.

125. Sur les representations lineaires des groupes clos, Comment. Math.Helv. 2, 269-283; cEuvres completes: Partie I, Groupes de Lie, vols.1-2, Gauthier-Villars, Paris, 1952, pp. 1149-1163.

126. Sur un probl eme d equivalence et la theorie des espaces metri ques gene-ralises, Mathematica 4, 114-136; cEuvres completes: Partie III, Divers,geometrie differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp.1131-1153.

127. Geometrie projective et geometrie riemannienne, Trudy I Vsesoyuz.Matem. S'ezda, Khar'kov, 1930, 179-190; Euvres completes: PartieIII, Divers, geometrie diff erentielle, vols. 1-2, Gauthier-Villars, Paris,1955, pp. 1155-1166.


128. La theorie des groupes fanis et continus et l'Analysis situs, Memorial Sci.Math. XLII, 2nd ed., Gauthier-Villars, Paris, 1952; Euvres completes:Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp.1165-1225.

1931

129. Geometrie euclidienne et geometrie riemannienne, Scientia (Milano),393-402.

130. Le parallelisme absolu et la theorie unitaire du champ, Rev. Metaph.Morale, pp. 13-28; Actualites Sci. Indust., no. 44, Hermann, Paris,1932; 2nd ed., 1974; Euvres completes: Partie III, Divers, geometriedifferentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1167-1185.

131. Sur la theorie des systemes en involution et ses applications a la Rela-tivite, Bull. Soc. Math. France 59, 88-118; £uvres completes: PartieII, Algebre. Formes differentielles, systemes differentiels, vols. 1-2,Gauthier-Villars, Paris, 1953, pp. 1199-1229.

132. Sur les developpantes dune surface regle, Bull. Acad. Roumaine 14,167-174; Euvres completes: Partie III, Divers, geometrie differentielle,vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1187-1194.

133. Le groupe fondamental de la geometrie des spheres orientees reelles,Assoc. Avanc. Sciences, Nantes, pp. 21-28; Tuvres completes: PartieIII, Divers, geometrie differentielle, vols. 1-2, Gauthier-Villars, Paris,1955, pp. 1195-1202.

134. Lecons sur la geometrie projective complexe, Gauthier-Villars, Paris;2nd. ed., 1950.

1932

135. Sur le groupe de la geometrie hyperspherique, Comment. Math. Helv.4, 158-171; Euvres completes: Partie III, Divers, geometrie differen-tielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1203-1216.

136. Sur la geometrie pseudo-conforme des hypersurfaces de l'espace de deuxvariables complexes. I, Ann. Mat. Pura Appl. (4) 11, 17-90; Euvrescompletes: Partie II, Algebre. Formes differentielles, systemes differen-tiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 1231-1304.

136a. Sur 1 a geometrie pseudo-conforme des hypersurfaces de-l'espace de deuxvariables complexes. II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 1, 333-354; Euvres completes: Partie III, Divers, geometrie differentielle, vols.1-2, Gauthier-Villars, Paris, 1955, pp. 1217-1238.

137. Sur les proprietes topologiques des quadriques complexes, Publ. Math.Univ. Belgrade 1, 55-74; cEuvres completes: Partie I, Groupes de Lie,vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 1227-1246.

138. Les espaces riemanniens symetriques, Verh. Internat. Math. Kon-gresses Zurich, vol. I, pp. 152-161; (Euvres completes: Partie I,

Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 1247-1256.


139. Sur l 'equivalence pseudo-conforme de deux hypersuf faces de 1'espace dedeux variables complexes, Verh. Internat. Math. Kongresses Zurich,vol. II, pp. 54-56; Euvres completes: Partie II, Algebre. Formesdifferentielles, systemes differentiels, vols. 1-2, Gauthier-Villars, Paris,1953, pp. 1305-1306.

1933

140. Les espaces metriques fondessur la notion d'aire, Exposes de Geometrie,vol. I, Hermann, Paris.

140a. La cinematique newtonienne et les espaces a connexion euclidienne,Bull. Math. Soc. Sci. Math. R. S. Roumanie 35 (1933), 69-73;cEuvres completes: Partie III, Divers, geometrie differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1239-1243.

140b. Observations sur: St. Golgb. Sur la representation conforme de 1'espacede Finsl er sur 1 'espace eucl id i en, C. R. Acad. Sci. Paris 196, 2 7 - 29.

141. Sur les espaces de Finsler, C. R. Acad. Sci. Paris 196, 582-586;cEuvres completes: Partie III, Divers, geometrie differentielle, vols. 1-2,Gauthier-Villars, Paris, 1955, pp. 1245-1248.

141 a. Observations sur le memoire precedent (lettre a D. D. Kosambi), Math.Z. 37, 619-622; Euvres completes: Partie III, Divers, geometriedifferentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1249-1252.

1934

142. Les espaces de Finsler, Exposes de Geometrie, vol. II, Hermann, Paris.142a. Remarques au sujet de la Communication de M. Andre Weil, C. R.

Acad. Sci. Paris 198, 1742-1743; cEuvres completes: Partie I, Groupesde Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 1257-1258.

143. Le calcul tensoriel en geometrie projective, C. R. Acad. Sci. Paris 198,2033-2037; Euvres completes: Partie III, Divers, geometrie differen-tielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1253-1257.

143a. La theorie unitaire d'Einstein Mayer, preprint; cEuvres completes: Par-tie III, Divers, geometrie differentielle, vols. 1-2, Gauthier-Villars,Paris, 1955, pp. 1863-1875.

1935

144. La methode du repere mobile, la theorie des groupes continus et lesespaces generalises, Exposes de Geometric, vol. V, Hermann, Paris;Euvres completes: Partie III, Divers, geometrie differentielle, vols. 1-

2, Gauthier-Villars, Paris, 1955, pp. 1259-1320.145. Sur les domaines bornes homogenes de 1'espace de n variables com-

plexes, Abh. Math. Sem. Univ. Hamburg 11, 116-162; cEuvrescompletes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris,1952, pp. 1259-1305.

145a. Remarques au sujet dune communication de M. L. Pontfjagin sur lesnombres de Betti des groupes de Lie, C. R Acad. Sci. Paris 200, 1280-1281.


146. Observations sur une Note de M. G. Bouligand, C. R. Acad. Sci. Paris201, 702; Euvres completes: Partie III, Divers, geometrie differentielle,vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 132 1.

147. Le calcul tensoriel projectif, Mat. Sb. 42, 131-147; Euvres completes:Partie III, Divers, geometrie differentielle, vols. 1-2, Gauthier-Villars,Paris, 1955, pp. 1 323- 1 339.

147a. Sur une degenerescence de la geometrie euclidienne, Assoc. Avanc.Sciences, Nantes, pp. 1 28- 1 30; this book, Appendix B.

1936

148. L a geometrie de l 'i ntegral e f F (x , y , y' , y") d x , J. Math. Pures Appl.15, 42-69; Euvres completes: Partie III, Divers, geometrie diferen-ti elle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1341-1368.

149. Sur les champs d acceleration uniforme en Relativite restreinte, C. R.Acad. Sci. Paris 202, 1125-1128; (Euvres completes: Partie III, Divers,geometrie diferentielle, vols. 1-2, Gauthier-Villars, Paris, 1955,pp. 1 369- 1 372.

150. La topologie des espaces representatifs des groupes de Lie, Exposes deGeometrie, vol. VIII, Hermann, Paris; Enseign. Math. 35, 177-200;Selecta. Jubile scientifaque de M. Elie Cartan, Gauthier-Villars, Paris,1939, pp. 235-258; cEuvres completes: Partie I, Groupes de Lie, vols.1-2, Gauthier-Villars, Paris, 1952, pp. 1 307- 1 330.

151. Le role de la theorie des groupes de Lie dans 1'evolution de la geometriemoderne, C. R. Congres Math. Internat. (Oslo), vol. 1, pp. 92- 1 03;Euvres completes: Partie III, Divers, geometrie diferentielle, vols. 1-2,Gauthier-Villars, Paris, 1955, pp. 1 373- 1 384.

1937

152. Les espaces de Finsler, Trudy Sen. Vektor. Tenzor. Anal. 4, 70-8 1; cEuvres completes: Partie III, Divers, geometrie diff'erentielle, vols.1-2, Gauthier-Villars, Paris, 1955, pp. 1385-1396.

153. Les espaces a connexion projective, Trudy Sem. Vektor. Tenzor. Anal.4, 1 47- 1 59; cEuvres completes: Partie III, Divers, geometrie diff'eren-tielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1 397- 1 409.

154. La topologie des espaces homogenes clos, Trudy Sem. Vektor. Tenzor.Anal. 4, 388-394; euvres completes: Partie I, Groupes de Lie, vols.1-2, Gauthier-Villars, Paris, 1 9 52, pp. 1331-1337.

155. Lecons sur la theorie des espaces a connexion projective, Gauthier-Villars, Paris.

156. L'extension du calcul tensoriel aux geometries non-affines, Ann. ofMath. (2) 38, 1-13; cEuvres completes: Partie III, Divers, geometriediferentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1411-1423.

157. La theorie des groupes finis et continus et la geometrie di fferentielletraitees par la methode du repere mobile, Gauthier-Villars, Paris; 2nded., 1951.


158. L e role de la geometrie analytique dans 1'evol uti on de la geometrie,Travaux du XIth Congres Internat. Philosophic Descartes),vol. VI, Paris, 147-153; Actualites Sci. Indust., no. 535, Hermann,Paris.

159. Les groupes, Encyclopedia Frangaise, vol. 1, 3rd part, I.66-1- I.66-8.160. La geometrie et la theorie des groupes, Encyclopedic Francaise, vol. 1,

3rd part, 1.88-12-1.90-2.161. La geometrie riemannienne et ses generalisations, Encyclopedic Fran-

caise, vol. 1, 3rd part, I.90-3 - I.90-8.161 a. Les problemes d equivalence, Seminaire de Math. expose D, l l jan-

vier 1937; Selecta. Jubile scientifique de M. 'lie Cartan, Gauthier-Villars, Paris, 1939, pp. 113-136; Euvres completes: Partie II, Algebre.Formes difjerentielles, systemes differentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 1311-1334.

161b. La structure des groupes infinis, Seminaire de Math., exposes G et H,1 er et 15 mars 1937, pp. 1-50; Euvres completes: Partie II, Algebre.Formes differentielles, systemes differentiels, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 1335-1384.

1938

162. Les representations lineaires des groupes de Lie, J. Math. Pures Appl.17, 1-12; Selecta. Jubile scientifique de M. 'lie Cartan, Gauthier-Villars, Paris, 1939, pp. 1 53- 1 64; Euvres completes: Partie I, Groupesde Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 1339-1351.

163. Les espaces generalises et 1 'integration de certaines classes d'equati onsdifferentielles, C. R. Acad. Sci. Paris 206, 1689-1693; cEuvres com-pletes: Partie III, Divers, geometrie differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1425-1429.

164. Lecons sur la theorie des spineurs. I, II, Exposes de Geometric, vol.XI, Hermann, Paris; English transl., Hermann, Paris and MIT Press,1966; 2nd ed., Dover, New York, 1981.

165. La theorie de Galois et ses generalisations, Comment. Math. Helv.11, 9-2 5; cEuvres completes: Partie III, Divers, geometrie diferenti elle,vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 123-139.

166. Familles de surfaces isoparametriques dans les espaces a courbure con-stante, Ann. Mat. Pura AppL (4) 27, 177-191; Euvres completes:Partie III, Divers, geometrie difjerentielle, vols. 1-2, Gauthier-Villars,Paris, 1955, pp. 1431-1445.

1939

167. Sur des families rejnarquables d'hypersurfaces isoparametriques dans lesespaces spheriques, Math. Z. 45, 335-367;uvres completes: PartieIII, Divers, geometrie diferentielle, vols. 1-2, Gauthier-Villars, Paris,1955, pp. 1 447- 1 479.


168. Sur quelques families remarquables d' hypersurfaces, C. R. CongresMath. de Liege, pp. 30-41; Euvres completes: Partie III, Divers,geometrie diferentielle, vols. 1-2, Gauthier-Villars, Paris, 1955,pp. 1481-1492.

169. L e calcul diferenti el absolu devant les probl emes recents de geometrieriemannienne, Atti Fondaz. Alessandra Volta 9, 443-461; cEuvrescompletes: Partie III, Divers, geometrie diferentielle, vols. 1-2,Gauthier-Villars, Paris, 1955, pp. 1493-1511.

1940

170. Sur un theoreme de J. A. Schouten et W. van der Kulk, C. R. Acad.Sci. Paris 211, 21-24; Euvres completes: Partie II, Algebre. Formesdiferentielles, systemes diferentiels, vols. 1-2, Gauthier-Villars, Paris,1953, pp. 1307-1310.

171. Sur les groupes lineaires quaternioniens, Vierteljschr. Naturforsch.Ges. Zurich, 85, 191-203; Euvres completes: Partie II, Algebre.Formes diferentielles, systemes diferentiels, vols. 1-2, Gauthier-Villars,Paris, 1953, pp. 251-263.

172. Sur des families d'hypersurfaces isoparametriques des espaces spheriquesa 5 et 9 dimensions, Univ. Nac. Tucum a n. Revista A 1, 5-22; Euvrescompletes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris,1952, pp. 1513-1530.

1941

173. Sur les surfaces admettant une seconde forme fondamentale donnee,C. R. Acad. Sci. Paris 212, 825-828; cEuvres completes: PartieIII, Divers, geometrie diferentielle, vols. 1-2, Gauthier-Villars, Paris,1955, pp. 1531-1534.

174. La geometria de las ecuaciones diferenciales de tercer orden, RevistaMat. Hisp.-Amer. (1) 1, 3-33; cEuvres completes: Partie III, Divers,geometrie diferentielle, vols. 1-2, Gauthier-Villars, Paris, 1955,pp. 1535-1565.

175. La notion d'orien tati on dans les diferentes geometries, Bull. Soc. Math.France 69, 47-70; cEuvres completes: Partie III, Divers, geometrie

vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1569-1570.

1942

176. Sur les couples de surfaces applicables avec conservation des courburesprincipales, Bull. Sci. Math. (2) 66, 55-85; cEuvres completes: PartieIII, Divers, geometrie diferentielle, vols. 1-2, Gauthier-Villars, Paris,1955, pp. 1591-1621.

177. Les surfaces isotropes d 'une quad ri que de 1 'espace a sept dimensions,preprint.


1943

178. Sur une classe d'espaces de Weyl, Ann. Sci. Ecole Norm. Sup. (3)60, 1-16; Euvres completes: Partie III, Divers, geometri e d ifferenti ell e,vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1621-1636.

179. Les surfaces q u i admettent une second e forme fondamensale d onnee,Bull. Sci. Math. (2) 67, 8-32; Euvres completes: Partie III, Divers,geometrie difjerentielle, vols. 1-2, Gauthier-Villars, Paris, 1955,pp. 1637-1661.

1944

180. Sur une classe de surfaces apparentees aux surfaces R et aux surfacesde Jonas, Bull. Sci. Math. (2) 68, 41-50; Euvres completes: PartieIII, Divers, geometrie diferentielle, vols. 1-2, Gauthier-Villars, Paris,1955, pp. 1663-1672.

1945

181. Les systemes diferentiels exterieurs et leurs applications geometriques,Actualites Sci. Indust., no. 994, Hermann, Paris.

182. Sur un probleme de geometrie difjerentielle projective, Ann. Sci. EcoleNorm. Sup. (3) 62, 205-231; Euvres completes: Partie III, Divers,geometrie differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955,pp. 167 3-1699.

1946

183. Lecons sur la geometrie des espaces de Riemann, 2nd ed., Gauthier-Villars, Paris; English transl. in Lie Groups, History, Frontiers andApplications, vol. 13, Math. Sci. Press, Brookline, MA, 1983.

184. Quelques remarques sur les 28 bitangentes d 'une quartique plane etles 27 droites dune surface cubique, Bull. Sci. Math. (2) 70, 42-45; Euvres completes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris, 1952, pp. 1353-1356.

1947

185. Sur 1 espace anallagmatique reel a n dimensions, Ann. Polon. Math.20, 266-278; cEuvres completes: Partie III, Divers, geometrie difjeren-tielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1701-1713.

185a. La theorie des groupes, Alencon, Paris.

1949

186. Deux theoremes de geometrie anallagmatique reelle a n dimensions,Ann. Mat. Pura Appl. (4) 28, 1-12; cEuvres completes: PartieIII, Divers, geometrie difjerentielle, vols. 1-2, Gauthier-Villars, Paris,1955, pp. 1715-1726.


List of Cartan's works in the history of scienceand his reminiscences

1931

187. Notices sur les travaux scientifiques, Paris; Selecta. Jubile scientifique deM. Elie Cartan, Gauthier-Villars, Paris, 1939, 15-112; .uvres com-pletes: Partie I, Groupes de Lie, vols. 1-2, Gauthier-Villars, Paris,1952, pp. 1-101; Gauthier-Villars, Paris; 2nd ed., 1974.

1937

188. Discours prononce a /'inauguration d'un buste eleve a la memoire deGaston Darboux a Nimes le dimanche 22 octobre 1933, Notices etDiscourrs Acad. Sci. Paris 1924-1936, pp. 437-478.

1939

189. Allocution a la Sorbonne 18 mai 1939, in Jubile scientifique de M.Elie Cartan celebre a la Sorbonne 18 mai 1939, Gauthier-Villars, Paris,1939, pp. 51-59; this book, Appendix C.

1941

190. Charles Maurain, Jubile de Charles Maurain, Paris, pp. 5- 1 4.191. Le role de la France dans le developpement des mathematiques, preprint;

English transl., this book, Appendix D*.

1942

192. Notice sur Tullio Levi-Civita, C. R. Acad. Sci. Paris 215, 233-235.1943

193. Notice necrologique sur Georges Giraud, C. R. Acad. Sci. Paris 216,516-518.

1946

194. Notice necrologique sur Antoine-Francois Jacques-Justin-GeorgesPerrier, C. R. Acad. Sci. Paris 222, 421-423.

195. Notice necrologique sur Thomas Hunt Morgan, C. R. Acad. Sci. Paris222, 705-706.

196. Notice necrologique sur Leon Alexandre Guillet, C. R. Acad. Sci. Paris222, 1149-1151.

197. Notice necrologique sur Simon Flexner, C. R. Acad. Sci. Paris 222,1265-1266.

198. Notice necrologique sur Louis Martin, C. R. Acad. Sci. Paris 222,1417-1419.

198a. Gaspard Monge. Sa vie, son a'uvre, C. R. Acad. Sci. Paris 223, 1049-1054.

*Added in Proof. The Serbian publication (without the preface): Saturn, 1940, No. 4-5,81-96, No. 6-7, 129-144. The publication of Cartan's French text: Publications de 1'InstitutMathematique (N.S.), Beograd 51 (65), 1992, 2-21.


199. Notice necrologique sur Paul Langevin, C. R. Acad. Sci. Paris 223,1069-1072.

200. L'a uvre scientifique de M. Ernest Vessiot, Bull. Soc. Math. France 75,1-8.

1948

201. Un centenaire: Sophus Lie, Les grands courants de la pensee mathema-tique, Cahiers du Sud, pp. 253-257; 2nd ed., vol. 1, Paris, 1962;English transl., Great currents of mathematical thought, vol. 1, Dover,New York, 1971, pp. 262-267.

202. Gaspard Monge: sa vie, son cruvre, Alencon, Paris.1949

203. La vie et l'a?uvre de Georges Perrier, Annuaire Bureau des Longitudes,Paris, c 1-c4.

Collections of Cartan's works

204. Selecta. Jubile scientifique de M Elie Cartan, Gauthier-Villars, Paris,1939.

205. Gruppy golonomii obobshchennykh prostranstv. Teoriya grupp i ge-ometriya. Metricheskieprostranstva osnovannye na ponyatii ploshchadi,Series of Monographs and Studies in Non-Euclidean Geometry, no. 1,Izdat. Kazan. Univ., Kazan, 1939.

206. Geometriya grupp Lie i simmetricheskie prostranstva, Izdat. Inostr.Literat., Moscow, 1949.

207. Euvres completes: Partie I, Groupes de Lie, vols. 1-2, 19 52; Partie II,Algebre. Formes differentielles, systemes diferentiels, vols. 1-2, 1953;Partie III, Divers, geometrie differentielle, vols. 1-2, 1955, Gauthier-Villars, Paris.

208. Prostranstva affinnoi, proyektivnoii konformnoi svyaznosti, Series ofMonographs and Studies in Non-Euclidean Geometry, no. 3, Izdat.Kazan. Univ., Kazan, 1962.

209. Euvres completes: Partie I, Groupes de Lie; Partie II, Algebre. Formesdiferentielles, systemes differentiels, Partie III, Geometrie diferentielle.Divers, vols. 1-2, C. N. R. S., Paris, 1984.

209a. On manifolds with an affine connection and the theory of general rela-tivity, Bibliopolis, Naples, 1986.

Cartan's scientific correspondence

210. Die Cartan Albert Einstein letters on absolute parallelism 1929-1932,Princeton Univ. Press, Princeton, NJ, 1979.

211. Lettres d'E. Cartan a G. Tzitzeica, A. Pantazi et G. Vranceanu, ElieCartan, 1869-1951, Hommage de l'Acad. Republique Socialiste deRoumanie, a l'occasion du centenaire de sa naissance, Editura Acad.R.S.R., Bucharest, 1975, pp. 83-116.

APPENDIX A

Rapport sur les Travaux de M. Cartanfait A la Faculte des Sciences de l'Universite de Paris

PAR

H. POINCARE'

..... Le role preponderant de la theorie des groupes en mathematiques a etelongtemps insoupconne; it y a quatre-vingts ans, le nom meme de groupe etaitignore. Cest GALois qui, le premier, en a eu une notion claire, mais c'estseulement depuis les travaux de KLEIN et surtout de Lie que l'on a commencea voir qu'il n'y a presque aucune theorie mathematique ou cette notion netienue une place importante.

On avait cependant remarque comment se font presque touj ours les progresdes mathematiques; c'est par generalisation sans doute, mais cette generalisa-tion ne s'exerce pas dans un sens quelconque. On a pu dire que la mathe-matique est l'art de donner le meme nom a des choses differentes. Le jourou on a donne le nom d'addition geometrique a la composition des vecteurs,on a fait un progres serieux, si bien que la theorie des vecteurs se trouvait amoitie faite; on en a fait un autre du meme genre quand on a donne le nomde multiplication a une certaine operation portant sur les quaternions. 11 estinutile de multiplier les exemples, car toutes les mathematiques y passeraient.Par cette similitude de nom, en effet, on met en evidence une similitude defait, une sorte de parallelisme qui aurait pu echapper a l'attention. On n'aplus ensuite qu'a' calquer, pour ainsi dire, la theorie nouvelle sur une theorieancienne deja connue.

II faut s'entendre, toutefois: it faut donner le meme nom a des chosendifferentes, mais a la condition que ces choses soient differentes quant "a lamatiere, mais non quant a la forme. A quoi tient cc phenomene mathema-tique si souvent constate? Et d'autre part en quoi consiste cette communautede forme qui subsiste sous la diversite de la matiere? Elle tient a cc que toutetheorie mathematique est, en derniere analyse, l'etude des proprietes d'ungroupe d'operations, c'est-a-dire d'un systeme forme par certaines operations

IActa Mathematlca 38 (1914), 137-145.

263

264 A. RAPPORT SUR LES TRAVAUX DE M. CARTAN, BY H. POINCARE

fondamentales et par toutes les combinaisons qu'on en peut faire. Si, dansune autre theorie, on etudie d'autres operations qui se combinent d'apres lesmemes lois, on verra naturellement se derouler une suite de theoremes corre-spondant un a un a ceux de la premiere theorie, et les deux theories pourrontse developper avec un parallelisme parfait; it suffira d'un artifice de langage,comme ceux dont nous parlions tout a l'heure, pour que cc parallelisme de-vienne manifeste et donne presque l'impression d'une identite complete. Ondit alors que les deux groupes d'operations sont isomorphes ou bien qu'ilsont meme structure.

Si alors on depouille la theorie mathematique de se qui n'y apparait quecomme un accident, c'est-a-dire de sa matiere, it ne restera que 1'essentiel,c'est-a-dire la forme; et cette forme, qui constitue pour ainsi dire le squelettesolide de la theorie, cc sera la structure du groupe.

On distinguera parmi les groupes possibles quatre categories principales,sans compter certains groupes etranges ou composites qui ne rentrent dans au-cune categorie, ou qui participent des caracteres de deux ou plusieurs d'entreelles. Cc sont:

I. Les groupes discontinus et finis, ou groupes de Galois; cc sont ceuxqui president a la resolution des equations algebriques, a la theoriedes permutations, etc.... .

IL Les groupes discontinus et infinis; cc sont ceux que l'on rencontredans la theorie des fonctions elliptiques, des fonctions fuchsiennesetc.... .III. Les groupes continus et finis ou groupes de LIE proprement Bits;cc sont ceux auxquels se rattachent les principales theories geome-triques, telles que la geometric euclidienne, la geometrie non-euclidienne, la geometric projective, etc.... .

IV. Les groupes continus et infinis, beaucoup plus complexes, beau-coup plus rebelles aux efforts du geometric. Its sont en connexionnaturelle avec la theorie des equations aux derivees partielles.

M. CARTAN a fait faire des progres importants a nos connaissances surtrois de ces categories, la 1

ere , la 3e , et la 4e 11 s'est principalementplace au point de vue le plus abstrait de la structure, de la forme pure,independamment de la matiere, c'est-a-dire, dans 1'espece, du nombre et duchoix des variables independantes.

Groupes continus et finis

Je commencerai par les groupes continus et finis, qui ont etc introduitspar LIE dans la science; le savant norvegien a fait commaitre les principesfondamentaux de la theorie, et it a montre en particulier que la structure deces groupes depend d'un certain nombre de constantes qu'il designe par lalettre c affectee d'un triple indice et entre lesquelles it doit y avoir certaines

A. RAPPORT SUR LES TRAVAUX DE M. CARTAN, BY H. POINCARE 265

relations. II a enseigne egalement comment on pouvait construire le groupequand on connaissait ces constantes. Mais it restait a discuter les diversesmanieres de satisfaire aux relations qui doivent avoir lieu entre les constantesc ; on pouvait supposer que les divers types de structure seraient extremementnombreux et extremement varies, de sorte que l'enumeration en serait a peupres impossible. 11 ne semble pas eu titre tout a fait ainsi, au moins en ce quiconcerne les groupes simples.

La distinction entre les groupes simples et les groupes composes est due aGALOIS et elle est essentielle, puisque les groupes composes peuvent toujourstitre construits en partant des groupes simples. 11 est clair que le premierprobleme a resoudre est la construction des groupes simples.

Vers 1890, KILLING a annonce que tous les groupes simples continus et fi-nes rentrent: soit daps quatre grands types generaux dej a signales par LIE, soitdans cinq types particuliers dont les ordres sont respectivement 14, 52, 78,133, et 248. C'etait la un resultat d'une tres haute importance; malheureuse-ment toutes les demonstrations etaient fausses; it ne restait que des apercusdenues de toute force probante.

II etait reserve a M. CARTAN de transformer ces apercus en demonstrationsrigoureuses; it su It d'avoir lu le memoire de KILLING pour comprendre com-bien cette tache etait difficile. La methode repose sur la consideration de1'equation caracteristique, et en particulier de la forme quadratique q/r (e)

qui est le coefficient de w' 2 dans cette equation; cette consideration per-met de reconnaitre si le groupe integrable, ou de trouver son plus grand sousgroupe invariant integrable, ou enfin de reconnaitre si le groupe est simpleou semisimple.

M. CARTAN a donne une maniere de former, dans chaque type, les groupeslineaires simples dont le nombre des variables est aussi petit que possible.

Une des plus importantes applications des groupes de LIE est l'integrationdes equations differentielles ordinaires ou partielles qui sont inalterees parles transformations dun groupe. M. CARTAN a applique' cette methode aucas des systemes d'equations aux derivees partielles don't l'integrale generalene depend que de constantes arbitraires. Les operations a faire sont toutesde nature rationnelle ou algebrique.

Groupes discontinus et finis

M. CARTAN a fait faire aussi un progres important a la theorie des groupesde GALOIS, en les rattachant a celle des nombres complexes. On sail qu'ondesigne par nombres complexes des expressions algebriques susceptibles desubir des operations qui peuvent titre regardees comme des generalisations del'addition et de la multiplication, et auxquelles on peut appliquer les reglesordinaires du calcul avec cette difference que la multiplication, quoique as-sociative, n'est pas commutative. La plus connu des systemes de nombres


complexes a recu le nom de quaternions et on en a fait des applicationsnombreuses en Mecanique et en Physique Mathematique.

Ces nombres complexes ont un lien intime avec les groupes de Lie et enparticulier avec les groupes lineaires simplement transitifs; it y a, a ce su-jet, un theoreme de M. POINCARE dont M. CARTAN a donne une nouvelledemonstration. La theorie des nombres complexes a ete poussee plus loinpar M. M. SCHEFFERS et MOLLIEN qui en ont entrepris la classification et ontles premiers mis en evidence l'importance de la distinction entre les systemesa quaternions et les systemes sans quaternions.

M. CARTAN est arrive a resoudre completement le probleme, par uneheureuse adaptation des methodes qui lui avaient reussi dans l'etude desgroupes de Lie. 11 a pris comme point de depart une equation caracteristiquequi n'est pas tout a fait la meme que celle qu'on envisage a propos des groupesde Lie, mais qui se prete a une discussion analogue. M. CARTAN a montrecomment on peut construire un systeme quelconque par la combinaison d'unsysteme pseudonul et de systemes simples et comment les systemes simplesse reduisent aux quaternions generalises; comment enfin les systemes Bits dela 2e classe se deduisent facilement de ceux de la 1 ere classe. Il a etudieaussi le cas ou les coefficients sont des nombres reels.

Ces resultats ne constituent pas, comme on pourrait titre tente de la croire,une simple curiosite mathematique. Its sont au contraire susceptibles d'appli-cations nombreuses. En particulier, ils se rattachent a la theorie des groupesde GALOIS; it est clair que les lois de la composition des substitutions d'ungroupe de GALOIS sont associatives, sans titre commutatives; elles peuventdone titre regardees comme les regles de la multiplication d'un systemed'unites complexes; et par consequent elles definissent un systeme de nom-bres complexes. Or si on applique a ce systeme le theoreme de M. CARTAN,on retrouve, de la fagon la plus simple et pour ainsi dire d'un trait de plume,les resultats que M. FROBENIUS avait obtenus par une tout autre voie et quiavaient ete regardes a juste titre comme le plus grand progres que la the' oriedes groupes de GALOis eut fait depuis longtemps.

On peut, par cette voie, reconnaitre quels sont les groupes lineaires lesplus simples qui sont isomorthes a un groupe de GALOis donne, ce qui nousconduit au probleme de l'integration algebrique des equations differentielleslineaires. M. POINCARE a eu 1'occasion d'appliquer les principes de M. CAR-TAN A l'integration algebrique d'une equation lineaire.

Groupes continus et infinis

La determination des groupes continus infinis presente beaucoup plus dedifficultes que celle des groupes finis et c'est la que M. CARTAN a deploye leplus d'originalite et d'ingeniosite. Il s'est restreint d'ailleurs a une certaineclasse de groupes infinis, la plus importante au point de vue des applica-tions, et celle sur laquelle l'attention de Lie avait surtout ete attiree, je veux


parler des groupes dont les transformations finies dependent de fonctions ar-bitraires d'un ou de pluseurs parametres, ou, plus generalement, de ceux oules variables transformees, considerees comme fonctions des variables prim-itives, constituent l'integral general d'un systeme d'equations aux deriveespartielles.

M. CARTAN s'est d'ailleurs servi, dans cette etude, de resultats importantsqu'il avait obtenus dans des travaux anterieurs relatifs aux equations auxderivees partielles et aux equations de PFAFF, travaux dont nous parleronsplus loin.

La theorie de la structure, telle que LIE 1'expose dans 1'etude des groupesfinis, n'est pas susceptible d'etre immediatement generalisee et etendue auxgroupes infinis. M. CARTAN lui substitue done une autre theorie de la struc-ture, equivalente a la premiere en ce qui concerne les groupes finis, mais sus-ceptible de generalisation. Si f est une fonction quelconque des variablesX, et si les Xtf representent les symboles de Lie, on aura identiquement:

df+1: x,fco;=o

les rvi etant des expressions de Pfaff dependant des parametres du groupe etde leurs differentielles.

Au lieu de faire j ouer le role essentiel aux symboles Xt. f ' , comme le faisaitLIE, M. CARTAN 1'attribue aux expressions de PFAFF co qui sont invariantespar les substitutions du groupe des parametres. Les relations qui definissentla structure se presentent alors sous une autre forme. Au lieu de relationslineaires entre les Xi.f' et leurs crochets, nous aurons des relations lineairesentre les covariants bilineaires des co et des combinaisons bilineaires de cesmeme expressons. Le coefficients de ces relations sont les memes dans lesdeux cas, quoique dans un autre ordre; ce sont les constantes c de LIE.

Sans sortir encore du domaine des groupes finis, M. CARTAN a illustre cettetheorie nouvelle en l'appliquant a des exemples concrets, et en particulier augroupe des deplacements de l'espace; it a montre comment elle se rattachaita la theorie classique du triedre mobile de M. DARBOUx et comment ellepermettait l'etude des invariants differentiels des surfaces et en particulier deceux de certaines surfaces imaginaires remarquables.

Voyons maintenant comment ces notions peuvent titre etendues auxgroupes infinis. La notion d'isomorphisme holoedrique peut titre facilementdefinie en ce qui concerne les groupes finis, parce que l'on n'a qu'a' faire cor-respondre une a une les transformations infinitesimales des deux groupes acomparer. Nous ne pouvons plus employer ce procede lorsque les transfor-mations infinitesimales sont en nombre infini; M. CARTAN donne donc unedefinition differente, quoique equivalente a la premiere dans le cas ou celle-cia un sens. Un groupe est le prolongement d'un autre quand it transformeles memes variables que cet autre et de la meme maniere et qu'il transforme


en meme temps d'autres variables auxiliaires. Par exemple, le groupe desdeplacements des points de l'espace aura pour prolongement le groupe desdeplacements des droites ou celui des cercles de l'espace. Deux groupes sontalors isomorphes quand deux de leurs prolongements sont semblables.

La theoreme fondamental de LIE peut alors titre etendu aux groupes infinis;on montre que tout groupe infini est isomorphe au groupe qui laisse invari-antes a la fois certaines fonctions U et certaines expressions de PFAFF 60 etc o. Les differentielles totales des U s'experiment lineairement en fonctionsdes c o, les covariants bilineaires des co (mais non ceux des cii) s'exprimentbilineairement en fonctions des co et CO. Les coefficients de ces relationslineaires ou bilineaires jouent le role des constantes c de LIE. Cc sont desfonctions des invariants U. Cc qui caracterise les groupes transitifs, c'estqu'il n'y a pas d'invariants et par consequent que les coefficients se reduisenta des constantes. Cc qui caracterise les groupes finis, c'est que les expressionsn'existent pas.

Les coefficients en question peuvent-ils titre choisis arbitrairement? Non,ils sont assujettis a certaines conditions que M. CARTAN determine et quepeuvent titre regardees comme la generalization des conditions de structurede LIE.

Les trois theoremes fondamentaux de LIE se trouvent donc etendus auxgroupes infinis, de sorte que M. CARTAN a fait pour ces groupes ce que LIEavait fait pour les groupes finis.

Cette analyse a mis en evidence des resultats tout a fait surprenants. Ungroupe fini est toujours isomorphe a un groupe transitif, par exemple a celuiqu'on appelle son groupe parametrique, et on aurait pu titre tente de croirequ'il en etait de meme pour les groupes infinis, puisqu'au premier abord lademonstration ne semblait mettre en oeuvre que la notion generale de groupe.Au contraire, M. CARTAN a montre qu'il existe les groupes infinis qui ne sontisomorphes a aucun groupe transitif.

Cc n'est pas tout: un groupe infini peut titre meriedriquement isomorphea lui-meme, un groupe infini peut n'admettre aucun sous groupe invariantmaximum, etc., .... La notion du prolongement normal permet ensuite aNi CARTAN de determiner tous les groupes isomorphes a un groupe infinidonne. Citons un resultat particulier. Les groupes qui ne dependent que defonctions arbitraires dun argument, s'ils sont transitifs, sont isomorphes augroupe general dune variable.

Etant donne un groupe defini par ses equations de structure, M. CARTANmontre qu'on peut determiner les equations de structure de tous ses-groupespar des procede purement algebriques et applique cette methode a des casparticuliers tels que celles du groupe general de deux variables ou it retrouve,par une voie nouvelle, quelques sous groupes deja connus et importants parleurs applications.

Si l'on se donne deux systemes differentiels et un groupe, on peut se deman-der s'il y a des transformations du groupe qui transforment un des systemes


dans l'autre et quelles elles sont; on peut se demander egalement s'il y adans le groupe des transformations qui n'altereront pas l'un de ces systemesdifferentiels et qui naturellement formeront un sous-groupe. L'etude de cesous-groupe a fait egalement l'obj et d'un memoire de M. CARTAN .

Enfin M. CARTAN s'est propose en ce qui concerne les groupes infinis, lememe probleme qu'il avait resolu pour les groupes finis, la formation de tousles groupes simples. 11 a montre qu'ici aussi, les groupes simples peuventse ramener a un nombre restreint de types; ceux qui sont primitifs et d'oul'on peut deduire tous les groupes transitifs simples se repartissent en sixgrandes classes; quant aux groupes simples qui ne sont isomorphes a aucungroupe transitif, ils peuvent etre deduits des precedents par des procedes desprocedes que M. CARTAN nous fait connaitre.

Le probleme propose se trouve donc entierement resolu.

Equations aux derivees partielles

Le probleme de l'integration d'un systeme d'equations aux derivees par-tielles a fait l'objet de travaux nombreux. M. CARTAN s'est place pourl'etudier a un point de vue particulier; it remplace le systeme d'equationsaux derivees partielles par le systeme correspondant d'equations de PFAFF,c'est-a-dire d'equations aux differentielles totales.

Dans la theorie des expressions de PFAFF, it y a une notion, introduitepar M. M. FROBENIUS et DARBOUX, qui joue un role extremement impor-tant, c'est celle du covariant bilineaire; nous avons dej a vu apparaitre cecovariant a propos de la theorie des groupes infinis. M. CARTAN en a donneune interpretation nouvelle a l'aide du calcul de GRASSMANN, et cette in-terpretation 1'a conduit a une generalisation. De chaque expression de PFAFF,it deduit une serie d'expressions differentielles qu'il appelle ses derivees; laderivee premiere est la covariant bilineaire; la derivee ne est n + 1 foislineaire. C'est en cherchant quelle est la premiere de ces derivees qui s'annuleidentiquement que l'on reconnaitra si, et jusqu' quel point, it est possi-ble de reduire le nombre des variables independantes sur lesquelles porte1'expression.

Cette consideration a permis a M. CARTAN de retrouver sous une formeextremement simple tous les resultats connus relatifs au probleme de PFAFFet un assez grand nombre de resultats entierement nouveaux.

Comment maintenant cela peut-il servir a la resolution d'un systemed'equations de PFAFF, et surtout a reconnaitre quel est le degre d'arbitraireque comporte l'integrale generale dun pareil systeme? C'est en se servant dela notion d'involution que M. CARTAN a resolu cette question. Un systemeest dit en involution si, jusqu' une certaine valeur de m, par toute mul-tiplicite integrale a m dimensions passe une multiplicite integrale a m + 1dimensions. M. CARTAN donne une maniere de reconnaitre si un systeme est


en involution pour les valeurs de m inf erieures a un nombre donne, et, parla, de savoir combien la solution generale contient de fonctions arbitraires de1, de 2, ... , de n variables.

On retrouve ainsi sous une forme nouvelle la theorie des caracteristiques deCAUCHY, celle des caracteristiques de MONGE, celle des solutions singulieres,etc., ... ; on retrouve egalement sous une forme plus simple tous les resultatsde M. RIQUIER.

M. CARTAN a applique' sa methode a un certain nombre de cas particulierso1 l'lntegration peat se faire par des equations dlfferentlelleS ordinaires. Il1'a egalement completee en s'aidant de la theorie des groupes qui lui etait sifamiliere; it a ainsi reconnu des cas ou l'on peut determiner les invariantsd'un systeme de PFAFF, sans en determiner les caracteristiques, c'est-a-dired'une facon rationnelle, et d'autres ou les caracteristiques s'obtiennent sansintegration.

Conclusions

On voit que les problemes traites par M. CARTAN sont parmi les plus im-portants, les plus abstraits et les plus generaux don't s'occupent les Mathema-tiques; ainsi que nous l'avons dit, la theorie des groupes est, pour ainsi diew,la Mathematique entiere, depouillee de sa matiere et reduite a une formepure. Cet extreme degre d'abstraction a sans doute rendu mon expose unpeu aride; pour faire apprecier chacun des resultats, it m'aurait fallu pourainsi dire lui restituer la matiere dont it avait ete depouille; mais cette resti-tution peut se faire de mille facons differentes; et c'est cette forme unqueque l'on retrouve ainsi sous une foule de vetements divers, que constitue lelien commun entre des theories mathematiques qu'on s'etonne souvent detrouver si voisines.

M. CARTAN en a donne recemment un exemple curieux. On connaitl'importance en Physique Mathematiques de ce qu'on a appele le groupede LORENTZ; c'est sur ce groupe que reposent nos idees nouvelles sur leprincipe de relativite .et sur Dynamique de l'Electron. D'un autre cote ,

LAGUERRE a autrefois introduit en geometrie un groupe de transformationsqui changent les spheres en spheres. Ces des groupes sont isomorphes, desorte que mathematiquement ces deux theories, l'une physique, l'autre geome-trique, ne presentent pas de difference essentielle.

Les rapprochements de ce genre se presenteront en foule a ceux quietudieront avec soin les travaux de Lw et de M. CARTAN. M. CARTAN n'ena pourtant signale qu'un petit nombre, parce que, courant au plus presse, its'est attache a la forme seulement et ne s'est preoccupe que rarement desdiverses matieres dont on la pouvait revetir.

Les resultats les plus importants enonces par M. CARTAN lui appartien-nent bien en propre. En ce qui concerne les groupes de Lw, on n'avait que

A. RAPPORT SUR LES TRAVAUX DE M. CARTAN, BY }t POINCARE 271

des enonces et pas de demonstration; en ce qui concerne les groupes de GA-LOIS, on avait les theoremes de FROBENIUS qui avaient ete rigoureusementdemontres, mais par une methode entierement differente; enfin en ce quiconcerne les groupes infinis on n'avait rien: pour ces groupes infinis, l'ceuvrede M. CARTAN correspond a ce qua ete pour les groupes finis l'ceuvre de LIE,celle de KILLING, et celle de CARTAN lui-meme.

APPENDIX B

Sur unede la geometrie euclidienne

PAR

M. ELIE CARTAN

Professeur a la Faculte des Sciences de Parisl

La geometrie dans un plan isotrope dif ere profondement de la geometrieplane classique; les lignes qui jouent dans un plan nonisotrope le role descirconf erences sont, dans un plan isotrope, des paraboles toutes tangentes enun meme point a la droute de l'infini. Si l'on prend pour axe des y uneparallele a la direction isotrope unique du plan, le groupe de la geometrieeuclidienne du plan isotrope est la forme:

fx' = x + a,(1)

y'=cx + hy + b ,

l'arc elementaire d s d'une courbe etant reduit a d x . La notion ordinaire decourbure disparait, mais it s'y substitue une pseudocourbure egale af (x) , lorsque la courbe est definie par y = f(x).711

(X)Le groupe (1) est un sous-groupe du plus grand groupe affine qui laisse

invariant le point a l'infini dans la direction Oy, a savoir:

fx'=kx +a ,(2) ly'=cx+hy +b;un autre sous-groupe invariant de ce dernier, a savoir le groupe

fx'=kx +a ,(3)

'y = cx +y +b ,

peut etre pris comme base d'une geometrie plane a direction isotrope privi-legiee. Dans cette geometrie, qui est en un certain sens une degenerescencede la geometrie euclidienne, on peut definir la longueur d'un vecteur parallele

'Assoc. Franc. Avanc. des Sciences, 59e session, Nantes, 1935, 128-130.

273

274 B. SUR UNE DEGENERESCENCE DE LA GEOMETRIE EUCLIDIENNE, BY E. CARTAN

a la direction isotrope comme etant la difference des ordonnees y' et y deson extremite et de son origine, mais la notion de longueur disparait pour lesvecteurs nonisotropes.

La geometrie fondee sur le groupe (3) est interessante; on voit tout de suitequ'etant donnee une ligne plane autre qu'une droite, on peut definir d'unemaniere intrinseque un element d'art ds par la formule

2 dxd2y - dyd2X(4) ds = = f"(x)dx2.

dxLe second membre est en effet le rapport de deux aires, faire du par-

allelogramme construit sur les deux vecteurs (dx, d y) et (d 2x , d2y), et

faire du parallelogramme construit sur les deux vecteurs (dx, d y) et (0, 1 ) .

Cet element d'arc est identiquement nul quand la ligne consideree est unedroite. Si 1'on attache a chaque point de la ligne deux vecteurs T et N, lepremier tangent a la ligne et de composantes ds , d , le second parallele aOy et de longueur 1, on a les formules de Frentit generalisees:

dMT dTT1V

dN=O.

(5) ds ds=k+

dsZ

/!f/

Le coefficient k- ds = - 2 X est la courbure. Les courbes deds f (X)

courbure nulle sont les paraboles tangentes a la droite de l'infini au point al'infini sur Oy. La courbure est du reste un invariant pour le groupe general2

Ce quit donne un certain intertit a la geometrie precedente, c'est qu'ellese presente d'elle-mtime quand on veut chercher des proprietes geometriquesintrinsequement attachees a une integrale f F(x, y, y' , ydx , ou F est

z

une fonction donnee de x, y, y' = d , y" _ une propriete est Bitedx

intrinseque si elle ne depend pas du choix des coordonnees x, y. Si la fonc-tion F se reduit a V7, le plus grand groupe qui laisse invariante l'integraleest precisement le groupe (3). Si F est de la forme y , o u A etB sont des fonctions de x, y, on a une geometrie que joue par rapport ala geometrie de groupe (3) le meme role que la geometrie riemannienne parrapport a la geometrie euclidienne, avec cette difference que 1'espace doit titreregarde comme engendre non par des points (x, y) mais par des elementslineaires (x, Y, y'); 1'espace est un espace d'elements lineaires a connexionacne, assimilable au voisinage de chaque element lineaire a un plan euclidienisotrope de groupe (3).

U n autre cas particulier interessant est celui d e l'integrale f /j7T d x quiest liee a la geometrie affine unimodulaire.

APPENDIX C

Allocution de M. Elie Cartan

A la fin de cette emouvante ceremonie, apres tous les eloges dont vousm'avez comble et que j'ai conscience de n'avoir qu'imparfaitement merites,permettez que ma pensee se reporte vers ceux qui ne sont plus et qui auraientete si fiers de les entendre. Je pense a mon pere et a ma mere, humblespaysans qui pendant leur longue vie ont donne a leurs enfants 1'exemple dutravail j oyeusement accompli et des charges vaillamment acceptees. C'est aubruit de 1'enclume reyonnant chaque matin des l'aube que mon enfance a etebercee, et je vois encore ma mere actionnant le metier du canut, aux instantsque lui laissaient libres les soins de ses enfants et les soucis du menage.

En meme temps qu'a' mes parents je pense a mes premiers maitres, lesinstituteurs de l'Ecole primaire de mon village de Dolomieu, M. Collomb, etsurtout M. Dupuis; ils donnaient a plus de deux cents garcons un enseigne-ment precis dont plus tard la valeur. Je suis oblige d'avouer-et jen'en ai pas honte-que un excellent eleve; capable d'enumerersans hesitation les sous-prefectures de n'importe quel departement, et aucunesubtilite des regles du participe passe ne m'echappait. Un jour un deleguecantonal qui s'appelait Antonin Dubost et qui devait plus tard devenir undes plus hauts personnages de l'Etat vint inspecter l'ecole; cette visite orientatoute ma vie. II fut decide que je me presenterais au concours des boursesdes lycees; M. Dupuis dirigea ma preparation avec un devouement affectueuxque je n'oublierai j amais. Tout cela me valut un beau voyage a Grenoble, ouje subis sans trop d'emoi des epreuves pas trop redoutables. Je f us recu bril-lamment, ce qui remplit M. Dupuis de fierte et grace a l'appui de M. Dubost,qui s'interessa pendant toute sa vie avec une affection toute paternelle a macarriere et a mes succes, je fus gratifie dune bourse complete au College deVienne.

A Page de dix ans je quittai donc joyeux le foyer paternel, sans me douterque bien peu de jours me suffiraient pour regretter ce que je perdais. Ilfallut m'adapter a la vie d'internat que je devais mener pendant plus de dixans. Apres cinq ans de college pendant lesquels je dus mettre les boucheesdoubles, ma bourse fut transferee au Lycee de Grenoble ou. j'achevai mesetudes classiques par la rhetorique et la philosophie, puis au Lycee Janson-de-Sailly, qui etait dans toute la fraicheur de sa premiere jeunesse, rayonnant du

275

276 C. ALLOCUTION DE M. ELIE CARTAN

succes que venait d'obtenir Le Dantec recu premier a dEcole Normale. J'eusa Janson des professeurs remarquables, Salomon Bloch en mathematiqueselementaires A, et en mathematiques speciales Emile Lacour dont to as su,mon cher Tresse, sans l'avoir connu comme professeur, depeindre la noblessede caractere. C'est dans cette classe que j'eus comme camarade, avec EugenePerreau qui devait entrer avec moi a dEcole Normale, Jean Perrin, plus jeuneque nous, et qui devait devenir une des plus grandes gloires de la sciencefrancaise.

C'est avec emotion, mon cher Tresse, que je t'ai entendu evoquer nosannees d'Ecole Normale. Je ne suis pas sur que le recul du temps n'ait pas em-belli le souvenir que to as garde de moi et du role que j'aurais jou aupres demes camarades. Ce que je me rappelle, c'est en effet une camaraderie frater-nelle et une collaboration qui s'est montree surtout assez etroite dans l'anneede preparation a l'agregation. Je vois encore les seances ou le soir, reunisdans une salle quelconque, nous ecoutions l'un de nous exposer la lecon qu'ildevait faire le lendemain. La les critiques etaient libres et franches et com-bien profitables. Je me rappelle particulierement une lecon sur l'intersectiondes quadriques qui nous frappa pour la maniere elegante et neuve dont laquestion etait concue; l'auteur de cette lecon etait Arthur Tresse.

Tu as parle tout a l'heure, mon cher ami, de l'admiration que nous pro-duisaient les cours de M. Emile Picard, qui excellait a nous ouvrir de vastesperspectives dans un domaine encore nouveau pour nous. A I'Ecole mtimec'est Jules Tannery qui exerga sur nous la plus profonde influence; par unesorte de transposition mysterieuse due a 1'ensemble de toute sa personne, ason regard peut-titre, le respect de la rigueur dont it nous montrait la necessiteen mathematiques devenait une vertu morale, la franchise, la loyaute le re-spect de soi-meme. Comme on 1'a dit deja, Tannery etait notre conscience:c'est pourquoi nous l'aimions, c'est pourquoi nous avons voue a sa memoireun culte fidele.

Nous admirions aussi I' elegance de certaines conferences de Kaenigs, laclarte de 1'enseignement de Goursat. A la Sorbonne c'etait la limpidite descours de Mecanique rationnelle d'Appell, 1'elegance incomparable des coursde Darboux. Les lecons qui nous produisaient l'impression la plus profondepeut-titre etaient celles d'Hermite, dont le visage et les yeux d'une beauteadmirable s'illuminaient comme s'il contemplait au sein de la Divinite cemonde eternel des nombres et des formes dont nous parlait tout a l'heureuM. Picard.

Tannery, Goursat, Appell, Darboux, Picard, Hermite, que de grands nomss'offraient a 1'admiration de notre jeunesse. Je n'ai pas parle du geant desMathematiques, Henri Poincare, dont les lecons passaient bien au-dessus denos tetes; it n'est aucune branche des mathematiques modernes qui n'aitsubi son empreinte, et vous comprendrez que je garde a sa memoire uneparticuliere reconnaissance puisque le dernier travail de sa vie si brusquementinterrompue a ete un rapport sur mon oeuvre scientifique. De cette illustre

C. ALLOCUTION DE M. ELIE CARTAN 277

pleiade de grands mathematiciens, vous seul, mon cher Maitre, nous restez;nous admirons toujours votre jeunesse et je me f elicite que mon age medonne encore le privilege d'entendre retracer ma carriere scientifique par lemaitre admire qui, it y a un demi-siecle, m'initiait a I'Analyse mathematique,presentait mes premieres notes a I'Academie et etait le rapporteur de monjury de these.

Apres ma these dont le suj et, to l'as peut-titre oublie, mon cher Tresse, mefut signale par toi a ton retour de Leipzig ou to avais ete I'eleve de SophusLie, je fus nomme maitre de conferences a Montpellier. Je garde le meilleursouvenir des quinze ans que j'ai passes en province, a Montpellier d'abord, aLyon, et a Nancy ensuite. Cc furent des annees de meditation dans le calme,et tout cc que j'ai fait plus tard est contenu en germe dans mes travauxmurement medites de cette periode. C'est a Nancy que je commengai a mefamiliariser avec les vastes auditoires. J'avais a y enseigner les elements del'Analyse aux eleves de l'Institut electrotechnique et de Mecanique applique' e.Institut encore jeune, mais dej a prospere sous la direction de l'homme audevouement admirable qu'etait Vogt, Cet enseignement m'interessait beau-coup et j'eus la satisfaction de sentir tout de suite le contact s'etablir avecles eleves. Je me trouvai ainsi prepare a 1'enseignement des mathematiquesgenerales qui devait m'titre confie un peu plus tard a la Sorbonne.

C'est un enseignement analogue que je donne a l'Ecole de Physique et deChimie depuis vingt-neuf ans. Dans la mesure oU je merite les eloges af-fectueux que votre amitie m'a prodigues, mon cher Langevin, je suis tresheureux d'avoir pu vous aider a realiser le dessein qui vous tient a coeur,celui de faire de l'Ecole technique que vous dirigez un veritable etablissementd'enseignement superieur en assurant aux eleves une culture theorique forte-ment organisee. La tache, la encore, ma ete rendue facile par le courantde sympathie qui n'a cesse d'unir le matre et les eleves, toujours attentifset desireux d'acquerir les connaissances dont ils reconnaissent eux-mtimesl'utilite pour leur carriere future. Cc n'est pas sans un vif regret que je quit-terai bientot, cette Ecole a laquelle me rattachent tant de liens; mon departne pourra affaiblir les sentiments d'admiration que pour le savantet l'homme qui la dirige.

Tu as retrace tout a 1'heure, mon cher Maurain, en termes qui m'ont par-ticulierement touche, venant de I'ami, du doyen affectueusement venere detous ses collegues, ma carriere de professeur a la Sorbonne. Cela a touj oursete pour moi une grande joie que d'enseigner; je me suis touj ours interesse acc que j'enseignais: c'est une condition necessaire et peut-titre suffisante pourinteresser ceux qui vous ecoutent. Si ma prochaine mise a la retraite ne mevieillit pas premeturement, it me sera agreable de donner de temps en tempsquelques series de lecons sur des sujets que je n'ai pas encore eu 1'occasiond'enseigner.

C'est a l'Ecole Normale que s'est exercee une grande partie de ma carrierede professeur; pendant quelque quatorze ans j'y ai eu tout mon service. 11

278 C. ALLOCUTION DE M. ELIE CARTAN

est vrai que j'y comprends les annees de guerre, pendant lesquelles je vousai accueilli a plusieurs reprises, mon cher Julia, lorsque grand blesse vousveniez vous reposer dans notre vieille Ecole des operations successives qu'onetait oblige de vous faire subir au Val de Grace. 11 est difficile d'imaginerun auditoire plus interessant que celui l'Ecole Normale; devant lui on peutaborder tous les problemes et j'en ai aborde un certain nombre. J'ai eteheureux d'entendre de vous, mon cher Bruhat, et de vous, mon cher Julia,l'opinion qu'ont bien voulu garder de moi mes eleves. Cc sont maintenantdes maitres; un grand nombre enseignent dans les Facultes. L'un d'eux, celuilque ses camarades de Janson envoyaient passer leurs colles chez Cartan, estl'un des plus jeunes membres de l'Academie des Sciences.

Nous, leurs aines, nous avons la grande joie de voir sortir de l'Ecole Nor-male des generations successives de brillants mathematiciens; nous sommesassures ainsi qu'elle n'abdique pas le role de pepiniere des mathematiquesqu'elle joue depuis longtemps et qui inspira autrefois a Sophus Lie l'idee delui dedier son grand traite sur la theorie des groupes. Et puisque, par unepensee touchante, le fils de Sophus Lie a voulu marquer cc Jubile par l'envoidu buste de son pere, ne serait-il pas naturel que la place de cc buste soit a labibliotheque des Sciences de l'Ecole Normale? 11 rappellerait aux promotionssuccessives a la fois le grand mathematicien norvegien et les normaliens quiont ete ses eleves a Leipzig et ont illustre l'Ecole, les Vessiot, les Tresse, lesDrach.

Mon cher Bruhat, vous avez parle en termes qui me sont alles au cceurde la dynastic normalienne des Cartan. Me permettrez-vous d'adjoindre auxdeux noms d'Henri Cartan et d'Helene Cartan les noms de deux autres nor-maliens qui m'ont ete tres chers? Le premier est celui de mon beau-frereAntoine Bianconi, cacique litteraire de la promotion de 1903, dont la mortsur le champ de bataille interrompit 1'ceuvre philosophique qu'il meditait etqui promettait d'etre importante. Le second est celui de ma plus jeune sceurAnna Cartan, dont le succes au concours d'entree a Sevres m'avait rempli dejoyeuse fierte; eleve elle aussi de Jules Tannery, dont elle ne pouvait parlersans emotion, elle a termine prematurement sa brillante carriere comme pro-fesseur au Lycee annexe de Sevres. 11 m'est doux de penser qu'elle est un peupresente ici, en voyant au milieu de nous la compagne de promotion a qui laliait une tendre affection, ma chere amie Madame la Directrice de l'Ecole deSevres.

Mon cher Julia, c'est avec empressement que je me suis associe a votreprojet de fonder pour les jeunes mathematiciens un cercle d'etudes, votreseminaire, ou ces jeunes gens, travaillant en collaboration, exposeraientchaque annee une question importante de Mathematiques. Vous nous avezdit a cc propos que les jeunes sentent; sans peut-titre trop se l'avouer, le be-soin de s'appuyer sur leurs aines. En entendant tout a l'heure Dieudonne,nous avons compris combien vous aviez raison.

C. ALLOCUTION DE M. ELIE CARTAN 279

Mon cher Dieudonne, les paroles que vous m'avez adressees me touchentau dela de toute expression. Elles montrent que vous avez 1'enthousiasmede la jeunesse, vertu que je vous souhaite de conserver toute votre vie. Cetenthousiasme ne vous a-t-il pas fait depasser la mesure? J'aurais certes mau-vaise grace a vous contredire, mais je suis assez age pour savoir ne pas tirerde vos eloges un orgueil dyplace, sachant tres bien que si j'ai les qualites quevous m'attribuez, it m'en manque un certain nombre d'autres qui m'auraientpermis de rendre plus de services a 1'enseignement et a la science; elles nesont sans doute pas dans ma nature, mais je n'ai peut-titre pas eu assez deferme volonte pour les acquerir.

Mon cher Demoulin, nous sommes lies par une vieille amitie et de nom-A

breux souvenirs communs; nous avons scouts ensemble les maitres dont jerappelais les noms tout a l'heure. Je suis tres sensible aux felicitations quevous m'apportez au nom des savants strangers. Je remercie particulierementtous ceux d'entre eux, et je les vois ici nombreux, qui ont tenu a assister enpersonne a cette csremonie. Leur presence m'est precieuse et l'empressementavec lequel des savants de nombreuses nations etrangeres ont bien voulus'associer a mon Jubile m'a vivement touche. Dans le monde trouble ou nousvivons, it est indispensable que la collaboration internationale, au moins dansle domaine scientifique, soit maintenue malgre tous les obstacles.

En meme temps qu'aux delegues strangers, j'adresse mes remerciementsaux amis, aux collegues, aux eleves qui ont bien voulu repondre a l'appel duComity j ubilaire. Je remercie les membres de ce Comity qui ont accepts dedonner leur concours a l'organisation de cette fete, et surtout mon collegueet ami Darmois qui, avec l'aide de mon sieve Ehresmann, a pris sur lui lapart la plus lourde de cette organisation.

Plusieurs des orateurs precedents, et j'en suis particulierement touche, onttenu a associer le nom de la compagne de ma vie a cette commemorationde ma carriere scientifique. Depuis plus de trente-six ans elle est la flammeardente qui anime le foyer familial. Nos enfants nous ont reserve de grandesjoies; la douleur ne nous a pas sty epargnee. Nous n'oublierons jamais1'empressement avec lequel le Comity a tenu a faire sienne la pieuse penseede rendre presente ici, grace au grand artiste qu'est M. Charles Munch, famede 1'enfant disparu dont toi, mon cher Tresse, vous, mon cher Julia, et vous,mon cher Dieudonne, avez su evoquer la memoire en termes si emouvants.La csremonie de ce matin, ou vous avez tenu a ne pas dissocier l'homme duprofesseur et du savant, nous a donne a ma femme et a moi les plus grandesjoies qui puissent encore nous titre reservees.

APPENDIX D

The Influence of France in the Developmentof Mathematics'

Like any science, mathematics is a common, international possession; it isthe commonwealth that belongs to all developed nations, the commonwealthto which every nation contributes according to its abilities. It would be un-acceptable if any well-regarded mathematician would decline to pay awed re-spect to the great foreign minds of the past: Galilei from Italy, Newton fromEngland, Euler from Switzerland, Abel from Norway, Leibniz, Gauss, andRiemann from Germany, to mention but the most significant. They openednew routes in different fields of the science that, without them, would notbe what it is today. However, I hope to make you realize that the Frenchmathematicians made one of the most noteworthy contributions to the de-velopment of mathematics, and that, when it comes to the number of greatmathematical minds, France does not take second place to any other nation.I am honored and pleased to be given this opportunity to talk about this par-ticular subject in front of a friendly audience and in a country tied with myown by many common memories.

In mathematics, as in any other science, there are two kinds of scientists:those who open royal avenues by coming up with new ideas, usually simpleones but nevertheless ones that have not occurred to anyone else; and thosewho, on the vast land cleared by the first, till their own gardens, often pickingtasty fruits, and sometimes collecting magnificent harvests. When it comesto the development of any science, the latter are not simply significant butrather indispensable; however, it is clear that the names of the former arethose that are remembered and honored. Those are the people about whomI speak today.

Joseph Bertrand tells us that, at a Fontainebleau reception for the Dutchambassador, King Henri IV took pleasure in recalling great Frenchmen who,by their achievements in literature and art, exceeded their foreign rivals."Those I myself admire," said the Dutchman, by training a mathematician

1 This talk was presented by Elie Cartan in the French Institute in Belgrade, Yugoslavia, onFebruary 27, 1940. The talk was translated from French into Serbian by Milorad B. Protk,published in 1940 in the Yugoslavian journal Saturn and in 1941 as a separate book withthe introduction written by Mihailo Petrovic (see (190)). For this Appendix the lecture wastranslated from Serbian into English by Dr. Jelena B. Gill, who also wrote all footnotes.

281

282 D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS

whose field was geometry, "but I must notice that, so far, France failed toproduce any mathematicians." "Romanus se trompe!" cried Henri IV and,having at once turned to one of the servants, asked that M. de la Bigottierebe brought in. The first great French mathematician, M. de la Bigottiere-whose real name was Francois Viete (1540-1603)-was the founder of mod-ern algebra. He was the first to realize that the procedure for solving specialnumeric equations would be simplified if the operational symbolism whosebeginnings can be traced back to the ancient times was applied to letters aswell; also, he deserves most of the credit for the systematic development ofthat idea, and he predicted its unbounded expansion. At the end of the six-teenth century, when Galilei and an advanced geometry school brought fameto Italy, it was Francois Viete who secured for France a distinguished placein the process of founding modern mathematics. I should tell you that, forquite some time, Viete was in contact with one of your first mathematicians,Marin Getaldic (1566-1626), who was born in Dubrovnik and who, in Paris,in the year 1600, published one of Viete's last works.

For France, the seventeenth century was particularly glorious. In the his-tory of mathematics, mechanics, and physics, three names from this periodespecially stand out: Descartes, Pascal, and Fermat.

A philosopher, mathematician, and physicist, Rene Descartes (1596-1650) is frequently considered the originator of a new era in the historyof the human mind. As a physicist, he witnessed a defeat of his attemptsto explain the world; however, his idea that all physical phenomena can beexpressed in terms of space and motion has retained its attractiveness untilthe present day, because the founder of the general theory of relativity him-self believed that it may be possible to interpret physics by using geometricterms (it was nothing but the past development of mathematics that enabledEinstein to carry his ideas further than Descartes could have). Even if wedeny him credit for the creation of analytical geometry (1637 ), we must notundermine his role in mathematics. It is known that Greek geometers freelyused numbers and computations in their thinking, but for them the numbershad not yet completely lost the geometric character they had in hellenisticscience; as the words "square" and "cube" stand for both the numbers andthe geometric forms it is clear that the common speech of today still showstraces of this double use. Descartes was the first to use abstract numberssystematically to represent geometric forms and to convert geometric reason-ing into computations. In that way he created an extraordinarily powerfultool. To him we must ascribe the growth of geometry that stemmed primar-ily from analytical and differential geometry; he enriched the latter with ageneral method for finding tangents of algebraically defined curves. Thanksto analytical geometry, mathematicians not only succeeded in understandinga space of any number of dimensions but also learned to think geometricallyin such a space. It is possible to say that it is in fact analytical geometry thattaught mathematicians to feel comfortable in, for example, a spheric three-

D. THE INFLUENCE OF FRANCE IN THE DEVELOPMENT OF MATHEMATICS 283

dimensional space, i.e., the one that, only recently, physicists started usingto explain physical phenomena. All of this represents, although remote, nev-ertheless unquestionable consequences of Descartes's ideas and results. Inalgebra, it is to him that we owe the rule about the signs. In pure geometry,he should be credited with a theorem that, having been independently discov-ered by Euler, now bears Euler's name. A result of analysis situs2, a scienceunknown at the time, this theorem establishes the relationships between thenumber of vertices, edges, and sides of a convex polyhedron. Finally, inmechanics, Descartes's principle of conservation of linear momentum pro-vides an illustration of the intuition that required nothing more than a properrefinement to bring about one of the basic principles of classic mechanics.

Even in his early youth, Blaise Pascal (1623-1662), a somewhat strangebut extraordinary genius, exhibited an unusual talent for geometry by writing,at the age of sixteen, Traite sur les sections coniques, a treatise about curvesthat are most frequently studied as flat conic section and play an importantrole in Kepler's planetary laws. Pascal used the results of his contemporaryGerald Desargues, who was one of the most significant French geometers andwho, alongside Pascal, was a forefather of projective geometry. By taking,in a way similar to Desargues's, the perspective as a starting point, Pascalsucceeded in reducing all properties of conic sections to a property that hecalled "L "hexagramme mystique": if a hexagon is inscribed into a cone, thethree points at which pairs of opposite sides cross each other always lie on astraight line. Even by this result Pascal demonstrated the creative power ofan eminent geometer.

As soon as Pascal the forefather of projective geometry established himself,Pascal the founder of mathematical probability took the stage. When hisfriend Chevalier de Mere asked him a couple of questions concerning a gameof chance, Pascal answered them by reducing all possible outcomes to thosemost basic. Pierre de Fermat, on the other hand, came up with the sameanswer but in a completely different way. The evolution of the principles ofmathematical probability is well illustrated in the letters exchanged betweenPascal and Fermat. The scope of this new research did not escape Pascal: "Byconnecting the exactness of a mathematical approach with the uncertaintyof chance," he was known to say, "the new science can rightly be given anastounding name Geometry of Chance." From the famous betting proof,it is known to what extent his research and thinking were influenced by hisinterest in this new geometry. It is also known that this geometry playedan instrumental role in the development of modern science, in which entireportions of physics are nothing but chapters of mathematical probability, andmany of the laws of physics are nothing but laws of chance.

Pierre de Fermat (1601-1665), whom we mentioned earlier, is one of thegreatest mathematical geniuses. He became a counselor of the parliament at

2The old name for topology.


the age of thirty and held that position until his death. Although his voca-tion did not predestine him for mathematical fame, he made sure to devoteenough time to his favorite avocation. Fermat is especially famous for hisresearch in arithmetic and number theory. On the margins of a copy of Dio-phantus's work about undefined equations (which was published in 1612 byBache de Meriziac, the author of Problemes plaisants et delectables) he wrotea number of important theorems without proofs; it is a matter of common be-lief that he was in possession of their proofs. The most famous among thosetheorems is the one frequently called Fermat's Last Theorem according towhich the sum of the nth degrees of two integers cannot equal the nth degreeof a third integer for any integer n that is greater than two. This theoreminspired a wealth of results whose authors, in spite of having at their dis-posal modern algebraic results that had been unknown to Fermat, have neverbeen able either to prove or disprove it. It has been believed for a long timethat, even if the theorem is wrong in general, it might in fact be wrong onlyfor some values for n ; however, it is by no means known if the number ofthe values for which it is wrong is finite or infinite. Through the researchprompted by this single theorem conducted in nearly all mathematicallydeveloped theories-Fermat influenced the growth of number theory. Hiscontemporaries readily recognized his extraordinary skills in that field. Inone of his letters, Pascal wrote that his own results in number theory weresurpassed by Fermat's and that his was but to admire them.

The first half of the seventeenth century was an era of strong advancementof integral and differential calculus. With respect to integral calculus (deter-mining areas and volumes, finding centers of gravity), it is enough to mentionCavalieri3 and de Roberval4. As Fermat's own research, however, went quitefar in this field, we are indebted to him for the classical integration proce-dures. On the other hand, once while trying to fight a tremendous toothacheby solving roulette problems, Pascal accidentally discovered a procedure forobtaining integrals of higher powers of trigonometric functions. The namesof those whom we have been talking about are found in differential calculusas well (the tangent problem). By his method "de maximis et minimis", Fer-mat introduced the notion of an infinitesimally small number. Lagrange andLaplace considered Fermat to be the actual founder of infinitesimal calculus,while Emile Picard5 believed Pascal's works about roulette to represent thebeginnings of integral calculus. Originally, Leibniz scribbled his formulae ofinfinitesimal calculus on a copy of one of Pascal's manuscripts, which, as hehimself put it, had suddenly showed him the way.

It would be unfair to conclude the account of these great minds with-out mentioning that, at the age of twenty-eight, Pascal constructed the first

3Bonaventura Cavaliers (1598-1647).4Gilles Personne Roberval (1602-1675).5Charles Emile Picard (1856-1941).


arithmetic machine, capable of adding and subtracting. Due to his workTraite de 1'equilibre des liqueurs, Pascal can be considered together withArchimedes-one of the founders of hydrostatics; this is why it comes asno surprise that the barrel he used to check what is today known as Pas-cal's Principle is displayed next to his death mask in the little chapel erectedin the churchyard of Port Royal. Finally let me mention the experimentsconcerning atmospheric pressure, which, it is suspected, he conducted un-der the influence of Mersenne6, the soul of a small group of philosophers,mathematicians, and physicists that, before the creation of the Academy ofSciences in 1666, represented the first small but lively academy.

Those were fortunate times when one and the same man could be accom-plished in philosophy, mathematics, and physics, and when a philosophersuch as Malebranche7 could have the extraordinary feeling that colors mightbe related to the number of vibrations of which light is composed'.

II

The second half of the seventeenth and the beginning of the eighteenthcentury were dominated by Christian Huygens (1629-1695) from the Nether-lands, Isaac Newton (1642-1727) from England, and Gottfried Wilhelm vonLeibniz (1646-1716) from Germany. It should be enough to mention thatthe last two are credited with the discovery or, rather, the systematization ofinfinitesimal calculus, while the first is famous for his works in differentialgeometry, rational and applied mechanics, and especially his works concern-ing the theory of light (in which he originated and developed an undulatorytheory as opposed to Newton's particle theory). In this period, a remarkablescientific revolution was triggered by Newton's proof that stars and objectson Earth move according to the same laws of mechanics, namely, that oneand the same law, the law of gravitation, explains the motion of planets, themoon, and comets as well as the existence of Earth's gravity, high and lowtide, and so on. It was Newton's genius that created an entirely new sciencecelestial mechanics. But even if the earliest beginnings of this science did takeplace in England, it was France that provided a particularly fertile soil forits future development. To realize this, it is enough to recall the names ofthose whose works contributed the most to its growth: Clairaut, d'Alembert,Euler, Lagrange, Laplace, Gauss, Cauchy, Poisson, Le Verrier, Tisserand, andfinally and especially-Henri Poincare.

I pause for a moment on the first of them, Clairaut. The second in a familyof twenty-one children, with a father who was a teacher of mathematics,Alexis Claude Clairaut (1713-1765) demonstrated talents similar to thoseof Pascal; however, unlike Pascal, his first works in no way revealed thesignificance of those that followed. He sent his first announcement to theAcademy of Sciences before reaching the age of thirteen, and addressed an

6Marin Mersenne (1588-1648).7Nicolas de Malebranche (1638-1715).


article about lines with double curvatures at the age of sixteen. He waseighteen when, against the existing rules, the king named him a member of theAcademy of Sciences, the Division of Mechanics. I shall ref rain from tellingyou about his research in the field of pure mathematics in general and aboutthe part connected with solving differential equations in particular the latterof which should be well known to all who studied differential equations andfocus instead on those results that made him famous. Newton and Huygenscame up not only with a theoretical proof that, instead of being a perfectsphere, the earth is a sphere flattened at the poles, but also with a way tocalculate the measure of flatness. However, when in 1701, at the Pyrenees,Cassini8 determined the degree of arc of the Paris meridian, their conclusionscame to be questioned. After debates that were occasionally confusing butalways lively, in 1736 the Academy of Sciences decided to launch, underthe guidance of de Maupertuis9, an expedition that would travel to Laplandto determine the degree of the Lapland meridian arc. Working under veryhard conditions, which were further complicated by snow and polar night,the team which included Clairaut as well came up with a numerical valuethat was remarkably larger than the one Cassini had obtained in France,hence proving beyond any doubt that the earth is indeed flattened at thepoles. Understandably, de Maupertuis won laurels for the success of theexpedition: with his head wrapped in a bear skin, his hand pressing againsta globe, he posed for a portrait. But Clairaut continued to think about apossible cause of the earth's polar flatness and tried theoretically to determinethe shape that a fluid planet would assume under the influence of Newton'sattraction. The results of his research were published in 1743 in La Theoriede la Figure de la Terre, the book that d'Alembert characterized as a classicalaccount of everything that had been done by that time, the account thatmarked an important date in the history of celestial mechanics. In addition,Clairaut explained the motion of the moon and in so doing contributed toNewton's lunar theory. He summarized his results from this field in Theoriede la Lune, a book published in 1732, to which, two years later, he addednumerical tables, which, as Fontaine had put it, made it possible to find out"every step that the moon makes in the sky". A few years later, by predictingthe next return of Halley's comet, Clairaut reached popular recognition andfame. After explaining that the perturbations caused by Saturn would delaythe return of Halley's comet for about one hundred days and the influenceof Jupiter would delay it for an additional five hundred and eighteen days,he predicted that its next passage through the perihelion would occur aroundApril 13, 1759, but cautioned that, due to numerous other factors that hehad to neglect, this date might be off by up to one month indeed, Halley'scomet passed through the perihelion on March 13, 17 59. Almost one century

8Jacques Cassini (1677-1756).9Pierre Louis Moreau de Maupertuis (1698-1759).


later, by determining the position of an until-then-unknown planet that hadbeen the main cause of the disturbance of Uranus, French astronomer LeVerrierl ° attained nearly the same glory.

III

The second half of the eighteenth century was dominated by Euler andLagrange and, in a somewhat lesser degree, by d'Alembert.

Leonhard Euler (1707-1783), "the prince of mathematicians", was bornin Basel and spent part of his life in St. Petersburg and Berlin. His geniusglowed in all areas of mathematics, and his work has had significant andlasting influence. I will always remember the delight I experienced whilereading his Introduction to the infinitesimal analysis, the book that was givento me as an award at the end of my final year of gymnasium: it openeda whole new world in front of me, preparing me to understand better thelectures I would attend at the Sorbonne and in l'Ecole Normale.

Jean Le Rond D'Alembert (1717-1783) left his trace in many differentareas of mathematics. A well-known algebraic theorem that bears his nameasserts that the total number of solutions (real and complex) of a rationalequation equals the highest degree of the variable. Although d'Alembert'sproof of this result was wrong, it should be mentioned that Euler's proof,based on completely different principles, was not without flaws. Only whenthe famous mathematician Gauss entered the mathematical scene was a cor-rect proof found, and only with Cauchy's appearance was a real and verysimple justification of this theorem established. In analysis I shall men-tion only the first correct formulation which came from d'Alembert-ofa partial differential equation describing vibrations of strings. And finally,it is well worth mentioning that, in mechanics, d'Alembert came up with aprinciple nowadays known as d'Alembert's principle which paved the wayfor Lagrange's analytical mechanics.

Joseph Louis Lagrange (1736-1813) was born in Torino, in a French fam-ily; although, like Euler, he spent a few years in Berlin, in 1787 he madehis permanent home in Paris, entitling France to consider him one of hervery own most celebrated minds. He is truly one of the most significantmathematicians of all times. He worked in all fields of mathematics. In thetheory of numbers he proved Fermat's theorem for the power four. In alge-bra, through developing a unique method for solving a polynomial equationby reducing it to an equation of a lower degree, he cleared a path for Abel,Gauss, and Galois; in addition, he demonstrated that polynomial equationsof the fifth degree cannot be solved in the way used for solving those of thethird and fourth degree. In analysis, he gave the method for solving partialdifferential equations of the first order and came up with the notion of a sin-gular solution. In function theory, he attempted but did not quite succeed inestablishing a rigorous foundation for infinitesimal calculus, the area whose

10Urbain Jean Joseph Lc Verrier (1811-1877).


principles had not yet been developed with desired exactness but whose con-sequences were nevertheless trusted. However, in spite of this lack of fullsuccess, his method of considering functions in an abstract way, indepen-dent of their geometric or mechanical meaning, had remarkable influence inpreparing the terrain for the modern theory of functions. Lagrange's talentfor generalizing became truly obvious in his works concerning the calculusof variations.

The calculus of variations was developed during the eighteenth century,through the works of Bernoulli and Euler, both from Switzerland. Its rootsare in some problems of geometry and mechanics, the simplest of whichmight be the problem of determining the shortest path between two pointson the same surface; here, the unknown quantity is not a number but, muchmore complexly, a line consisting of infinitely many points. De Maupertuiswas the one who, by his Principle of Least Action, reduced the problem ofdetermining a trajectory of a particle in a given force field to a problemof maxima and minima, giving special importance to this kind of calculus.It should not be forgotten, however, that by that time Fermat had alreadyreduced the laws of optics to a similar principle, according to which the pathchosen by light is the shortest in terms of time. By applying the infinitesimalvariation on an unknown line and by showing how that variation can becalculated, Lagrange introduced a general method into a theory in whichnearly every problem required a special procedure in order to be solved.

I shall omit Lagrange's work in celestial mechanics and, instead, devotemore time to his most significant work, Mecanique Anal ytique (17 8 8). Galilei,Descartes, Huygens, Leibniz, Newton, and d'Alembert gradually developedall of the grand principles of modern mechanics. But the problem of deter-mining the trajectory of a system governed by given forces was frequentlycomplicated by the necessity to take into account unknown relations betweenthe forces. With ingenious intuition, in the case without friction Lagrangecompletely removed the difficulty and gave a general procedure for determin-ing equations that would give the trajectory in question: to achieve this itis enough to determine the active force of that system as well as the workof that force for an infinitely small movement of the system. Aside frompractical importance, this wonderful creation has remarkable philosophicalimportance because it completely illuminates everything that is, from thepoint of view of mechanical properties, important in a system of particles.In this respect, Lagrange's genius is equal to that of Descartes, the creator ofanalytical geometry.

The so-called Lagrange's equations in Mechanique Analytique representedan analytical model for various mechanical explanations of certain physicaltheories. From that point of view this work has great philosophical signifi-cance; but, although it is the most important work of the nineteenth century,it created the impression that everything can be explained by the principles ofmechanics an impression as erroneous as Descartes's belief that everything


can be explained in terms of geometry which is the reason that, today, it iscompletely abandoned. Nevertheless, it illustrates the ability of mathematicsto provide physicists with the tools they require to carry out their theories.

Some of the extraordinary minds were inclined to see danger in the man-ufacturing of structures (similar to the one created by Lagrange) that offeredinsights into infinite arrays of phenomena; they feared that such structuresmight cause a loss of connection with reality. For instance, the great geome-ter Poncelet, known for his works in mechanics, avoided using Lagrange'smethod and, instead, preferred following to the last detail the influences andinteractions of various forces in order to determine, step by step, their actualworks. The same type of skepticism prevented Poncelet from using analyticalgeometry and prompted him, instead, to examine directly relations betweenvarious geometric figures by applying principles of classic geometry. Withrespect to accepting the latest results, there are indeed two kinds of minds,both equally important for the development of science and both found amonggreat French mathematicians.

IV

Visible as early as the end of the eighteenth century, the French superiorityin mathematics became especially clear during the French Revolution and atthe beginning of the nineteenth century. Among the great names of that eraone must include Monge, Laplace, and Legendre.

Pierre Simon de Laplace (1749-1827) owed his reputation to his researchin celestial mechanics, summarized in his charming treatise Exposition duSysteme du Monde. The peculiar result stating that even the finest detailsof almost all celestial phenomena can be explained evolved into scientificdeterminism, according to which, in order to be able to determine positionsand velocities of cosmic particles at a given time, it is enough to know theirpositions and velocities at any other time, provided it is known, in addition,which principles regulate the forces modeled after the forces of Newton'sgravitation that the particles are governed by. For a long time mathemati-cal physics developed according to this result; only recently, electromagnetismand atomic physics succeeded in proving it to be wrong. Still, this result hadstrong influence on the development of science. A very significant treatise,Theorie Analytique des Probabilities (1812), is another one for which we aregrateful to Laplace; the most important part of this work deals with the appli-cation of the notion of probability in the theory of least squares, the possibil-ity of which had been indicated by Legendre. While studying the inclinationof an ellipsoid, Laplace introduced spherical functions by means of whichone can express any function dependent on a point on a sphere. We shouldnot forget Laplace's famous equation which is satisfied by Newton's potentialfunction; this equation is of extraordinary importance in many problems ofanalysis, geometry, mechanics, and physics.


Adrien Marie Legendre (1752-1833) is responsible for the rejuvenation ofnumber theory, previously successfully treated by Euler. Although Euler wasthe first one to publish the reciprocity law in arithmetic, Legendre explainedit clearly and partially proved it; the law is named after Legendre. Gauss wasthe third mathematician discover this same law, but the first one to constructa correct and complete proof. Legendre's significant work of several years,Sur les Integrates Elliptique, a tract in two volumes, was published in 1825and 1826. There he presented a complete study of integrals involving squareroots of fourth-degree polynomials and developed different forms that can begiven to them. Although with this work Legendre became a forefather of themarvelous theory of elliptical functions, he let Jacobi and Abel take creditfor its founding. Finally, let us mention his Elements de Geometrie (1794), awork which had numerous editions and which, in schools of the Anglo-Saxoncountries, soon replaced Euclid's theory; in the history of the non-Euclideangeometries, this work had definitive importance.

Gaspard Monge (1746-1818) was one of the best French geometers. Thereare two reasons why. First, by founding modern projective geometry, hejoined the long process of development of perspective, the theory whose prin-ciples had been known to Italian renaissance painters, which Desargues andPascal applied to the theory of conic sections, and which, following the pre-vious two, the French geometer de la Hire" expanded to the theory of polesand polars of a circle. Monge systematized projective geometry and enrichedit with constructions on surfaces that are not flat. On the other hand, byhis treatise Applications de 1'A nal yse a la Geometrie he gave a substantialboost to differential geometry, the field that was separated from Descartes'sanalytical geometry by Euler's and Meusnier's significant works concerningthe properties of surfaces; it is Monge to whom we are indebted for the no-tion of measure of curvature, as well as for its application in stereometry;it was his idea to characterize a vast family of surfaces by obtaining themas a solution set of a single partial differential equation. He managed tointegrate the equation of minimal surfaces, surfaces which have been andstill are an object of important research, and which had been obtained firstin Plateau's experiments. Monge presented his theories during his lecturesat l'Ecole Normale-the school founded in 1795 as a convent as well asat 1'Ecole Polytechnique (at which Lagrange and Laplace taught as well). Iam pleased to have a chance to mention Dupin12, for he was one of the nu-merous students with whom Monge worked; Dupin is known for his workDeveloppement de Geometrie, in which he introduced the notions of conju-gated tangents and indicatrix at a point of a surface; also, Dupin can beconsidered a creator of a new branch of geometry.

1 1 Phillipe de la Hire (1640-1718).12Francois Pierre Charles Dupin (1784-1873).


V

The most remarkable names in France during the first half of the nine-teenth century were those of Fourier, Cauchy, Poncelet, and Galois. Althoughquite different from each other, they all cleared new paths in science.

Jean Baptiste Joseph Fourier (1768-1830) can be considered the founderof mathematical physics. I shall neglect his important results in algebra andinstead tell you about Theorie Mathematique de la Chaleur, the work he didnot publish until 1822 but which must have been in his thoughts since at least1807. With this work Fourier opened up a new field in mathematical analysis."Unknown to the ancient geometers, and for the first time used by Descartesfor researching curved lines and surfaces," Fourier says, "analytical equa-tions are by no means limited to these general phenomena. Since mathemat-ical analysis determines the most diverse relations and measures time, space,forces and temperature, it is safe to say that it is as wide and rich as Natureitself. It always follows the same paths and gives the same interpretations, inthat way certifying about the unity, simplicity and stability of the Universe."It should not be forgotten that, according to Fourier, the richest source ofall mathematical discoveries lies in the study of nature. As, for instance, themathematical theory of heat had a significant influence on the development ofpure mathematics, we may say that Fourier's viewpoint was correct. Createdby Fourier to help him integrate frequently encountered partial differentialequations, the theory of trigonometric series prompted incredibly many arti-cles, all of which were trying to establish a rigorous foundation for this theoryas well as to complete and further develop it. The basic problem that neededto be solved was determining which functions can be represented in the formof a Fourier series. As even many of Fourier's own examples were peculiar,it did not take much to make the mathematicians truly puzzled, in a wayin which a musician would be puzzled upon discovering that, by combiningfinite or infinite numbers of pure sounds and their various multiples (har-monics), it is possible to create any disconnected sequence of sounds. Theseunusual results forced mathematicians to check once more and specify thenotion of a function and to start thinking, bit by bit, about the foundationsof their own science. This is what brought about unbelievable consequenceswhich have not yet fully presented themselves. Group theory -a field whichso frequently failed mathematicians and which caused many paradoxes that, Iam afraid, have not yet been successfully resolved was one of the branchesof mathematics that eventually evolved from these efforts; another branchthat had its origins in the same efforts is the theory of functions of one realvariable, a creation of French mathematics from the end of the nineteenthand the beginning of the twentieth century.

Augustin Cauchy (1789-1857 ), an extraordinarily fruitful theorist, wassuccessful in all areas of mathematics: number theory, geometry, analysisand celestial mechanics. Unlike Euler, he did not explore series without first


finding out whether they made sense, that is, whether they were convergent;in that way, we may say, he opened up the era of exactness. Cauchy discov-ered the general rule later found independently by J. Hadamard 13 -whichexplains how to determine those values of the variable for which a powerseries is convergent. Creation of a theory of functions with a complex (orimaginary) variable is another of Cauchy's great accomplishments. For morethan three centuries, imaginary quantities were a scandal in mathematics.They were encountered for the first time in the sixteenth century, by Italianalgebraists, in the formula for the roots of a third-degree equation in theparadoxical case when all of the roots are real. But, once researchers gotadjusted to these new quantities and learned how to use them, it was easy todetermine important results concerning real numbers, some of which couldnot have been obtained in any other way. Sometime toward the end of theeighteenth century, the Swiss mathematician Argand explained the secret ofimaginary quantities by finding their importance in the possibility of express-ing a vector in a plane whenever one needed to give not only the length of thevector but its orientation as well. When Cauchy started representing a pointin the plane by just one imaginary (or, better, complex) quantity instead oftwo real coordinates, he got the idea of a function with a complex variable,a function which would assign one point in the plane to another point inthe plane. In this way Cauchy created a whole new world. The elements ofthat world are perfectly organized: just as Cuvier14 was able to reconstructa creature from the antediluvial era from just one piece of its skeleton, amathematician became able to reconstruct one of Cauchy's functions, pro-vided he knew its values at every point of the arc, no matter how small the arcmight be. The perfect order in this world, its marvelous harmony, and withthe exclusion of number theory -a long sequence of theorems determiningproperties of functions and their numerous applications, all leave the mostmagnificent impression.

As Cauchy created the right conditions for more discoveries than he couldhave possibly anticipated, the significance of his opus should be measured bythe length of the sequence of works concerning functions of a complex vari-able. A single theorem from this sequence, whose beauty is in its simplicity,was nearly enough to immortalize the name of Liouvillels. Another theoremon the same subject named after Emile Picard, perhaps the greatest amongthe living mathematicians opened vast and until-then hidden horizons, andcreated a stream of articles that has not yet ceased.

By using a viewpoint different from Cauchy's, the German mathematicianWeierstrass also developed a theory of functions of a complex variable. Fora long time it had been believed that the viewpoint one chose was irrelevant,

13Jacques Salomon Hadamard (1865-1963).14Georges Cuvier (1769-1832), a French naturalist.15Joseph Liouville (1809-1882).


but Bore116 demonstrated in one of his most charming results that thiswas not true, and that Cauchy's viewpoint penetrates deeper into the heartof the matter. Borel indeed took out of the plane so much that no circle,regardless of how small, was left intact, and yet inside of what remainedhe managed to construct a function that, although satisfying all of Cauchy'srequirements, did not satisfy Weierstrass's definition, that conditions the ex-istence of a function of a complex variable by the existence of an intact por-tion of the plane. By starting his celebrated collection of monographs aboutthe theory of functions the collection whose past and present contributorsinclude mathematicians from all countries Borel himself contributed a lotto the development of functions of a complex variable.

With Jean Victor Poncelet (1788-1867) we enter the era of pure geom-etry. Poncelet is considered the founder of projective geometry, the fieldwhose subject is studying those properties of objects that do not change inprojections. He is the one who discovered the new and very useful notionof transformations by means of reciprocal polars, the transformations whichmake it possible to derive one flat figure from another, with a provision that,peculiarly, the sides of the new figure correspond to the vertices of the old one,and vice versa. Frequently, a transformation of this type makes it possible toexplore the properties of some figure by reducing them to the easier-to-exploreproperties of another. Somewhat later, Gergonne17 used this to derive theduality principle, a principle very important in projective geometry. Finally,Poncelet was the one who discovered the continuity principle, according towhich if a figure had a certain property, it will retain the same property evenafter being deformed, provided that the ratios between its various elementswere taken into account. By many simple examples Cauchy proved that thisprinciple, as formulated by Poncelet, was wrong; however, if formulated in aslightly different and much more precise way, this principle is in fact correct.Being very helpful, this principle is frequently used. In geometry, Poncelet'sinfluence was remarkable: in Germany, Steiner and Staudt owe the existenceof their works to Poncelet; in France, Chasles18, the first member of thedepartment of higher geometry at the Sorbonne, was the most outstandingrepresentative of modern pure geometry. To Chasles we are indebted forthe important historical monument L'Apercu historique sur le Developpementde la Geometrie, which led to the correction of a certain number of wrongopinions.

Before ending our discussion of Poncelet, I note that he played an impor-tant role in developing applied mechanics, which he taught for a long time,first in Metz and then at the Sorbonne.

16Emile Borel (1871-1956).17 Joseph Diez Gergonne (1771-1859).18Michel Chasles (1793-1880).


Evariste Galois (1811-1832) is one of the most unusual figures in thehistory of science. Having twice failed the entrance exam at l'Ecole Poly-technique, in 18 31 he was accepted to l'Ecole Normale, only to leave it ayear later. Taking an active part in politics earned him several months inprison; not quite twenty-one years old, he was killed in a duel triggered by aninsignificant quarrel. He had presented his mathematical discoveries in equa-tion theory to the Academy of Sciences in two different announcements, butboth of them were later lost; fortunately, he had also published them in sev-eral small articles in Bulletin de Ferussac in 1830 and also talked about themto his friend Chevalier in a letter written shortly before his death. Some otherresults, discovered among his papers, were published in 1846, in Liouville'smagazine.

The significance of his work can be explained quickly. Tartaglia, Cardano,and Ferrari, Italian algebraists of the sixteenth century, used the second andthird roots to solve equations of the third and fourth degree; however, allefforts to solve equations of higher degrees in the same way were in vain. Byshowing that some classes of equations can indeed be solved in that same way,Lagrange, Abel, and Gauss contributed a great deal to this problem. Abel firstshowed, in 1826, that a general equation of the fifth degree cannot be solvedby means of radicals. In that way it became clear that the problem, with whichmathematicians had wrestled since the sixteenth century, had not been wellformulated. The glory for solving it belongs to Galois, for he showed thateach equation determines a certain number of permutations of its roots, thepermutations forming a so-called group; although applied to the roots, thesepermutations do not disrupt their rational interactions (the meaning of theterm "rational interactions" needs an additional explanation). The natureof that group determines the basic properties of the equation, whether it ispossible to find its roots or not, and, in a general case, the nature of auxiliaryequations whose solving would result in solving the original equation. Bystarting from his own idea, Galois easily found the results of his predecessorsand successfully incorporated them into his own result.

The theory of substitution groups, i.e., groups of permutations of a certainnumber of objects, which was founded by Cauchy, demonstrated its full valuethrough Galois's works. Galois improved its important aspects and demon-strated how basic was the role of ordinary groups. Moreover, he enrichednumber theory by introducing new classes of imaginary quantities (Galois'simaginary numbers), each of which was tied to a power of a prime number,Galois's name is frequently encountered not only in the theory of equationsbut also in modern algebra. The letters he sent to his friend Chevalier makeit clear that in analysis he had as many important results as in algebra andthat his works on Abel integrals were twenty-five years ahead of those of thefamous German mathematician Riemann. Although it makes me sad to thinkhow much science lost by Galois's early death, I must also say that, as EmilePicard once put it, "When confronted with such a short and turbulent life,


one's respect for the extraordinary mind which left so deep trace in sciencegets even greater."

It was Galois's theory that made it possible to explain the miracle whichallowed imaginary quantities to appear in the formula for solving a third-degree equation with real roots; indeed, it became possible to show that, ifan equation has all roots real and if it can be solved by means of radicals, thenit can be solved by means of square roots only. By using the same theory, itcan also be shown that some of the ancient problems such as the problemof doubling a cube or the problem of trisecting an angle cannot be solvedwith a ruler and a compass. By his significant work Traite des Substitutions,Jordan19 erected a monument in honor of Galois.

Being both simple and profound, Galois's main idea permitted applica-tions in areas other than algebraic equations. Emile Picard and Ernest Ves-siot, for example, considered it highly important in integration of linear dif-ferential equations. It is noteworthy that Drach and Vessiot attempted toextend Galois's theory to solving the most general differential equations butencountered difficulties that could be overcome only if the original theorywere altered or if, at least, some of its magnificent simplicity were sacrificed.

The development of science after Galois demonstrated the growth of theimportance of groups in the most diverse branches of mathematics andphysics. Norwegian mathematician Sophus Lie, the founder of the theoryof groups of transformations, introduced them into analysis and geometry.A great admirer of Galois, he dedicated his momentous opus about groups oftransformations (in 1889) to l'Ecole Normale Superieure. Indeed, the mostsignificant results concerning developing, refining, extending, and finding newapplications of Galois's theory were made in France. Poincare claimed thatthe notion of group had already existed in the spirit of geometry; the axiomthat two geometric figures are equal to each other if each of them is equal toa third is in fact identical to the statement that there is a group that regulatesgeometry, more precisely a family of procedures by which one figure turnsinto another that is equal to the first. It is extraordinarily important thatgroup theory is capable of giving us all concrete, connected meanings thatcan be given to the expression "equal figures"; as it was shown in 1872 by thegreat German mathematician Felix Klein, exactly this implies the existenceof infinitely many geometries, each ruled by a special group, as well as by thefact that each geometry can be investigated independently, without resortingto elementary geometry. This framework encompasses projective geometry,the field in which two figures are considered equal if one of them can beobtained from the other by a sequence of projections.

VI

Since Galois's death one century has passed. During that period mathe-matics has developed remarkably, innumerable volumes have been written,

19Camille Jordan (1838-1922).


some of which, I must say, take undeserved space in libraries. Some of thetheories, just formulated at the time of Galois, have since been profoundlyexplored, and some of them have penetrated other areas of mathematics; ina word, as mathematics, like other sciences, has been constantly and dramat-ically changing, it became difficult for a mathematician, no matter who hemight be, to have true insight into its current state. There are fewer and fewerminds capable of making significant discoveries in either pure or appliedmathematics. It is rare to encounter a genius similar to that of the French-man Andre Ampere (1775-1836), who was also a physicist, the founder ofelectrodynamics, and a remarkable mathematician (he and Monge share thecredit for creating the theory of partial differential equations of the secondorder). The Frenchman Gabriel Lame (1795-1870) was an analyst, geometer,and the founder of elasticity theory, while the Frenchman Simeon Poisson(1781-1840) is famous for his works in analysis and mathematical physics;Augustin Fresnel (17 88-1827 ), the creator of physical optics whose workshad finally ensured, at least until the appearance of quantum physics, a tri-umph of the modular theory of light can be considered a mathematician aswell.

Instead of giving you a long, and likely tedious, list of names, let us focuson just a few of the greatest contemporary French mathematicians, those whowere my professors and to whom I am honored and happy to have a chanceto pay respect.

Soon after being admitted to l'Ecole Polytechnique, Charles Hermite(1822-1901) wrote to the well-known professor Jacobi-who, along withAbel, was one of the founders of the theory of elliptical functions and senthim an article about classifying Abel's transcendental functions, the func-tions related to integration of the most general algebraic differentials. Jacobi,who was once, under similar circumstances, kindly received by Legendre,congratulated the young Hermite on his marvelous results. That was only thebeginning of regular correspondence between these two great mathematicians.It was Jacobi to whom, at the age of twenty-four, Hermite sent his discover-ies in advanced algebra, the discoveries that ultimately secured him a placeamong the most prominent geometers. Building on the most famous Gauss'sresults, he confidently approached the algebraic theory of shapes in their mostgeneral form and introduced continuous variables into number theory, a fieldcharacterized by discontinuity. The fact that he was the one who introducedquadratic forms with indefinite conjugate terms, today known as Hermite'sforms, is the reason that his name is one of the most frequently found inworks from quantum physics. In 1873, Hermite became famous by discover-ing the transcendentality of e , the base of Neper's logarithm (the existence oftranscendentals, the numbers that satisfy no algebraic equation whose coeffi-cients are rational numbers, had first been demonstrated by Joseph Liouville).As Hermite's result made a strong impression, some expected him to provetranscendentality of n, and thus, consequently, to destroy forever the hope


that a circle can be squared with a ruler and a compass; however, havingfound inspiration in Hermite's method and having devised a way to modifyit properly, Ferdinand Lindemann, a German mathematician, came up witha proof instead, securing the honor for himself.

Hermite always left a profound impression on his listeners. "No one willever forget the sermon-like sound of Hermite's lectures," said the well-knownmathematician Painleve2°, "or the feeling of beauty and revelation that onehad to experience while listening to him talk about a marvelous discovery orsomething that was still waiting to be discovered. His word had the abilityto open vast horizons of science; it conveyed affection and respect for highideals." Every time I had a chance to listen to Hermite, I had before me animage of quiet and pure joy caused by contemplations about mathematics,joy similar to the one that Beethoven must have felt while feeling his musicinside of himself.

Gaston Darboux (1847-1917) was an analyst and geometer at the sametime. Although he was the initiator of some results in analysis, I shall not talkabout that part of his work because it was his work in geometry that broughthim recognition. He surely was not one of the geometers who avoided tar-nishing the beauty of geometry by flattering analysis, and neither was he oneof the analysts inclined to reduce geometry to calculations without any con-cern for or interest in their geometric meanings. In this respect he followedin Monge's footsteps, connecting fine and well-developed geometric intuitionwith skilled applications of analysis. All of his methods are extraordinar-ily elegant and perfectly suited for the subject under investigation. Whileteaching in the department of higher geometry at the Sorbonne, where hesucceeded Michel Chasles, he frequently and with reverence spoke about thetheory of triple orthogonal systems, with pleasure stressing the importance ofLame's works; not less frequently he spoke about the theory of deformationsof planes, the theory which originated in Gauss's Disquisitiones circa Super-frcies Curvas and which, even before Darboux, was a subject of significantworks of French mathematicians, among whom Ossian Bonnet certainly de-serves a mention. Finally, Darboux demonstrated the usefulness of a systemof local coordinates, i.e., coordinates connected with the investigated figurerather than independent of it. Thanks to the theory of groups, Elie Cartanfurther developed this approach and adapted it to the most diverse spaces cre-ated as a consequence of general relativity theory. Darboux had tremendousinfluence on the development of geometry; of his numerous students and fol-lowers, I shall mention only the well-known Roumanian geometer Tzitzeica,one of the founders of the Mathematical Reviews of the Balkan Union, aman whose recent death is still mourned in the world of science. Classic inits field, Darboux's work Theorie des Surfaces is a splendid monument erectedin honor of both analysis and geometry.

20Paul Painleve (1863-1933).


A story has it that, when a young German mathematician expressed hispuzzlement over Lagrange's refusal to recognize Gauss as the greatest Germangeometer, Lagrange told him, "No, he cannot be the greatest German geometer for he is the greatest European geometer!" In the same spirit one couldsay that Henri Poincare (1854-1912) was not only a great mathematicianbut mathematics itself. It is impossible to find a branch of mathematics-abranch of physics even-in which he did not leave a trace or which he didnot rejuvenate or from which he did not infer a completely new field. Aftercreating Fuchsian functions21

, he used uniform functions with the same parameter to express the coordinates of a point on an algebraic surface, and inthat way obtained the result which, before him, was known only for some special classes of surfaces. He solved the uniformization problem in a way that,at the time, was quite brave. He was a forerunner of the theory of functionswith several complex variables. Also, he created the theory of differentialequations in a real field; due to that theory, he was then able to restore themethods of celestial mechanics, to study periodic solutions of problems ofthis field, and to investigate stability problems. In analysis situs, the part ofgeometry interested only in those properties of objects that are not affected bycontinuous transformations, Poincare authored several treatises that wouldbecome the starting point for nearly all later results in that field. At the Sorbonne, by lecturing on all areas of mathematical physics, he influenced theideas triggered by Michelson's experiment22

• With his early death, sciencelost one of its most prominent leaders. Translated to many languages, hisscientific-philosophical works La Science et ['Hypothese and La Valeur de laScience are well known to the entire world. In some ways-one of which iswell illustrated by Poincare's words, "Thought is only a flash in the middleof a long night, but the flash that means everything"-Poincare can be compared with Pascal. It will take a long time to develop all of Poincare's ideasand to explore all of the paths that he had paved by his rich and diverse work.

Finally,! would like to mention Paul Appell and Edouard Goursat-thefirst of whom is the author of Traite de Mecanique Rationnelle, and the secondof Traite de Calcul Differentiel et Integral-and also, once again, Emile Picard, the last living from that celebrated generation. Two years ago, togetherwith the great German mathematician David Hilbert, Emile Picard receiveda gold medal from the Mittag-LeIDer Institute, and only several weeks ago,at the celebration of the fifty years since Picard was elected a member of theAcademy of Sciences, Emile Borel talked about his scientific opus. I alreadymentioned the famous theorem named after him, as well as those among hisworks that developed Galois's theory. His work concerning algebraic functions with two variables represents the foundation of algebraic geometry, a

21 It was Poincare himself who named them this way after the German mathematician LazarusFuchs; nowadays, these functions are called automorphic.

22Also known as the Michelson-Morley experiment.


branch of geometry especially well developed in Italy. It is true that theviewpoint of Italian geometers from the past century was more clearly de-fined than that of Emile Picard, but, as Emile Borel put it, algebraic geometrywould be certainly crippled without Picard's contributions.

VII

The glory of French mathematics created by the greatest results of Hermite,Darboux, Poincare, and Picard has not darkened. Indeed, the flame is asstrong as it has ever been. As the time is short, to justify this statement I amforced to limit myself to just a few names.

Gabriel Kaenigs was a fine geometer, the elegance of some of his works canbe compared with that of Darboux. By creating new transcendentals, PaulPainleve solved a problem that even to Poincare seemed unapproachable;Poincare characterized Painleve's results in analysis by saying: "Mathemat-ics is a well-ordered continent whose countries are united; the work of PaulPainleve is a magnificent island in an ocean." But this judgment is some-what incomplete because Painleve who, for a long time, taught mechanicsat l'Ecole Polytechnique-also remarkably advanced mechanics; besides, histheoretical research prompted development of aviation in such a measurethat one may say that, thanks to Painleve, aviation is an exclusively Frenchcreation.

The results of Jacques Hadamard were numerous and significant: in arith-metic, he worked on the Riemann's function related to the complicated prob-lem of distribution of prime numbers; in geometry, he researched geodesiclines with opposite curvatures; in analysis, he published works about par-tial differential equations in mathematical physics. Also, he gave a strongstimulus to the calculus of variations and functional analysis, the new sci-ence founded by the Italian mathematician Volterra. Finally, his seminar atCollege de France, where all foreign mathematicians wished to present theirlatest results, influenced international collaboration in mathematics. As he isstill young, I may say with certainty that his work is far from finished.

The research of functions with complex variables has always been verysuccessful in France. Here I mention Emil Borel; the short-lived analystFatou; Paul Montel, famous for his theory concerning families of normalfunctions; Gaston Julia, known for his works about elevation of rationalfunctions; and so forth.

The theory of functions with real variables is of almost exclusively Frenchorigin. Set up by Camille Jordan's Traite d'Analyse (which, like Emile Pi-card's treatise of the same name, had international influence), founded by theworks of Emile Borel, Henri Lebesgue (who defined measure of a set), RendBaire (who introduced integrals which today bear his name), and Denjoy (thecreator of the totalization theory), it introduced unexpected harmony into afield that had been neglected for a long time, testimony to the daring andtalent of its creators.


I cannot but mention Maurice Frechet's theory of abstract spaces, Bouli-gand's infinitesimal geometry, and Elie Cartan's works in analysis and geom-etry, the last of which I am not qualified to judge.

Institut Henri Poincare is born from new French enthusiasm for researchin the field of mathematical physics. Emile Borel, the soul of probabilitytheory, started a praise-deserving series of publications in this field, similar tothe one in function theory -a series in which Frechet, Paul Levy, and GeorgesDarmois presented their excellent results. The Department of TheoreticalPhysics is headed by Louis de Broglie, the creator of wave mechanics, whorestored atomic physics and reconciled the undulatory and corpuscular theoryof light. I should not forget to mention the Institut of Mechanics, headed byHenri Villat, known for his results in hydrodynamics, who is also editor of theinternationally known collection Memorial des Sciences Mathematiques andeditor-in-chief of Journal de Mathematiques Pures et Appliquees, a journalwhich, nearly a century ago, was started by Liouville and which for quitesome time was edited by Camille Jordan.

The account of French mathematical activity would be incomplete with-out a mention of l'Ecole Polytechnique and l'Ecole Normale. For more thana century, great French mathematicians have owed their education to oneof the two institutions; in the last half-century that marvelous role belongedalmost exclusively to l'Ecole Normale, which, even a good fifty years ago,Sophus Lie considered a nursery of French mathematics. Young talents frommany countries have been coming here to get the same education as theirFrench colleagues. That is why it is difficult not to consider Georges Tzitze-ica, whom I already mentioned, to be a French mathematician. For the samereason, I am inclined to include among French mathematicians my goodfriend Mihailo Petrovic, a doyen of Yugoslav mathematics, who is widelyrecognized for his great originality in inventing the spectral method ' i n arith-metic, algebra, and analysis, and also for creating general phenomenology,the field which systematically examines the problems of existence of analyt-ical molds that could be used to present simultaneously several apparentlydifferent physical theories. I hope that you will not object if I credit hisresults to the accomplishments which mathematics owes to France.

Thanks to l'Ecole Normale, young mathematicians are ready to replacethe older ones. One might say that it is too early to mention names, butsome of them are nevertheless already well known. I shall mention onlyJacques Herbrand, whose works, mercilessly interrupted by his early death,were announcing a great mathematician, perhaps similar to Evariste Galois.

Ladies and gentlemen, it is time for me to finish this talk, for I have alreadyused a great deal of your kind attention. In conclusion, I would like to makejust one remark of general nature.

More than any other science, mathematics develops through a sequence ofconsecutive abstractions. A desire to avoid mistakes forces mathematiciansto find and isolate the essence of the problems and entities considered. Car-


ried to an extreme, this procedure justifies the well-known joke according towhich a mathematician is a scientist who neither knows what he is talkingabout or whether whatever he is indeed talking about exists or not. Frenchmathematicians, however, never enjoyed distancing themselves from reality;they do know that, although needed, logic is by no means crucial. In math-ematical activity, like in any other type of human activity, one should finda balance of values: there is no doubt that it is important to think correctly,but it is even more important to formulate the right problems. In that re-spect, one can freely say that French mathematicians not only always knewwhat they were talking about, but also had the right intuition to select themost fundamental problems, those whose solutions produced the strongestinfluence on the overall development of science.

Bibliography

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[Ya 1) L M. Yaglom, Tangential metric in two-parameter famil y of curves on a plane, TrudyScm. Vektor. Tenzor. Anal. 7 (1949), 341-36 1. (Russian)

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‰lie Cartan (1869-1951)

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