Remembering Shoshichi KobayashiHyperbolic Manifolds and Holomorphic Mappings and the Kobayashi...

Remembering ShoshichiKobayashiGary R. Jensen, Coordinating Editor

Shoshichi Kobayashi was born January 4,1932, in Kofu City, Japan. He grew upamidst the colossal devastation of WorldWar II. After completing a BS degreein 1953 at the University of Tokyo, he

spent the academic year 1953–1954 on a Frenchscholarship at the University of Paris and theUniversity of Strasbourg. From there he went tothe University of Washington in Seattle, where hereceived a Ph.D. in 1956 for the thesis “Theoryof connections,” (Annali di Mat. (1957), 119–194),written under the direction of Carl B. Allendoerfer.There followed two years as a member of theInstitute for Advanced Study in Princeton and thentwo years as a research associate at MIT. In 1960he became an assistant professor at the Universityof British Columbia, which he left in 1962 to go tothe University of California at Berkeley, where heremained for the rest of his life.

Shoshichi published 134 papers, widely admirednot only for their mathematical content but for theelegance of his writing. Among his thirteen books,Hyperbolic Manifolds and Holomorphic Mappingsand the Kobayashi metric in complex manifoldscreated a field, while Foundations of DifferentialGeometry, I and II, coauthored with Nomizu, areknown by virtually every differential geometrystudent of the past fifty years. In their articlesbelow, Shoshichi’s student, Toshiki Mabuchi atOsaka University and former Professor TakushiroOchiai at the University of Tokyo summarize

Gary R. Jensen is professor emeritus of mathematics atWashington University in St. Louis. His email address [email protected].

Personal and family photos of S. Kobayashi provided by andused with permission of the Nomizu family.

DOI: http://dx.doi.org/10.1090/noti1184

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many of Shoshichi’s mathematical achievements.Several of his students relate their experiences ofworking under Professor Kobayashi’s supervision.Some Berkeley colleagues write of their personalexperiences with Shoshichi.

His brother, Hisashi, recounts some details ofShoshichi’s early education. Shoshichi’s wife anddaughters tell a fascinating story of his childhood,his studies abroad, and major life experiencespreceding his move to Berkeley.

He died August 29, 2012, of heart failure on aflight from Tokyo to San Francisco.

Toshiki MabuchiMany of Professor Kobayashi’s books are known asstandard references in differential geometry, com-plex geometry, and other related areas. Especially,Foundations of Differential Geometry, Vols. I and II,

Toshiki Mabuchi is professor of mathematics at OsakaUniversity. His email address is [email protected].

1322 Notices of the AMS Volume 61, Number 11

coauthored by Nomizu, are very popular withmathematicians as well as with physicists. Heand Nomizu received the 2007 MSJ PublicationPrize from the Mathematical Society of Japan forthis work. His books Hyperbolic Manifolds andHolomorphic Mappings (1970) and TransformationGroups in Differential Geometry (1972) also influ-enced many mathematicians. His mathematicalachievements range across differential geometry,Lie algebras, transformation groups, and complexanalysis. The most important ones are:1. the Kobayashi intrinsic pseudo-distance,2. the Kobayashi hyperbolicity and measurehyperbolicity,3. projectively invariant distances for affine andprojective distances,4. the study of compact complex manifolds withpositive Ricci curvature,5. filtered Lie algebras and geometric structures,and6. Kobayashi-Hitchin correspondence for vectorbundles.

In (1) and (2) we see his outstanding creativ-ity. Kobayashi’s distance decreasing property forholomorphic mappings plays a very important rolein (1), while the generalized Schwarz lemma iscrucially used in (2). His works now give us a funda-mental tool in the study of holomorphic mappingsbetween complex manifolds. For instance, Picard’ssmall theorem follows easily from (2). Recently,the above results were being generalized by himto almost complex manifolds.

On the other hand, (4) has led succeeding mathe-maticians to Frankel’s Conjecture and Hartshorne’sConjecture. Among them, his joint work with T.Ochiai, “Characterization of complex projectivespaces and hyperquadrics,” J. Math. Kyoto U. 13(1972), 31–47, was effectively used in Siu-Yau’sproof of Frankel’s Conjecture. It should also benoted that the method of reduction modulo p inMori’s proof of Hartshorne’s Conjecture becamea clue to the Mori theory on the minimal modelprogram of projective algebraic manifolds.

The Kobayashi-Hitchin correspondence for vec-tor bundles in (6) states that a holomorphic vectorbundle E over a compact Kähler manifold is stablein the sense of Mumford-Takemoto if and only ifE admits a Hermitian-Einstein metric. Kobayashiand Lübke proved the “if” part, while the “only if”part, conjectured by Kobayashi and Hitchin, wasproved finally by Donaldson and Uhlenbeck-Yau.

Since 1995 Professor Kobayashi regularly at-tended our annual workshop on complex geometryat Sugadaira, Japan.

Takushiro Ochiai 1

The life and academic achievements of ShoshichiKobayashi give a definition of what a greatmathematician should be. He left us numerousmanuscripts of high originality, comparable togreat musical compositions. His books, taken asa whole, harmonize splendidly into a great sym-phony. Though aware of my inability to reach theheight of his talent, I dare to write this article tobring attention to the fine personal character andremarkable academic achievements of ShoshichiKobayashi, outstanding mathematician, mentor,and colleague.

Kobayashi published papers in academic jour-nals every year without fail, starting from his virginpaper in 1954 until his last days. Among a totalof 134 papers, there are 85 single-author papersand 49 collaborative works. A unique featureof his single-author publications is their brevity.There are 34 papers of less than five pages, 28papers of less than ten pages, and 15 papers ofless than 28 pages. Every one of them is deepand rich in content with transparent and easilycomprehensible explanations, as is evidenced byhis invention of the concepts of the Kobayashidistance, hyperbolic complex manifolds, and theHermitian-Einstein vector bundles. Inspired byChern’s result, which further improved on thegeneralization of the classical Schwarz’s lemmaby L. Ahlfors (“The holomorphic maps betweenHermitian manifolds” by S. S. Chern), Kobayashibecame interested in Schwarz’s lemma and readall the related papers one by one. He especiallyadmired Carathéodory’s point of view and devotedhimself to and took advantage of the holomorphicmaps, which he admitted led him to the discoveryof the Kobayashi distance.

Expanding all his papers, paying attention to thehistoric background and development, Kobayashisubsequently published thirteen self-containedbooks, every one of which is easily comprehensibleby graduate students. Professor Kobayashi oncesaid to me: “When I write books, I prepare oneor two years for the first manuscript and makecertain to give lectures based on it for one or twoyears in order to deepen the contents before finallycompleting the final manuscript.”

To convey to the public his philosophy of mathe-matics, or rather of differential geometry, I includehere passages from his essays in Japanese as pub-lished in well-known magazines on mathematical

Takushiro Ochiai is professor emeritus of mathematicsat the University of Tokyo. His email address is [email protected] is a translation into English of some part of my article“Kobayashi-sensei wo shinonde,” which appeared in SugakuSeminar, 2013.

December 2014 Notices of the AMS 1323

First row: T. Ochiai, S. Kobayashi, S.-T. Yau;Second row: H. Mori, S.-S. Chern. Tokyo, 1977.

sciences which deeply reflect his mathematicalachievements. The transparency, beauty, and open-mindedness of his viewpoints, which he sharpenedwell to the limit, express my personal and eternalyearning.

“Differential geometry is one viewpoint overmathematics, and a method, in itself. When variousfields in mathematics reach the point of being wellunderstood, the phenomenon called ‘algebraiza-tion’ occurs. I think it is possible to see differentfields of mathematics from a differential geometricviewpoint as well as from an algebraic viewpoint.The raison d’etre of differential geometry is tooffer a new viewpoint and powerful methods ratherthan being considered like a theory of numbers.Moreover, the concepts and methods understoodfrom a geometric point of view are so natural(different from artificially created nonsense) thatdevelopment beyond anticipation later occurs inmany cases. Differential geometry can producelimitless developments by carrying its methodsinto all the fields of mathematics. It is especiallyimportant, then, to know how to make use of thedifferential geometric method to make connectionsin fields such as the theory of functions (one orseveral complex variables), algebraic geometry,topology, a differential equation theory.” (SuuriKagaku, August 1965)

“Wonderful theorems in mathematics are provenwith breakthroughs of originality which causeeveryone to understand them. A mathematiciangets the greatest feeling of happiness when he findssuch a new idea. A problem, which is solved merelyby understanding and following a routine process,is a petty problem, and a theorem whose proofdoes not use any idea which comes involuntarily,is really tedious. It seems that any problem whichdoes not open a new field, or any result which hasno application in any other field of mathematics,disappears after a while. An eternal life is given onlyto a beautiful result.” (Sugaku Seminar, December1965)

“Some percentage of the work in mathemat-ics consists of the creation of suitable notation.Suitable notation makes calculations easy to han-dle, makes formulas look beautiful and easy tomemorize, and makes theorems apparent at aglance. Suitable notation is important not onlyfrom a passive role to make descriptions beautifuland easy, but also from an active role, even tosuggesting what should be done next. We cannotexplain in one word what kind of notation issuitable or what kind of theorem is good. It is amathematical sense that will take it in somehow.”(Sugaku Seminar, September 1967)

“When you study subjects in mathematics whichare not restricted to calculus, I would like tourge you to study the history of its developmenttogether by all means. Since modern mathematicsrequires too much stringency, lectures as wellas textbooks rarely touch historical background.However, I think that one can understand thesubject more deeply by getting to know the historyand ‘why this concept was produced.’ I want peoplewho become school teachers to study the historyof mathematics by all means.” (Suuri Kagaku,February 2001)

Through my acquaintance with ProfessorKobayashi for many years, I am convinced thathis passion for mathematics was motivated byhis love for human beings and for scholarship. Iam extremely fortunate to have collaborated withProfessor Kobayashi on some of his mathematicalwork.

Joe WolfI met Shoshichi Kobayashi when we both arrivedat Berkeley in September of 1962. We got to knoweach other quite well for two reasons. First, thedifferential geometry group at Berkeley, just thenforming under Professor Chern’s guidance, wasvery cohesive, both socially and mathematically.Second, we shared an office where we had manyinformal conversations about differential geometry,mathematics in general, and academic life. I alwayslearned some interesting mathematics when Shoand I talked. Years before, I had read Nomizu’s shortpaperback Lie Groups and Differential Geometry,and I was thrilled to learn that Shoshichi had justcompleted his monumental work on differentialgeometry with Nomizu. At various times during the1960s the geometry group at Berkeley included S.-S.Chern, Sho Kobayashi, Phil Griffiths, Jim Simons,Alfred Gray, Peter Gilkey, Jeff Cheeger, BlaineLawson, Nolan Wallach, Manfredo do Carmo, Wu-YiHsiang, Hung-Hsi Wu, me, and many others. It was

Joe Wolf is professor emeritus of mathematics at theUniversity of California, Berkeley. His email address [email protected].


very collaborative and did not draw distinctionsbetween big shots, young academics, and graduatestudents. And there were many famous andinfluential visitors attracted by Chern, includingGene Calabi and Fritz Hirzebruch. So it was a kindof mathematical heaven. During this time Shoshichiconstructed his pseudo-metric, with the associatednotion of Kobayashi hyperbolicity, and also hisreproducing kernel methods for irreducibility ofunitary representations.

When the building that currently houses themath department (Evans Hall) was built, math wasto have floors seven, eight, nine, and ten. Sho and Iwent into the building to choose our offices beforethe elevators were installed. We walked up thestairs together, and at some point I realized that hewas speeding up. I couldn’t keep pace, but he hadto stop at the seventh floor and I managed to getto the eighth. So his new office was on the seventhfloor and mine was on the eighth. Later, when Showas math department chair, the chancellor’s officeinformed him that we were losing our space on theseventh floor. With Sho in charge we came out ofthese “space wars” pretty well, retaining most ofthe seventh floor.

It is hard to realize that Sho is no longer withus. It was a great privilege to have been a friendand colleague of Shoshichi Kobayashi.

Hung-Hsi WuI first met Sho in 1962 at one of the AMS SummerInstitutes in Santa Barbara. I had just finishedmy first year as a graduate student at MIT, andhe told me he was on his way to Berkeley. Weended up being colleagues for forty-seven yearswhen I myself got to Berkeley in 1965. Althoughas colleagues we could not help but run into eachother often, I think it was in the ten or so yearsfrom 1980 to 1990 that I had extended contactwith him every week when he drove me home aftereach differential geometry seminar late in the dayon Friday. We had to walk a bit before we could getto his car and that gave us even more of a chance tochat and gossip. I am afraid the intellectual qualityof the conversations was not particularly high,but the entertainment value was off the charts. Itwas most enjoyable. However, the one thing thathas stuck in my mind about Sho all these yearsis probably the fortuitous confluence of eventssurrounding the discovery of the Kobayashi metricin 1966.

In the summer of 1966, Professor Chern andI attended the AMS Summer Institute on entirefunctions in La Jolla, and Professor Chern gave a

Hung-Hsi Wu is is professor emeritus of mathematics atthe University of California, Berkeley. His email address [email protected].

M. Berger, S. Kobayashi, W. Klingenberg, K. Yano,S.-S. Chern, 1970s.

lecture on his new result on the volume decreasingproperty for holomorphic mappings from (basi-cally) the unit ball in Cn into an Einstein manifoldof the same dimension with negative curvaturein a suitable sense. This is a generalization ofthe famous Ahlfors-Schwarz lemma on the unitdisc, and it is to Professor Chern’s credit thathe recognized it as the special case of a generaltheorem about holomorphic mappings on complexmanifolds. At the time, the idea of putting thesubject of holomorphic functions in a geometricsetting was very much on his mind. In the followingfall, he gave a similar talk in one of the first lecturesof the Friday geometry seminar, but this time hehad a manuscript ready. The main ingredientof his proof is basically a Weitzenböck formulafor the volume form; his observation was thatnegative curvature (in one form or another) of thetarget manifold limits the behavior of holomorphicmappings. This is the beginning of what PhillipGriffiths later called hyperbolic complex analysis.Of course both Sho and I were in the audience,and within two or three weeks Sho came up with ageneralization using more elementary methods.

At the time I was fascinated with Bloch’stheorem (in one complex variable) and was tryingto understand why there would be a univalentdisc for holomorphic functions into the unit disc.Professor Chern’s paper contains a reference to apaper of Grauert-Reckziegel which can also be saidto be an application of the Ahlfors-Schwarz lemma.Upon reading it, I got the idea that Bloch’s theoremwas a consequence of the phenomenon of normalfamilies and, as a result, I could prove a qualitativegeneralization for Bloch’s theorem for holomorphicmappings between complex manifolds. I wroteup my findings and, as it was the tradition thenamong the geometers at Berkeley, put copies of mymanuscript in my geometry colleagues’ mailboxes.


Nomizu and Kobayashi, 1954.

In a few days, Sho came up with the manuscriptof his announcement that all complex manifoldscarry an intrinsic metric, the metric that now bearshis name. Sho recognized that with the availabilityof the Ahlfors-Schwarz lemma, the constructionof the Carathéodory metric could be “dualized” todefine the Kobayashi metric. His paper took thefocus completely out of the holomorphic mappingsthemselves and put it rightfully on the Kobayashimetric of the target complex manifold in question.This insight sheds light on numerous classicalresults (such as the little and big Picard theorems)and inaugurates a new era in complex manifolds.

Robert GreeneBerkeley in the 1960s was an earthly paradise forpeople interested in real and complex geometry.When I arrived in early 1965 as a new graduatestudent, Professor Chern was the unquestionedleader of the geometry group, and indeed of thefield itself, and Kobayashi occupied a special placeas an organizer and presenter of the field as awhole. He was, after all, the author with K. Nomizuof The Book, Foundations of Differential Geometry,which was to function for many years as theveritable bible of the subject. It was a summit allthe new students in geometry were determinedto climb. When Volume II appeared in 1969, thewhole became a definitive work indeed, a positionwhich the passage of time has not dimmed as faras the field up to that point is concerned.

After qualifying examinations, I began workwith Hung-Hsi Wu as my dissertation advisor, ahappy association that turned later into a long-termcollaboration. Almost as soon as Wu had acceptedme as a student, he went on sabbatical in Englandfor a term, so he asked Kobayashi to take me under

Robert Greene is professor of mathematics at the Uni-versity of California, Los Angeles. His email address [email protected].

his wing. The proposal was that I should readHelgason’s Differential Geometry and SymmetricSpaces under Kobayashi’s guidance. Kobayashi wasthe soul of politeness in dealing with my obviousdisaffection from the book I was supposed to begoing through. And in spite of this somewhatrocky start—I think Kobayashi never quite entirelyforgave me for not liking Helgason’s book—webecame friends.

Impressive though Kobayashi and Nomizu’sgreat survey book was and is, it was anotherof Kobayashi’s books, Hyperbolic Manifolds andHolomorphic Mappings, that demonstrated bestthe elegance with which he could present a subjectfrom its beginnings. When his “little red book,” aswe young people thought of it, appeared it had abig influence on us, flowing along as it did like acrystal stream. It was seductive, and indeed it didconvince people that hyperbolic manifolds had agood bit more vitality as an independent subjectthan it in fact did in the long run, vital though theKobayashi metric became and remains as a tool.The hyperbolic idea swept through Berkeley like aCalifornia wildfire in the hills. I did not work onhyperbolic manifolds directly at that time, but laterin the mid-1970s Wu and I developed a refinementof the strictly negative curvature criterion, so Idid get into the hyperbolic act eventually. Andthe general circle of ideas did influence me in thedirection of thinking about complex manifolds.In this somewhat indirect way, Kobayashi shapedmy mathematical life, since complex geometry hasremained my principal mathematical interest.

Gary R. JensenFor me, Shoshichi Kobayashi was the ideal thesisadvisor and mentor. Without his help and guidanceat many stages, I would never have become aprofessor of mathematics. Here is the story.

After four semesters of course work at Berkeley,I could imagine only Professor Kobayashi assomeone I might talk to. When I asked him tobe my advisor, his response was cautious butnot cold. After all, I had taken no courses indifferential geometry other than the requiredcurves and surfaces course. He suggested that Ispend the summer reading his new book, writtenwith Nomizu, Foundations of Differential Geometry,Vol. I. Thus began my lifelong love of differentialgeometry.

At the beginning of the fall semester he agreedto be my advisor. He handed me a list of problemshe and James Eells had edited for the proceedingsof a recent conference in Kyoto, Japan, with anindication of two or three problems that mightinterest me. A week later we agreed that I look atthe problem proposed by Eells and Sampson: Does


any simply connected, compact Riemannian spaceof nonnegative curvature admit a Ricci parallelmetric? During that academic year I read papersand got nowhere. When I finally told ProfessorKobayashi that I felt I was making no progress,in fact, that I really had no idea of how to begintrying to solve this problem, his reply was simpleand profoundly helpful. “Why don’t you just lookat the dimension four homogeneous case first,” hesaid. It is embarrassing to remember that for ayear this idea had not occurred to me. Along withthis advice he suggested a paper by S. Ishihara onfour-dimensional homogeneous spaces. At last, apaper I understood and saw how to apply to myproblem.

Graduate school at Berkeley in the mid-sixtieswas wonderful. In our meeting at the beginningof the fall 1967 semester, Professor Kobayashiquietly mentioned that this would be my last year.“But I don’t have enough results for a thesis,” Ireminded him. Yes, he agreed, but he stated againthat this would be my last year. I got the messageand it was a powerful motivator. In our weeklymeetings I started presenting partial results, bitsand pieces that at first seemed to hit a wall, butsoon began yielding to the assault. By the end ofDecember I had found all homogeneous Einsteinspaces of dimension four.

Meanwhile, Professor Kobayashi had raised theissue of a job for next year. Strangely enough, Iwas very vague on this point, no doubt due to mysubconscious desire to remain a graduate studentfor the rest of my life. He introduced me to somemathematicians from a university in the East, andI told them I would like to join their department.Weeks passed with no communication from thatdepartment. I didn’t give it much thought, butone day Professor Kobayashi asked me, with someanxiety, whether I had heard anything. Hearingthe answer and hearing that I had not applied foranything else, he took me by the arm and escortedme to the library’s collection of notebooks ofavailable jobs. Together we picked out a few. Hetold me to write a letter of application to eachand to find two more people to write letters ofrecommendation for me. It still frightens me tothink what might have become of me if he hadnot intervened so effectively at that time to makesure that I had applied for jobs. In early March Iinterviewed at Carnegie-Mellon and accepted theiroffer.

In June my family and I headed out for Pittsburghwith a copy of the manuscript of Foundations ofDifferential Geometry, Vol. II, in the trunk of the car.After reading it I struggled to find new researchproblems. I wrote to Professor Kobayashi that Ineeded more contact with differential geometers. Iasked him if a postdoctoral fellowship somewhere

might be possible. In early March a call came fromWashington University in St. Louis asking me if Iwould be interested in coming there to interview fora one-year postdoc position connected to a specialyear in symmetric spaces. Professor Kobayashi hadsuggested my name to them for this position.

At the end of my postdoc year, I accepted atenure-track offer from Washington University.Professor Kobayashi’s mentoring continued. Hewas instrumental in arranging a visiting researchposition for me at Berkeley during the summer of1971. In one conversation that summer he directedmy attention to his 1963 Tôhoku Math. J. paper“Topology of positively pinched Kaehler manifolds,”which formed the basis of my best-known paperof the 1970s, “Einstein metrics on principal fibrebundles,” a paper that probably tipped the tenuredecision in my favor.

Myung H. KwackI am very fortunate to have had Professor ShoshichiKobayashi as my teacher, mentor, and advisor.He was instrumental in my work as a mathe-matician, and I am grateful for his support andencouragement throughout my career.

My father left Korea to study in the US in 1954right after the Korean War. He left the rest of hisfamily of four: a wife and three children. Afterseven years of separation, he brought his familyto the US I had just finished high school and asemester of college in Seoul, Korea, and I foundmyself enrolled in San Francisco State Collegeunable to communicate in a foreign culture. Ifound that I enjoyed studying calculus textbooks.I transferred to Berkeley, getting a BA degree inmathematics in 1965, and then entered into thegraduate program at Berkeley. I still felt sociallyuncomfortable, as my language skills were notmuch improved. In addition, only one or two girlswere in mathematics courses. It was a pleasantsurprise that I passed the qualifying examinationin 1967. I met Professor Kobayashi and somehowfelt comfortable with him and had enough courageto ask him to be my thesis advisor.

With his usual warmth and understandingnature, Professor Kobayashi was able to makeme feel at ease during visits to his office tolearn and ask questions about mathematics. Heintroduced me to the concepts of hyperbolicmanifolds. He suggested the problem of extendingthe classical Big Picard Theorem to hyperbolicmanifolds and gave me many articles dealing withrelated problems. Many times I rushed to his officehaving “solved” the problem and started writing

Myung H. Kwack is professor emeritus of mathe-matics at Howard University. Her email address [email protected].


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my “solution” on the blackboard until I could nolonger continue, having come to a gap or an errorin my “proof.” Professor Kobayashi always hadtime and patience to listen to my wrong “proof”and never suggested that I should try to checkmy “solutions.” Instead, he encouraged me as I satdiscouraged after discovering my error. Withouthis encouragement I would not have been able tofind a proof of a generalization of the Big PicardTheorem and experience the deep pleasure ofdiscovering a mathematical theorem which hascome to be known in hyperbolic geometry asKwack’s Theorem.

Professor Kobayashi understood not only thedifficulty of doing mathematical research but alsothe challenges faced by a young girl from a foreigncountry at the time when girls were not encouragedto study. Professor Wu at Berkeley once said thatProfessor Kobayashi had the most female thesisstudents, making him the envy of the faculty ofthe Department of Mathematics at Berkeley.

Professor and Mrs. Kobayashi came to visitHoward University while I was there when Pro-fessor Kobayashi gave a week of lectures. Hecontinued to support and encourage me to domathematical research by sending many articles,his as well as many related ones, especially when-ever I mentioned any interest in some problems.Furthermore, whenever I come to the Bay Area tovisit my parents, he and his wife would take meto lunch and any gatherings of mathematicians hewas attending.

I will be grateful to Professor Kobayashi forthe rest of my life for enabling me to become amathematician.

Eriko ShinozakiI first met Professor Kobayashi twenty years ago,during my junior year of college, when he wasa visiting professor at International ChristianUniversity in Tokyo, Japan. I had always wanted tostudy mathematics at university and had started asa math major at ICU. I began to doubt this decision,however, because out of all the math coursesoffered at ICU at that time none felt comfortable.

High school was where I always felt confidentwhen studying mathematics and always knew whatI was doing. I had to choose between geometry,algebra, and analysis after the fall of my junioryear, but at that time I never considered geometryas my area of study. I was even thinking aboutchanging to a chemistry major. I took ProfessorKobayashi’s class, since it was a required subject,not knowing that it would totally change my life.Thus, in my third year of mathematics studies, Ithanked God for giving me this opportunity tomeet Professor Kobayashi. I knew he would soon beleaving after the second term (ICU’s academic yearis divided into three terms), so I fully focused ongeometry, solving everything I could, and visitedhis office almost every day.

After the end of the term exams, for the firsttime since high school, I finally achieved a grade atICU with which I was familiar. When I got my examsback from Professor Kobayashi, he asked me if Iwas willing to continue in differential geometry. Totell the truth, all I wanted to do was ask him howI could possibly study differential geometry, dueto the fact that there was no differential geometryteacher at ICU!

Subsequently, he kindly advised me to continuemy differential geometry studies under Ms. MakikoTanaka (currently a professor at Tokyo ScienceUniversity), who was at the time helping us, the ICUstudents, with our calculus recitations. Fortunatelyfor me, she was also a differential geometry majorstudying under Professor Kobayashi’s colleague,Professor Nagano at Sophia University. ProfessorKobayashi also offered to let me continue mydifferential geometry studies under his guidance,as well as advise me on my senior thesis, whichwas to be submitted the following year.

During most of my senior year he was at TokyoUniversity evaluating its mathematical curricula,enabling me to meet with him often on the Hongocampus. I even met him on the day he was returningto Berkeley. We worked on the thesis, then walkedto Ueno station together and took the train to Naritaas we continued our mathematical discussions onthe train, shaking hands and saying good-bye atthe airport.

Eriko Shinozaki is a mathematics teacher in a Yokohama highschool. Her email address [email protected].


Since that time, for almost twenty years, Icontinued to meet with Professor Kobayashi eachtime he visited Japan for meetings, seminars,university evaluations, and special lectures. Whatwe worked on together changed over the years.During my senior year Professor Kobayashi helpedme choose a thesis on the topic of conjugateconnections, helped me find a theorem and thensent it to the Tokyo Journal of Mathematics underthe title “Conjugate connections and moduli spacesof connections.” However, since I decided to workfull-time as a mathematics teacher in a Japanesehigh school rather than pursue graduate studies(finding a job was very competitive in Japan in1995), it became difficult for me to continue withmy research.

One day Dr. Kobayashi asked me if he couldgive my topic to another of his graduate studentsand asked me to help him translate and type, usingLATEX, his book Differential Geometry of Curves andSurfaces into English instead. I thanked him verymuch for allowing me the opportunity to continueworking in the field of mathematics with himand appreciated his confidence in my translationand understanding of his work. Unfortunately, weweren’t able to complete the translation together,but I plan to continue working on it with ProfessorMakiko Tanaka.

Hisashi Kobayashi 2

I realize how different was the world aroundShoshichi, the first son of our family, from thataround me, the third son, although we were raisedby the same parents. I knew pretty much about hiscareer, but I was not well aware of what Shoshichifelt or thought in his youth.

He published several essays in a Japanesejournal, Mathematical Seminar, and elsewhere, butI read them for the first time only after he hadpassed away. Shoshichi was a quiet person likeour mother and did not say much even to us, hisyounger brothers, about his memories and storiesof younger days. I wish I had read these essayswhile he was alive; then I could have asked moredetails. Since my childhood, Shoshichi has beenmy role model, teacher, and the person I respectedmost among those I have personally known. Hehas been my hero, so to speak.

Soon after his birth our parents moved to Tokyoand opened a futon (Japanese bed) store in Koenji,

Hisashi Kobayashi is professor emeritus of electrical engi-neering and computer science at Princeton University. Hisemail address is [email protected] is an excerpt of the speech “Rediscover my brotherShoshichi,” presented at the memorial reception held May 25,2013, at the University of Tokyo. The complete speech is athttp://www.shoshichikobayashi.com.

January 18, 1950. L to R back: Shoshichi (18),Toshinori (15). L to R front: Mother Yoshie (41),Kazuo (2), Father Kyuzo (45), Hisashi (11).

Suginami-ku. By the time I grew to the age whenI could remember things, our parents moved thestore to Kyodo, Setagaya-ku, Tokyo. ApparentlyShoshichi liked mathematics since his childhood,but he had some difficulty with a homeworkassignment given when he was a fifth- or sixth-grader, in which the student was asked to computethe volume of the cone that can be created froma fan shape. He recalls this incident in his essay“Deeply impressed by a beautiful theorem” in theMay 1973 issue of Mathematics Seminar.

In April 1944 Shoshichi entered Chitose MiddleSchool in Tokyo. This school emphasized militarytraining, but discussions with the older studentsbegan his thinking of going on to high school. Itwas a big surprise for me to find that it was notuntil he entered middle school that he developedthe idea of going to a high school, despite the factthat he was talented enough to be inspired by thePythagorean Theorem.

In the spring of 1945 when he became a second-year student of the middle school, our familyevacuated from Tokyo and moved to Minamisaku,Nagano-Ken, and the war ended soon after. Thanksto the kind people of Hiraga Village, the familystayed there until the fall of 1948. When Shoshichibecame a fourth-year student of Nozawa MiddleSchool3 there, the mathematics teacher was Mr. Mu-neo Hayashi. His encounter with Mr. Hayashi turnedout to be a giant step to nurture the mathematicianShoshichi. He writes about that period in anotheressay, “The mathematician I luckily encountered:Muneo Hayashi, math teacher in middle school.”

It seems that Mr. Hayashi’s encouragementallowed Shoshichi to gain confidence to apply to

3Shoshichi’s classmate Noboru Naito wrote in detail of thisschool and Mr. Hayashi in “Newsletter No. 2, August 2013”at the http://www.shoshichikobayashi.com.


http://www.shoshichikobayashi.com

Shoshichi, age 3.

Ichikoh (high school), and our parents came torealize that their first son was talented enough toadvance to Ichikoh and then Todai (University ofTokyo). Shoshichi must have been the pride andemotional mainstay of our parents, who had losteverything in the war.

In 1948 Shoshichi succeeded in entering Ichikohin his fourth year at middle school (skipping thefifth year), and the following year he enteredthe University of Tokyo under the new educationsystem just introduced. In 1951 he advanced to themathematics department, the Division of Sciences,and studied with the late Professor Kentaro Yano(1912–1993), who encouraged Shoshichi to striveto win the French Government’s Scholarship. I wasin a junior high school at that time, and I rememberwell that Shoshichi attended French classes atAthénée Français and Institut Français du Japonon his way home after a day of studying at Todai.

He was told that the examination for the FrenchGovernment Scholarship was stiff and advised tomake a first trial when he was a senior at Todai.To his and Professor Yano’s surprise, he made itin his first attempt.

After a year of study in France, he moved to theUniversity of Washington in Seattle, where he gothis Ph.D. in less than two years. I was wonderingall these years why he did not go to Princeton orHarvard, whereas he advised me to go to Princetonlater. He explains his situation at that time in

another essay, “My teachers, my friends and mymathematics: the period when I studied in theUnited States” in Mathematics Seminar, July 1982.

In recapitulating Shoshichi’s life from his child-hood until his marriage, I believe that his encounterswith his seniors at Chitose Middle School motivatedShoshichi to think about going to a high school.Mr. Muneo Hayashi at Nozawa Middle School dis-covered Shoshichi’s talent in mathematics and tooktime personally to nurture it. Professor KentaroYano encouraged him to study in France. Dr. Kat-sumi Nomizu egged Shoshichi on to study in the USProfessor Allendoerfer hired him as an assistant.Shoshichi’s friends and all of these wonderfulencounters served as the sources of energy thatdrove Shoshichi to work as a mathematician forover fifty-five years. He led a fruitful life, blessedwith a wonderful spouse and family.

Mei and Yukiko Kobayashi

Shoshichi Kobayashi was born January 4, 1932, dur-ing the Japanese New Year’s holiday celebrationsin the seventh year of the reign of the EmperorShowa.4 Shoshichi was the eldest of five sons bornto his parents, Yoshie and Kyuzo.

Several months after the arrival of Shoshichi,the family moved to Tokyo, where business oppor-tunities were more promising. Within a few years,Kyuzo had saved enough to open his own futonstore.

When the war ended, Shoshichi’s math teacher,Mr. Muneo Hayashi, advised his parents to allowtheir son to apply for Ichiko (number one highschool) in Tokyo. Shoshichi passed the entranceexam in his fourth year at middle school.

At the end of his first year in Ichiko, Shoshichitook the entrance exam for admission to theUniversity of Tokyo. He passed it on his first tryand entered the university after only one year atIchiko.

At the University of Tokyo (Tokyo-daigaku orTodai) Shoshichi found that he was among thetop students in mathematics. At the end of hissophomore year he declared his major to bemathematics and registered for a program toreceive high school teaching certification. At thistime he was unaware that people could make aliving “just doing mathematics.” The situation soonchanged.

Mei Kobayashi is Shoshichi’s daughter. Her email addressis [email protected].

Yukiko Kobayashi is Shoshichi’s wife. Her email address [email protected] name Shochichi is an abbreviation for Showa-shichi-nen (seventh year of reign of Emperor Showa, 1932). Showais the postmortem name of Hirohito.


When Shoshichi started looking for a majoradvisor, his assigned undergraduate advisor toldhim about a charismatic young professor namedKentaro Yano, who would be returning from theInstitute for Advanced Study at Princeton thefollowing academic year. Just hearing about some-one returning from Princeton seemed exotic andexciting to Shoshichi. Sure enough, Professor Yanolived up to and exceeded all expectations. Advisorand advisee made quite an odd but inseparablepair. Yano was a vivacious chain smoker whoenjoyed drinking and clubbing, while Shoshichiwas a quiet bookish type, a nonsmoker who drankmodestly and only during social occasions. Wordscannot express the admiration, love, and respectthat Shoshichi had for his thesis advisor. Their kin-ship lasted until Yano’s death during the year-endholiday season in 1993. Under Yano’s guidanceShoshichi was bitten by the research bug.

Soon it became clear that postgraduation studiesin France would be better than remaining in Tokyo.Shoshichi enrolled in night classes at AthénéeFrançais to learn French to prepare for the examfor a fellowship to study abroad. The effortpaid off handsomely. Shoshichi won a fully fundedfellowship to study at the Universities of Strasbourgand Paris for one year (1953–54). He was elated bythe news.

The fellowship required recipients to pay fortheir boat fare and transportation to Paris. Thiscost Shoshichi’s parents nearly a fifth of what theyhad paid for a house three years before. Costsfor room, board, and tuition in France plus asmall stipend would be covered by the Frenchgovernment. The fare for the return trip at theconclusion of the year was also included in thepackage.

On what was the hottest summer day in Tokyorecorded at that time, Shoshichi boarded a Frenchpassenger ship with other fellowship recipientsand waved to his relatives who had travelled tothe port of Yokohama to wish him well duringhis year abroad. No one imagined that Shoshichi’sdeparture would be permanent. He would returnfor his first brief visit to Japan in 1965 as ahusband, father, and visiting professor from theUniversity of California at Berkeley, courtesy of aSloan Fellowship.

The journey to France was quite an adventure.Shortly after the ship passed through the SuezCanal, Shoshichi began running a dangerously highfever and was admitted to the hospital quartersonboard. He was only semiconscious when theship arrived in France. The first months of his

The family at Yokohama to wish Shoshichi bonvoyage to France, August 20, 1953.

postgraduate experience were spent in a Frenchhospital recovering from typhoid.5.

The French mathematical community was veryfriendly. Shoshichi attended seminars organized byfamous professors.6 He discovered that he mightbe able to make a living “just doing mathematics.”One day in Paris he met Katsumi Nomizu, ayoung Japanese mathematician who had justfinished his Ph.D. under Chern at the University ofChicago. Nomizu suggested that Shoshichi returnto Japan via the United States. He mentionedtwo great mathematicians from whom Shoshichicould learn the art of mathematics: Chern atChicago and Allendoerfer at the University ofWashington. Shoshichi sent a personal statementtogether with reprints of his recent publicationsto these two universities. A few weeks later hereceived an acceptance letter from the Universityof Washington stating that all expenses would becovered upon arrival. From Chicago he received athick envelope of application forms. His decisionwas not difficult. The French authorities agreed tocover his boat fare to the United States in lieu oftravel costs back to his parents’ home in Japan.

Shoshichi was overwhelmed by the generosityand kindness of the members of the universitycommunity in Seattle. On his first day at theuniversity, the department chair, Professor CarlBarnett Allendoerfer, made time to greet Shoshichiand inform him of special arrangements to have hisstipend paid in advance at the beginning of eachmonth for just the first year. The administrationhad heard from more senior foreign students ofthe difficulties in securing a loan after arriving withlittle money. Shoshichi took an immediate liking

5See Shoshichi’s account of his first five weeks in France inhis article “My memory of Professor Henri Cartan” in theNotices of the AMS 57, no. 8, 954–9556The seminars of Henri Cartan, Andre Lichnerowicz, Mar-cel Berger, and others in Paris. He worked with CharlesEhresmann in Strasbourg.


Wedding of Yukiko and Shoshichi.

to the professor. Over the course of a few monthshe found Allendoerfer to be a great statesman andscholar and was very happy when Allendoerferagreed to be his doctoral thesis advisor.

Later that year Shoshichi was best man at thewedding of his friend Akira Ishimaru. The bride’smaid of honor was Yukiko Ashizawa, who was alsofrom Tokyo but from a different section of thecity. She was on leave from her duties as a teacherat Rikkyo Elementary School in Tokyo. Had it notbeen for the war, it is doubtful that Shoshichiand Yukiko’s paths would have crossed in Tokyo.They were formally engaged before he left for theInstitute for Advanced Study in Princeton for hisfirst postdoc in 1956, one year after his arrival inthe United States. Shoshichi returned to Seattle thefollowing spring for Yukiko’s graduation and fortheir wedding on May 11, 1957.

After the wedding the couple drove to theUniversity of Chicago, which had indisputablybecome a leading international center of geometricresearch under the leadership of Shiing-ShenChern. Shoshichi was excited over the prospect ofmeeting Chern in person. Although Shoshichi hadheard many stories of the great mathematicianwhile in Paris, he was even more impressed by

Chern’s kindness. Shoshichi enjoyed the open andproductive collaborative atmosphere as a summerresearch associate sponsored by Chern.

In the fall the Kobayashis returned to Princetonto complete his second and final year at theinstitute. Eleven months following their marriage,the couple welcomed their first child, Sumire. In thefall of 1958 Shoshichi and Yukiko packed up foranother move, this time to MIT, where Shoshichiwould be a postdoctoral research associate. Justas they settled in and learned that they wouldsoon be parents to a second child, Shoshichireceived a tenure-track offer from the Universityof California at Berkeley. Chern would be movingto Berkeley, and he was working to establish aworld-class center for mathematical research. Heasked whether Shoshichi would be interested injoining the faculty. There could be no greaterprivilege.

Since United States visa regulations at the timerequired applicants to leave the country during ad-ministrative processing for a green card, Shoshichiaccepted a very generous offer from the Universityof British Columbia in Vancouver, Canada. In 1962Shoshichi and his family received visa clearance,and he joined the faculty at Berkeley. A yearlater Volume I of The Foundations of DifferentialGeometry was completed and published. In 1966Shoshichi became a full professor at Berkeley.He remained on the faculty at Berkeley after hisretirement, first as professor in graduate studiesand then as professor emeritus.


Remembering Shoshichi KobayashiHyperbolic Manifolds and Holomorphic Mappings and the Kobayashi...

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