Lagrange Space

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FINSLER-LAGRANGE GEOMETRY.

Applications to dynamical systems

Ioan Bucataru, Radu Miron

July 6, 2007


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In memory of three great Finslerists:Shiing-Shen Chern, Mendel Haimovici and Makoto Matsumoto.

This work has been supported by grants CEEX ET 3174/2005-2007 andCEEX M III 12595/2007.


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Contents

Part I Differential Geometry of Tangent Bundles

1 Tangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Tangent and cotangent bundles of a manifold . . . . . . . . . . 3

1.2 Tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Vertical subbundle . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Vertical and complete lifts . . . . . . . . . . . . . . . . . . . . . 13

1.5 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Nonlinear Connections . . . . . . . . . . . . . . . . . . . . . . 19

2.1 Nonlinear connections on a manifold . . . . . . . . . . . . . . . 20

2.2 Local representations of a connection . . . . . . . . . . . . . . . 23

2.3 Nonlinear connections on the tangent bundle . . . . . . . . . . 24

2.4 Characterizations of nonlinear connections . . . . . . . . . . . . 27

2.5 d-tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6 Curvature and torsion of a nonlinear connection . . . . . . . . . 32

2.7 Dynamical covariant derivative . . . . . . . . . . . . . . . . . . 33

2.8 Autoparallel curves . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.9 Symmetries of a nonlinear connection . . . . . . . . . . . . . . 37

2.10 Homogeneous connections and linear connections . . . . . . . . 40

3 N-Linear Connections . . . . . . . . . . . . . . . . . . . . . . . 43

3.1 N-linear connections . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 Berwald connection . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Horizontal and vertical covariant derivatives . . . . . . . . . . . 47

3.4 Torsion of an N-linear connection . . . . . . . . . . . . . . . . . 48

3.5 Curvature of an N-linear connection . . . . . . . . . . . . . . . 50

3.6 N-linear connections induced by a complete parallelism . . . . 52

3.7 Structure equations of an N-linear connection . . . . . . . . . . 54

3.8 Geodesics of an N-linear connection . . . . . . . . . . . . . . . 57

3.9 Homogeneous Berwald connection . . . . . . . . . . . . . . . . 59


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4 Second Order Differential Equations . . . . . . . . . . . . . . 61

4.1 Second order differential vector field . . . . . . . . . . . . . . . 614.2 Nonlinear connections and semisprays . . . . . . . . . . . . . . 63

4.3 Berwald connection of a semispray . . . . . . . . . . . . . . . . 66

4.4 Jacobi equations of a semispray . . . . . . . . . . . . . . . . . . 68

4.5 Symmetries of a semispray . . . . . . . . . . . . . . . . . . . . . 70

4.6 Geometric invariants of an SODE . . . . . . . . . . . . . . . . . 72

4.7 Homogeneous SODE . . . . . . . . . . . . . . . . . . . . . . . . 73

Part II Finsler-Lagrange geometry

5 Finsler Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.1 Finsler metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Geometric objects of a Finsler space . . . . . . . . . . . . . . . 82

5.3 Geodesics of a Finsler space . . . . . . . . . . . . . . . . . . . . 86

5.4 Geodesic spray and symmetries . . . . . . . . . . . . . . . . . . 90

5.5 Cartan nonlinear connection . . . . . . . . . . . . . . . . . . . . 94

5.6 Finsler linear connections . . . . . . . . . . . . . . . . . . . . . 99

5.7 Geodesic deviation and symmetries . . . . . . . . . . . . . . . . 105

5.8 Two dimensional Finsler space . . . . . . . . . . . . . . . . . . 108

5.9 Three dimensional Finsler space . . . . . . . . . . . . . . . . . . 1105.10 Randers spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.11 Ingarden spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.12 Anisotropic inhomogeneous media . . . . . . . . . . . . . . . . 122

6 Lagrange Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.1 Lagrange metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.2 Geometric objects of a Lagrange space . . . . . . . . . . . . . . 133

6.3 Variational problem . . . . . . . . . . . . . . . . . . . . . . . . 135

6.4 Canonical semispray . . . . . . . . . . . . . . . . . . . . . . . . 137

6.5 Symmetries and Noether type theorems . . . . . . . . . . . . . 140

6.6 Canonical nonlinear connection . . . . . . . . . . . . . . . . . . 143

6.7 Almost Kahlerian model of a Lagrange space . . . . . . . . . . 148

6.8 Metric N-linear connections . . . . . . . . . . . . . . . . . . . . 151

6.9 Almost Finslerian Lagrange spaces . . . . . . . . . . . . . . . . 155

6.10 Geometry of -Lagrangians . . . . . . . . . . . . . . . . . . . . 159

6.11 Gravitational and electromagnetic fields . . . . . . . . . . . . . 163

6.12 Einstein equations of Lagrange spaces . . . . . . . . . . . . . . 166


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7 Generalized Lagrange spaces . . . . . . . . . . . . . . . . . . . 173

7.1 Metric classes on T M . . . . . . . . . . . . . . . . . . . . . . . 1747.2 Metric nonlinear connections and semisprays . . . . . . . . . . 1797.3 Metric N-linear connections . . . . . . . . . . . . . . . . . . . . 1827.4 Regular metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 1857.5 Variational problem for regular GL-metrics . . . . . . . . . . . 1897.6 Deformations of Finsler metrics . . . . . . . . . . . . . . . . . . 1907.7 Connections for a deformed Finsler metric . . . . . . . . . . . . 1957.8 New metric classes . . . . . . . . . . . . . . . . . . . . . . . . . 1987.9 Nonholonomic Finsler frames . . . . . . . . . . . . . . . . . . . 200

Part III Dynamical systems

8 Dynamical Systems. Lagrangian Geometries . . . . . . . . . 207

8.1 Riemannian mechanical systems . . . . . . . . . . . . . . . . . . 2098.2 Finslerian mechanical systems . . . . . . . . . . . . . . . . . . . 2118.3 Nonlinear connection of a Finslerian mechanical system . . . . 2148.4 Metric N-linear connection of a Finslerian mechanical system . 2158.5 Electromagnetic tensors of a Finslerian mechanical system . . . 2178.6 Almost Hermitian model of a Finslerian mechanical system . . 2198.7 Lagrangian mechanical systems . . . . . . . . . . . . . . . . . . 2208.8 Almost Hermitian model of a Lagrangian mechanical system . 226

8.9 Generalized Lagrangian mechanical systems . . . . . . . . . . . 227

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246


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Preface

The present book is devoted to Finsler-Lagrange geometry and its applica-tions to dynamical systems. The geometries of Finsler, Lagrange and Hamil-

ton spaces were studied by different schools lead by M. Matsumoto (Japan),S.S. Chern (USA), P.L. Antonelli (Canada), L. Tamassy (Hungary), R. Miron(Romania) and others. Remarkable applications of these geometries are dueto G.S. Asanov (Russia), R.G. Beil and R.M. Santilli (USA), R.S. Ingarden(Poland), K. Kondo and S. Ikeda (Japan).

Among the very few books regarding Finsler geometry, this book presents,from our point of view, the geometry of Finsler spaces as a subgeometry ofLagrange spaces, which can be viewed as a subgeometry of the differentialgeometry of tangent bundle. Hence, the geometry of the tangent bundle isa natural framework for the Finsler-Lagrange geometry we develop in thisbook, while Finsler-Lagrange geometry is presented as a natural framework

for applications.This monograph is a natural and necessary continuation of the authors

work on the theory of Lagrange spaces published by Kluwer, in the FTPHseries, in the volumes [21], [132], [124], [130] or by Hadronic Press in the vol-ume [125]. It contains some new important chapters as: a geometrical theoryof connections, a geometrical theory of systems of differential equations anddynamical systems. Following some R.M. Santillis ideas from his treatise ofAnalytical Mechanics, [166], [167], we define the most general concept of La-grangian (and in particular Finslerian) mechanical system, where the externalforces depend also of velocity coordinates. Thus, the corresponding dynamicalsystem can be introduced only on the phase space as a canonical semispray.

The geometry of this semispray is the geometry of the considered Lagrangianmechanical system.

Differential geometry of the total space of the tangent bundle of a mani-fold has its roots in various problems from Differential Equations, Calculusof Variations, Mechanics, Theoretical Physics and Biology. Nowadays, it is adistinct domain of differential geometry and has important applications in thetheory of physical fields and special problems from mathematical Biology.

This book is devoted to the geometry of Finsler and Lagrange spaces and


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xii Preface

their applications to the geometrical theory of Finslerian and Lagrangian dy-

namical systems, as being an intrinsic part of the geometry of the total spaceT M of a differentiable manifold M.

The differential manifold T M has a special geometric structure: it is ori-entable, admits a globally defined vector field, the Liouville vector field C,possesses an integrable distribution, the vertical distribution V T M and anintegrable tensor field of (1,1)-type, the tangent structure J. We refer to theseas to the natural geometric structures on T M. Due to these aspects of thegeometry of the manifold T M we expect that some other geometric structures,like connections, systems of second order differential equations, metric struc-tures and symplectic structures, cannot be studied on T M without requiringsome compatibility conditions with the natural geometric structures on T M.

However, it is not possible to study these geometric structures on T M by us-ing methods borrowed from the geometry of the base manifold M. This is thereason we have to introduce new concepts that are specific to T M: semispraysand nonlinear connections and to investigate the geometrical properties of thephase space T M, determined by the configuration space M, [166].

Indeed, the geometry of a system of second order differential equationsis the geometry of a semispray. A semispray S is a globally defined vectorfield on T M such that JS = C. If a semispray S is given, then one canassociate to it different geometric objects like nonlinear connections and N-linear connections. Based on these entities we can develop the differentialgeometry of the pair (T M , S ). Such a geometry is imposed by a geometric

study of systems of second order differential equations, SODE, that appear inthe theory of dynamical systems or the theory of mechanical systems.

Different metric structures on T M are induced, in physical examples, eitherby regular Lagrangians, by Finsler metrics, or by generalized Lagrange metricswhich at their turn can be induced by Ehlers-Pirani-Schield axiomatic systemor by different metric structures from Relativistic Optics. Metric geometry ofT M that corresponds to these metric structures determines the backgroundof Lagrangian geometries we discuss in this book. Within these geometries,there are some particular aspects we want to emphasize:

1) One can build the geometry of a Lagrange space from the principles ofAnalytical Mechanics.

2) The geometry of a Finsler space is a particular form of the Lagrangegeometry with specific properties due to the homogeneity condition. Hence,one can build this geometry from the principles of Theoretical Mechanics.

3) One cannot study the geometry of generalized Lagrange spaces usingmethods from Riemannian geometry, one has to approach it as metric geom-etry on T M.

4) Geometric properties from Calculus of Variations can be obtained bymeans of an associated semispray and its differential geometry.


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Preface xiii

5) A geometric theory for Lagrangian dynamical systems can be obtained

if we use differential geometry of the manifold T M. This problem is studiedand published for the first time in this monograph.

6) It is easy to extend this theory to the differential manifold TkM, thespace of accelerations of order k > 1.

These considerations prove that the present book is a new introduction inFinsler-Lagrange geometry, having new openings for applications.

Historically speaking, a systematic study of the differential geometry oftangent bundles started in 1960s and 1970s with the work of P. Dombrowski[76], S. Kobayashi and K. Nomizu [101], K. Yano and S. Ishihara [194]. An im-portant contribution to the geometry of tangent bundle is due to M. Crampin[66] and J. Grifone [80] who associated a nonlinear connection on the tangent

bundle to a system of second order differential equations on a manifold. Also,we refer to work of V. Oproiu [154] for the geometry of the tangent bundle.Since then, theories of vertical and complete lifts and of nonlinear connectionshave been studied using the modern apparatus of differential geometry. Forthe theory of different lifts from a manifold to associated vector bundles werefer to the work of V. Cruceanu [71] and K. Yano and S. Ishihara [194]. Forthe theory of nonlinear connections and compatible linear connection we referto the work of R. Miron and M. Anastasiei [130] and [131]. A rigorous investi-gations of the differential geometry of the total space of a vector bundle can befound in R. Miron and M. Anastasiei [130] and [131]. Also the geometry of thetotal space of a covector bundle appears in the book of R. Miron, D. Hrimiuc,

H. Shimada and S. Sabau [138]. The notion of generalized Lagrange spaceswas introduced and studied by R. Miron in [119]. Systematic studies regard-ing geometric objects and covariant derivatives one can associate to a systemof second order differential equations have been done by M. Crampin et al.[67], O. Krupkova [105], B. Lackey [107], W. Sarlet [168], P.L. Antonelli andI. Bucataru [21]. Geometric theories for systems of higher order differentialequations were proposed by G.B. Byrnes [58], I. Bucataru [53], M. Crampin,W. Sarlet and F. Cantrijn [68], M. de Leon and P.R. Rodrigues [109]. R. Mironin [124] introduced and studied the notion of higher order Lagrange spaces.

In the first part of this book, we introduce the geometry of the first ordertangent bundle of a manifold. Within this context, we investigate using

a modern apparatus of differential geometry geometric objects such aslifts, nonlinear connections, semisprays and N-linear connections. We needthese concepts for the second part, where a metric compatibility of all thesestructures is studied.

Since most of the geometric objects we study in this book live on the totalspace of the tangent bundle of a manifold, the first chapter is dedicated tothe geometry of the tangent bundle. In this chapter, we introduce and pay aspecial attention to natural geometric objects that live on the total space of the


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xiv Preface

tangent bundle, such as vertical distribution, Liouville vector field and tangent

structure. Natural relations between the base manifold and the tangent spaceare studied using vertical and complete lifts and the natural projection of thetangent bundle.

In the geometry of the tangent bundle, and particularly in Finsler andLagrange geometries, an important role is played by the notion of horizontaldistribution and its associated concept of nonlinear connection. Despite thefact that a nonlinear connection lives on the total space of the tangent bundleof a manifold, it can be introduced in the same manner as a linear connection,using the concept of parallel transport. We shall see in the second chapter thata parallel transport always defines a connection, which in general is nonlin-ear. If additional conditions are required for the parallel transport we obtain

homogeneous connection and linear connections.The geometry of horizontal distributions and the associated nonlinear con-

nections are studied in chapter two. Using geometric structures like almostproduct structure, almost complex structure, adjoint structure, connectionmap, horizontal lift, we provide characterization for the existence of a nonlin-ear connection. The relation between the integrability of these structures andthe integrability of the nonlinear connection is studied. Then we prove thateach connection, generated by a parallelism on the base manifold, induces anonlinear connection on the tangent space. We also prove that each nonlinearconnection on the tangent space is the lift of such a connection from the basemanifold.

The disadvantage of nonlinearity for nonlinear connections induced bya parallel transport can be removed by considering its lift to an N-linear con-nection on the tangent bundle. The price we pay for this is that sometimes itis not easy to go back to the base manifold and study its geometry. However,all geometric objects we derive from a nonlinear connection on the tangentbundle are linear. An important object we derive from a nonlinear connectionis a linear connection on the tangent bundle that preserves the vertical andhorizontal distributions. This is the so called Berwald connection studied forthe first time by L. Berwald in [44]. In chapter three we pay attention to theclass of linear connections on tangent bundles that preserve both horizontaland vertical distributions. For such linear connections we determine the struc-

ture equations and study their geodesics. We determine the relation betweenthe geodesics of a nonlinear connection and the geodesics of an associatedN-linear connection.

The geometry of systems of second order differential equations, which westudy in chapter four, is closely related to the geometry of nonlinear con-nections. We shall see that all geometric invariants, which characterize thesolutions of a system of second order differential equations, can be determinedfrom an associated nonlinear connection. This theory is called KCC-theory


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Preface xv

by P.L. Antonelli in [23], based on work of its initiators, D.D. Kosambi [103],

E. Cartan [61] and S.S. Chern [64]. A recent treatment of KCC-theory canbe found in the work of P.L. Antonelli and I. Bucataru [20] and [21]. In thistheory, integral curves of a system of SODE are viewed as geodesics for anassociated linear connection on the tangent bundle. Hence we can use thetheory developed in chapter three for the geometry of a system of SODE.

In the second part of the book we study the geometry of the total spaceof the tangent bundle if a metric d-tensor, which is a metric structure onthe vertical subbundle, is given. If such a metric tensor is derived from thefundamental function of a Finsler space or Lagrange space, then its geometryis the geometry of the corresponding Finsler or Lagrange space, studied inchapters five and six. For the geometry of Finsler and Lagrange spaces as

a subgeometry of the geometry of the tangent bundle, we refer to work ofR. Miron [119], [121] and [130] and J. Kern [96]. Important contributions tothe geometry of Finsler spaces were obtained by M. Abate and G. Patrizio[1], D. Bao, S.S. Chern and Z. Shen [32], A. Bejancu [39], L. Berwald [43],H. Busemann [57], E. Cartan [62], M. Haimovici [81], M. Matsumoto [114],R. Miron and M. Anastasiei [130], H. Rund [162]. The complex Finsler andLagrange geometry by Gh. Munteanu in [149].

For Finsler and Lagrange spaces we give conditions that uniquely de-termine the geometric objects as semispray, nonlinear connection and N-linear connections. All linear connections that are usually associated with aFinsler space, Berwald, Cartan, Chern-Rund, and Hashiguchi connections are

uniquely determined using a system of axioms for each of them. The canoni-cal nonlinear connection of a Finsler or Lagrange space is uniquely determinedby two compatibility conditions: one with the metric structure and one withthe symplectic structure of the space. For both Finsler and Lagrange spacesif the canonical semispray is given, we derive the whole family of nonlinearconnections that are compatible with the metric structure. This is part of thesymplectic geometry we develop for Finsler and Lagrange spaces.

For a Finsler space, its geodesics with the arclength parameterization co-incide with the integral curves of the geodesic spray, with the autoparallelcurves of the canonical nonlinear connection; even more, they coincide withgeodesic curves of Berwald, Cartan, Chern-Rund, or Hashiguchi connection.

Consequently, we can use the theory developed in chapters two, three andfour to investigate these geodesics, their variation and symmetries. A specialattention is paid to the Noether-type theorems of Finsler geometry. If theFinsler space is two or three-dimensional, we study also the stability of thegeodesics as it has been done in [22].

If the metric structure, which we refer to as a generalized Lagrange metric,is not reducible to a Finsler or Lagrange space, in general, there are no canon-ical semisprays and nonlinear connection one can associate to such a space.


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xvi Preface

However, a compatibility condition between the metric structure, semisprays

and induced nonlinear connection is studied. We focus our attention to theparticular cases when a generalized Lagrange metric has a canonical semispray.These are the regular generalized Lagrange metrics studied by R. Miron, [119],S. Watanabe and F. Ikeda, [192], and J. Szilasi [179].

Important applications of the geometry of Finsler and Lagrange spaces aredue to G. Randers [159], P.L. Antonelli, R.S. Ingarden and M. Matsumoto[23], R.G. Beil [37], R.M. Santilli [166], S. Ikeda [175], G.S. Asanov [28], A.K.Aringazin [30], S. Rutz [164].

The theory we developed in the first two parts of the book has good appli-cations for a geometric study of dynamical systems determined by mechanicalLagrangian systems. In the last part of the book we investigate Riemannian,

Finslerian and Lagrangian mechanical systems, whose evolution curves aregiven, on the phase space T M, by Lagrange equations of the form:

d

dt

L

y i

L

xi= Fi(x, y), y

i =dxi

dt= xi,

where Fi(x, x) are the external forces of the system. If Fi are the componentsof a globally defined d-covector field on T M, then one can associate to themechanical system = (M,L,Fi) a globally defined vector field S on T M,which will be called the canonical semispray, or the dynamical system asso-ciated to . The geometry of the Lagrangian mechanical system is the

geometry of the phase space manifold T M endowed with the semispray S. Allgeometric objects one can derive from S, such as a nonlinear connection, anN-linear metric connection, will be used to study the system . The stabilityof is investigated as the stability of the integral curves of S. The dissipativecase is studied and we provide examples of dissipative Lagrangian MechanicalSystems.

If one lift the above mentioned theory to T M, we obtain an almost Her-mitian model (T M ,G,F) of the considered mechanical system . The geo-metric theory of can be deduced from that of the almost Hermitian space(

T M ,G,F). The theory presented in this chapter is based on the papers [55],

[140].

In our opinion, the book is useful to a large class of readers: graduatestudents, mathematicians, physicists and to everybody else interested in thesubject of Finsler-Lagrange geometry and its applications.

We are aware that we could have not reached this form of the book withoutthe enormous work we refer to in the Bibliography. We want to address ourthanks to all authors mentioned there and to everybody else we forgot tomention, without any intention, in our Bibliography.

We would like to acknowledge the direct improvement of this book that has


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Part I

Differential Geometry of

Tangent Bundles


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Chapter 1

Tangent Bundles

We start by studying the geometry of the tangent bundle (T M , , M ) over asmooth, real, n-dimensional manifold M. This geometry is one of the mostimportant field of modern differential geometry. The tangent bundle T Mcarries some natural geometric object fields like: the Liouville vector field

C, the tangent structure J, the vertical distribution V T M. They allow usto introduce the notion of a semispray and a nonlinear connection in thenext chapters. By studying the compatibility of semisprays and nonlinearconnections with a metric structure, we shall develop later a metric geometryof the tangent bundle. Particular cases of this geometry are given by thegeometry of Lagrange and Finsler spaces.

In this book all geometric objects and mappings are considered to be ofC-class, and we shall express this by using the words differentiable orsmooth.

We shall see that a rich geometry of the manifold T M can be developedfrom the notion of a nonlinear connection. It is more natural for a nonlinearconnection to be defined on the tangent space of a manifold rather than on thebase manifold. However, in chapter two we shall define a nonlinear connectionas usually one does for a linear connection, starting from a parallel transport.Then, we shall lift this to what is usually called a nonlinear connection onthe total space of the tangent bundle of a manifold. There, we shall studyrelations between a nonlinear connection and some natural geometric objectsone can define on the tangent bundle of a manifold.

1.1 Tangent and cotangent bundles of a manifold

In this section, the tangent and cotangent bundles over a real, finite dimen-sional manifold are presented.

We consider M a real, n-dimensional manifold, with A = {(U, )I}


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4 Chapter 1. Tangent Bundles

an atlas of C-class on M. For every local chart (U, ) at p

U

M

we denote by (xi)i=1,n the local coordinates induced by , which means that(p) = (xi(p)) Rn. We shall denote this by = (xi) or (U, = (xi)). If wehave two local charts (U, = (xi)) and (V, = (xi)), then their compatibilitymeans that 1 : (xi) (UV) xi(xj ) (UV) is a diffeomorphism.This will imply that rank(xi/xj) = n.

For each point p M, we introduce now the tangent space TpM at p tothe manifold M. Let Cp(M) = { : I R M, is smooth and (0) = p}.Two curves and Cp(M) have a contact of order 1 or the same tangent lineat the point p if there is a local chart (U, = (xi)) at p such that d0( ) =d0( ). The relation contact of order 1 does not depend on the localchart we choose and it is an equivalence on Cp(M). An equivalence class will

be denoted by []p and it will be called a tangent vector at the point p M.The set of all tangent vectors at the point p M will be denoted by TpMand it is called the tangent space to the manifold M at p M. We considerthe union of tangent spaces at all points of M, T M = pMTpM. Considerthe canonical projection : T M M defined by ([]p) = p. Clearly, is asurjection and 1(p) = TpM, p M.

The set T M carries a natural differentiable structure, induced by that ofthe base manifold M, such that the natural projection is a differentiablesubmersion. This differentiable structure will be described bellow. First, wepresent the structure of locally trivial vector bundle one can introduce on T M.

For a local chart (U, = (xi)) on M, we define U : 1(U)

U

Rn

through U([]p) = ((0), d0()). The n-dimensional Euclidean space Rn iscalled the typical fibre. The mapping U is a bijection and satisfies pr1U = ,which means that the following diagram is commutative:

1(U)

%%LL

LLLLLLLLL

U// U Rnpr1

U

The pair (U, U) is called a trivialization chart. As pr1 U = , the mapU preserves the fibres. Consequently, we obtain that the restriction of the

bijection U to the fibre at p, 1

(p) = TpM, U,p : TpM Rn

is also abijection. If we consider the composition

U,p 1V,p = d(p)( 1) : Rn Rn,

we obtain a linear isomorphism of the typical fibre Rn which can be identifiedwith an element of the Lie group GL(n,R). The following functions

gU V : p U V gU V(p) = U,p 1V,p Gl(n,Rn)


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1.1. Tangent and cotangent bundles of a manifold 5

are called structural functions. It is a straightforward calculation to check

that the following properties are true for the structural functions:

1) gU V(p) gV W(p) = gUW(p), p U V W;

2) gU U(p) = IdRn , p U;

3) g1U V(p) = gV U(p), p U V.Here U, V and W are domains of local charts.

The pair (1(U), ), where = ( IdRn) U, is a local chart onT M to which we refer to as an induced local chart. Next, on T M, weshall consider only induced local charts. Therefore, a differentiable atlas

AM = {(U, )I} of the differentiable manifold M determines a differen-tiable atlas AT M = {(1(U), = (IdRn)U)I} on T M. ThereforeT M is a differentiable manifold of dimension 2n and the canonical projection is a differentiable submersion. This implies also that (T M , , M ,Rn, GL(n,R))is a differentiable vector bundle, with typical fibre Rn and structural groupGL(n,R). We call this fibre bundle the tangent bundle of the manifold Mand we refer to it sometimes by (T M , , M ).

Let us fix now a local chart (U, = (xi)) at a point p M. Any curve Cp(M) is represented in the given local chart by xi = xi(t), t I,(p) = xi(0). Then the tangent vector []p is determined by the coefficients

xi = xi(0), yi = dxi

dt

t=0

.

Then the coordinates of []p induced by the local chart (1(U), ) are given

by ([]p) = (xi, yi) R2n, []p 1(U). Such an induced local chart

will be denoted by (1(U), = (xi, yi)). With respect to the induced localcoordinates, the canonical submersion has the expression : (xi, yi) (xi).At each point p M, the fibre of this vector bundle is TpM, which is a linearn-dimensional space isomorph with the typical fibre Rn. This isomorphism isinduced by a local chart and it is explained bellow.

Every local chart (U, ) at p

M induces an isomorphism U,p : TpM

Rn, such that (U,p1V,p)([]p) = d(p)(1)([]p), where (V, ) it is anotherlocal chart at p M. In local coordinates, if = (xi) and = (xi), then (U,p1V,p)(y

i) = yj(xi/xj). We remark here that there is no canonical way ofdefining isomorphisms between tangent spaces TpM and R

n, and consequentlywe cannot define natural isomorphisms between two tangent spaces TpM andTqM.

Let (U, = (xi)) be a local chart at p M and U,p : TpM Rn theinduced isomorphism. If {ei}i=1,n is the natural basis ofRn, we denote by


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/xi

|p =

1U,p(ei). Then

{/xi

|p

}i=1,n is called the natural basis of TpM.

Consider a vector []p TpM, with (t) = (xi(t)). Then, one can expressthe vector as []p = y

i/xi|p, where yi = dxi/dt|t=0. This is equivalent toU,p([]p) = y

iei.

For every p M, one can define the cotangent space Tp M at p to Mas the dual space of the tangent space TpM. Hence, T

p M = {p : TpM

R, p is linear}. We consider the union TM = pMTp M of cotangent spacesto M.

The space TM carries a differentiable structure of C-class and dimen-sion 2n. The set (TM, , M,Rn, GL(n,R)) is a vector bundle, called thecotangent bundle. The canonical submersion : TM M is definedby () = q if and only if

Tq M. Every local chart (U, = (x

i)) at

q M induces a natural basis dxi|q of the cotangent space Tq M such thatdxi|q(/xj|q) = ij.

The local coordinates on the cotangent space TM are denoted by (xi, pi),which means that a covector q Tp M can be expressed as q = pidxi|q. Thecanonical submersion : TM M has the local expression : (xi, pi) (xi).

If (U, = (xi)) and (V, = (xi)) are local charts around p M, the localcoordinates (xi) and (xi) are related by xi = xi(xj ), with rank(xi/xj) = n.The corresponding change of coordinates on T M, induced by (1(U), =(xi, yi)) and (1(V), = (xi, yi)) is given by:

xi = xi(xj), rank

xixj

= n,

yi =xi

xjyj.

(1.1)

We call (1.1) the change of induced local coordinates formula on T M. Sinceyi/yj = xi/xj we have that the Jacobian of 1 is always positive (itis equal to det(xi/xj)2), so T M is an orientable manifold.

The change of local coordinates formula on TM (the corresponding for-mula of (1.1) on TM) is:

xi = xi(xj), rank

xi

xj

= n,

pi =xj

xipj .

(1.2)

A geometric object defined on the tangent bundle or cotangent bundle of amanifold has to be invariant under the change of coordinates (1.1) or (1.2).


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1.2. Tensor fields 7

1.2 Tensor fields

Since at each point p on a manifold M we have defined a linear space TpMone can use these linear spaces to define tensors. Then, if p varies on themanifold, we can define tensor fields. Consider F(M) the set of C realfunctions defined on the manifold M.

If Xp = []p TpM is a vector on M it acts on functions nearby p Maccording to the following formula:

Xp(f) =d

dt(f )|t=0 = dx

i

dt

t=0

f

xi,

where (U, = (xi)) is a local chart at p. A vector field on M is a smooth mapX : M T M such that X = IdM, which means that p M, Xp TpM.The set of all vector fields over the manifold M is denoted by (M) and it isan F(M)-module. We recall here that a vector field X (M) can be viewedalso as a derivation on the ring F(M) of real functions on M. This meansthat if X (M) is a vector field, then one can define X : F(M) F(M),through X(f)(p) = Xp(f). Vector field X has the following properties:

1) X(af + bg) = aX(f) + bX(g), f, g F(M), a, b R;

2) X(f g) = X(f)g + f X(g), f, g F(M).

The above two properties can be used as defining axioms for a vector field ona manifold. If we consider also the Lie bracket of two vector fields [X, Y](f) =X(Y(f)) Y(X(f)), then (M) is an infinite dimensional real algebra. Avector field X (M) can be expressed locally as X = Xi(/xi), where Xiare functions defined on the domain of a local chart (U, = (xi)). Conse-quently, the Lie bracket of two vector fields X, Y (M) has the followinglocal expression:

[X, Y] =

Xi

xi, Yj

xj

=

Xi

Yj

xi Yi X

j

xi

xj.

For a vector field X

(M) and a fixed point p

M, there is an open subset

U in M, an open interval I in R that contains 0 and a map : I U M,such that d(t, q)/dt|t=0 = Xq, q U. For each fixed t I, map t =(t, ) : U M t(U) M is a local diffeomorphism on M. It has theproperties that 0(q) = q, q U and t s = s+t, s, t I, such thats + t I. The family of diffeomorphisms t is called the one-parameter groupof transformations (or the flow) induced by X.

An 1-form on the manifold M can be defined as a smooth map : M TM such that = IdM. The set of 1-forms on M is denoted by 1(M)


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and it is an

F(M)-module. One can define the set of 1-forms also as 1(M) =

{ : (M) F(M), is F(M)-linear}. The set of q-forms is denoted byq(M), where a q-form is a q-F(M)-linear and skew-symmetric map:

: (M) (M) q-times

F(M).

A tensor field of (r,s)-type is an F(M)-linear map

T : 1(M) 1(M)

r-times

(M) (M)

s-times

F(M)

If one fix a local chart (U, = (xi)), then a tensor field T determines nr+sfunctions defined over U,

Ti1i2irj1j2js := T

dxi1 , . . . , dxir ,

xj1, . . . ,

xjs

.

The nr+s functions Ti1i2irjij2js are called the components of the tensor field T

with respect to the local chart (U, = (xi)). Under a change of coordinatesxi xj(xi) on M the components of a tensor field T transform as follows:

Ti1i2irj1j2js =xi1

xk1

xir

xkr

Tk1k2krl1l2lsxl1

xj1

xls

xjs

.(1.3)

If u T M, we denote by TuT M the tangent space at u to T M. Thisis a 2n-dimensional vector space and the natural basis induced by a localchart (1(U), = (xi, yi)) at u is {/xi|u,/yi|u}i=1,n. After a change ofcoordinates (1.1) on T M, the natural basis changes as follows:

xi

u

=xj

xi(u)

xj

u

+yj

xi(u)

yj

u

, rank

xi

xj

= n,

y i

u

=xj

xi(u)

yj

u

.

(1.4)

A vector Xu TuT M has the form Xu = Xi(u)(/xi)|u + Yi(u)(/yi)|uwith respect to the natural basis. Under a change of coordinates (1.1) on T M,the coordinates of a vector Xu TuT M change as follows:

Xi =xi

xjXj,

Yi =xi

xjYj +

yi

xjXj.

(1.5)


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1.3. Vertical subbundle 9

We denote by (T M) and

F(T M) the set of all vector fields on T M and the

set of all real differentiable functions on T M, respectively. Then (T M) isan F(T M)-module finite generated and it is an infinite dimensional, real Liealgebra with respect to the Lie bracket.

We remark here that the tangent manifold T T M carries two natural pro-jections. One is the natural projection of the tangent bundle (T T M , , T M )and the second one is the linear map induced by . In local coordinatesthe two projections are expressed as follows:

: (x,y ,X ,Y) T T M (x, y) T M, and : (x,y ,X ,Y) T T M (x, X) T M.

If a section for the first projection defines a vector field on T M, we shallsee later that a section for both projections defines an important second ordervector field that will be called a semispray.

If T is a (1,1)-type tensor field and X is a vector field then the Frolicker-Nijenhuis bracket of T and X is a (1, 1)-type tensor field [T, X], defined asfollows:

[T, X](Y) = [T(Y), X] T[Y, X].The Frolicker-Nijenhuis bracket of two (1, 1)-type tensors K and L is a vectorvalued 2-form [K, L] and is defined as follows:

[K, L](X, Y) = [K(X), L(Y)] + [L(X), K(Y)] + (K

L)[X, Y]

+(L K)[X, Y] K[X, L(Y)] K[L(X), Y] L[X, K(Y)] L[K(X), Y].

In particular,

1

2[K, K](X, Y) = [K(X), K(Y)] + K2[X, Y] K[X, K(Y)] K[K(X), Y].

The vector valued 2-form NK = (1/2)[K, K] is called the Nijenhuis tensor ofK. It is a (1, 2)-type tensor field, skew symmetric with respect to its vectorarguments. Its vanishing is equivalent to the integrability of the tensor (1, 1)-type tensor K.

1.3 Vertical subbundle

For a natural study of the geometry of the tangent bundle of a manifold thereare two bundles one usually can associate, the vertical subbundle and the pull-back bundle of the tangent bundle through its projection. Even though thereis a natural isomorphism between these two bundles, there are authors thatprefer one or another. The vertical subbundle appears in work of R. Miron and


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M. Anastasiei, [130] and A. Bejancu [39]. The pull-back bundle is preferred

by D. Bao, S.S. Chern and Z. Shen, [32] and M. Crampin, E. Martinez andW. Sarlet, [67]. In our work we shall use the vertical subbundle of the tangentbundle.

The natural submersion : T M M determines a simple foliation ofdimension n (and codimension n) on the manifold T M. The leafs of this foli-ation are the tangent spaces TpM =

1(p), they are embedded submanifoldsof dimension n in T M. I f (xi, yi) are local coordinates on T M, then yi arecoordinates for the leafs of the foliation, while xi are transverse coordinatesof the foliation. One can see this from the change of coordinates on T M,given by formula (1.1). The relation between the geometry of this foliationand the geometry of Finsler spaces was recently studied by A. Bejancu and

H.R. Farran in [42].This foliation induces a regular, n-dimensional and integrable distribution

V : u T M VuT M TuT M, where VuT M are tangent spaces to the leafsof the foliation. We call this the vertical distribution of the tangent bundle.As : T M M is a submersion it follows that ,u : TuT M T(u)M isan epimorphism of linear spaces, for u T M, where ,u is the linear mapinduced by at u T M. The kernel of this linear map, ,u, is exactly thevertical subspace, which means that VuT M = Ker(,u), u T M. If wedenote by V T M = uT MVuT M, then V T M is a subbundle of the tangentbundle (T T M , , T M ), which we call the vertical subbundle. This subbundlehas the fibre, at each point u

T M, an n-dimensional vector space VuT M,

generated by {/yi|u}. Consequently, most of the geometric objects that liveon this bundle transform, under a change of induced coordinates, in a similarway with corresponding geometric objects from the base manifold M. Suchgeometric objects will be called distinguished or d-geometric objects, forshort.

The set of all vertical vector fields on T M is denoted by v(T M). A verticalvector field has the form X = Xi(x, y)/yi. As for any two vertical vectorfields X, Y, their Lie bracket is a vertical vector field, [X, Y] v(T M), wehave that v(T M) is a real Lie subalgebra of (T M). An important verticalvector field is C = yi(/yi). One can check using (1.1) and (1.4) that thevector field C is globally defined on T M. We call this vector field, the Liouville

vector field.We consider now the pull-back bundle ((T M) = T M T M , (), T M)

of the tangent bundle (T M , , M ) through its projection . For each u T M,its fibre at u is (T M)u = T(u)M, which is an n-dimensional linear space,generated by {/xi|(u)}. This fibre T(u)M is isomorph with VuT M, thefibre of the vertical subbundle at point u T M.

Consider now Tu T M the cotangent space of T M at u T M and denoteby {dxi|u, dyi|u} the natural cobasis. In other words, {dxi|u, dyi|u} is the dual


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1.3. Vertical subbundle 11

basis of

{/xi

|u,/y

i

|u

}. After a change of local coordinates (1.1) on T M,

the dual basis changes as follows:dxi =

xi

xjdxj , rank

xi

xj

= n,

dyi =xi

xjdyj +

yi

xjdxj.

(1.6)

For each point u T M, we have from the above formula (1.6) that dxi|u spanan n-dimensional subspace Vu T M ofT

u T M. This way we determine a regular,

n-dimensional, integrable distribution V : u T M Vu T M Tu T M.

Definition 1.3.1 A vector field Y (T M) is a symmetry of the verticaldistribution V T M if [X, Y] v(T M), X v(T M).

Proposition 1.3.1 A vector field Y = Yi(x, y)(/xi) + Yi(x, y)(/yi) is asymmetry of the vertical distribution if and only if the coefficient functions Yi

depend on position only, which means that Yi = Yi(x).

One can reformulate the above proposition by saying that a vector fieldY (T M) is a symmetry of the vertical distribution if and only if the vectorfield Y is projectable, which means that Y = Y

i(x)(/xi) is a vector fieldon the base manifold M.

The total space of the vertical subbundle V T M is an embedded sub-manifold in T T M. The canonical inclusion : V T M T T M preservesthe linear structure of the fibres. We look now for a canonical submersionJ : T T M V T M that preserves also the linear structures of the fibres andmakes the following sequence exact:

0 V T M T T M J V T M 0.(1.7)

The mapping J is therefore a morphism of vector bundles from T T M to V T Msuch that Ker J = Im = V T M. This mapping is called the tangent structureof the tangent bundle (or the vertical endomorphism ) and it is defined asfollows:

J =

y i dxi, or

J

xi

=

y iand

J

y i

= 0.

(1.8)

Using the transformation rules (1.4) and (1.6) one can check that the (1, 1)-type tensor field J is globally defined on T M. For the tangent structure J wehave the following properties:


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1) J2 = 0;

2) Ker J = Im J = V T M .

The Nijenhuis tensor field of the tangent structure J is given by the followingformula:

NJ(X, Y) = [JX ,JY] J[X ,JY] J[JX ,Y], X, Y (T M).(1.9)

A direct calculation shows that NJ = 0, which means that the tangent struc-ture J is integrable. From (1.8) one can also see that the tangent structureJ has constant coefficients with respect to the natural basis and cobasis and

therefore it is integrable.The tangent structure J acts also linearly on vector fields that live on the

tangent bundle T M. We can also consider J, the cotangent structure thatacts on 1-forms that live on the tangent space T M. The cotangent structureJ is defined by

J = dxi y i

, or

J(dyi) = dxi and

J(dxi) = 0.(1.10)

It is globally defined on T M and it has similar properties with the tangentstructure. Similarly as for the tangent structure, we have that NJ = 0,and the cotangent structure J is integrable. From (1.10) one can see thatcotangent structure J has constant coefficients with respect to the naturalbasis and cobasis and therefore it is integrable. The two structures J and J

are related through the following formula:

J(df)(X) = J(X)(f),(1.11)

where f is an arbitrary function on T M. More generally, one can prove thatfor a vector field X (T M), the Frolicker-Nijenhuis brackets of X and thetangent and cotangent structures are related by

[X, J] = [X, J].

One can extend the action of the cotangent structure J to arbitrary k-forms.If is a k-form on T M and X1,...,Xk are vector fields on T M we define J

through

(J())(X1,...,Xk) = (J(X1),...,J(Xk)).(1.12)

With this extension, we have now that J is an F(T M)-linear map on k(M).


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1.4. Vertical and complete lifts 13

1.4 Vertical and complete lifts

In the geometry of the tangent bundle it is important to extend some geometricobjects that live on the base manifold M to the tangent space T M. This taskcan be done using the lifting process. There are two natural important typesof lifts from M to T M, the vertical and the complete lifts, [71] [194].

For a function f F(M), we define fv = f , the vertical lift of f. Afunction f F(T M) is said to be a basic function if it is the vertical lift of afunction f F(M), which means that f = fv. A basic function f F(T M)is constant along the leafs of the vertical foliation.

For every u T M, we define the linear map lv,u : T(u)M TuT M as

lv,u Xi((u)) xi

(u)

= Xi((u)) y i

u

.

We can see that lv,u : T(u)M VuT M is a linear isomorphism. It is calledthe vertical lift of vectors of the tangent bundle. We may also think to thevertical lift lv as an F(M)-linear map between (M) and (T M). In thiscase lv is defined as follows: for every vector field X = X

i(/xi) (M),(lvX)(u) = lv,u(X(u)). The vertical lift of a vector field X (M) will bedenoted also by Xv (T M). It can be written as Xv = (Xi)v(/yi) v(T M) for X = Xi(/xi) (M).

Consider X a vector field on the base manifold M and t its one-parameter

group. Then, the vertical lift Xv has as one-parameter group, the vertical liftvt = t .

For a function f F(M) we define fc = yif/xi its complete lift. Theset of all complete lifts of functions from the base manifold M is a subring of

F(T M). A vector field on T M is perfectly determined if one knows its actionon complete lifts of functions. This means that if X, Y (T M) such thatX(fc) = Y(fc), f (M), then X = Y.

The complete lift Xc of a vector field X = Xi/xi (M) is defined asfollows:

Xc = (Xi)v

xi+ (Xi)c

y i.(1.13)

Consider X a vector field on the base manifold M and t its one-parametergroup. Then, the complete lift Xc has as one-parameter group the completelift ct = y

i(t/xi).

The vertical lift v of an 1-form = idxi 1(M) is defined by v =

(i)vdxi. The complete lift c of an 1-form is defined as c = (i)

cdxi +(i)

vdyi.

Next results show that vertical and complete lifts of functions and vectorscan be characterized using tangent and cotangent structures.


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Proposition 1.4.1

1) A function f F(T M) is the vertical lift of some function from thebase manifold M if and only if J(df) = 0.

2) A function f F(T M) is the sum of a complete lift and a vertical liftof some functions from the base manifold M if and only if dJ(df) = 0.

Proof.1) As J(df) = (f/yi)dxi we have that J(df) = 0 if and only if

f/yi = 0, which means that f is constant along the leafs of the verticaldistribution. Consequently f is the vertical lift of some function from the basemanifold M.

2) We have the following expression for dJ(df):

dJ(df) = 2

fy iyj

dyj dxi + 12 2f

y ixj

2

fxiyj

dxj dxi.Using this expression we have that dJ(df) = 0 if and only if

2f

y iyj= 0 and

2f

y ixj=

2f

xiyj.(1.14)

First equation from (1.14) implies that f has the form f(x, y) = Ai(x)yi+(x).

If we ask for this function to satisfy second equations (1.14), then we obtainAi/x

j = Aj/xi. Last equation is equivalent to the fact that Ai is the

gradient of some function F(M), which means that Ai = /xi. Weconclude now that dJ(df) = 0 is equivalent to the following form of f,

f = c + v, where , F(M). q.e.d.Proposition 1.4.2

1) A vector field X (T M) is the vertical lift of some vector field fromthe base manifold M if and only if J(X) = 0 and LXJ = 0.

2) A vector field X (T M) is the sum of a complete and a vertical liftof some vector fields from the base manifold M if and only if LXJ = 0.Proof. With respect to the natural basis we have:

(LXJ)

xi

=

X,

y i

J

X,

xi

= Xj

y i

xj+ Xj

xi Yj

y i

yj,

which is zero if and only if

Xj

y i= 0, which is equivalent to Xj = Xj(x) and

Xj

xi=

Yj

y i, which is equivalent to Yj =

Xj

xiyi + Vi(x).


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1.5. Homogeneity 15

Consequently, we have that

(LXJ)

xi

= 0 if and only if X = Xi(x)

xi+

Xi

xj(x)yj

y i+ Vi(x)

y i.

We remark here that condition (LXJ)(/yi) = 0 is satisfied if X is the sumof a complete and a vertical lift. q.e.d.

For the vertical and the complete lifts we have the following properties:1) (f g)c = fv gc + fc gv, f, g F(M);2) (f X)v = fvXv, (f X)c = fvXc + fcXv, X (M), f F(M);3) J(Xc) = Xv, [Xv, Yv] = 0, [Xv, Yc] = [X, Y]v, [Xc, Yc] = [X, Y]c;

4) (f )v = fvv, (f )c = fvc + fcv,

1(M),

f

F(M);

5) J(c) = v, (df)v = d(fv), (df)c = d(fc), 1(M), f F(M).A vector field X (T M) is called basic if there is a vector field X (M)

such that (X) = X. Sometimes we say that vector fields X and X are -related. In local coordinates we have that a vector field X = Xi(/xi) +Yi(/yi) is basic if the coefficients Xi are basic functions. As an example wehave that the complete lift Xc of a vector field X (M) is basic.

According to Proposition 1.3.1, we have that a vector field X (T M) isa symmetry of the vertical distribution if and only if X is a basic vector field.We have also that the complete lift Xc of any vector field X from the basemanifold is a symmetry of the vertical distribution.

1.5 Homogeneity

When studying the geometry of a manifold and its tangent bundle, there aregeometric objects defined along curves. First question one has to answer is ifthese objects depend on the parameterization of such curves. The indepen-dence of parameterization is equivalent to certain homogeneity of the discussedgeometric objects.

For a curve c : t I R c(t) M, if we change the parameterizationt t(s), then the tangent vector dc/dt|t=0 changes according to dc/dt =dc/ds ds/dt. An affine transformation of parameter t(s) = as + b, a R fora curve c implies the following change of the coordinates of the tangent vectordc/dt:

xi = xi, yi = ayi.(1.15)

Therefore, the change of coordinates (1.1) on the total space T M preservesthe above transformation for the coordinates of a vector.

Since it is important to decide if geometric objects that can be definedalong curves on M depend or not on their parameterization, we shall studytheir behavior under transformation (1.15).


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Denote by T M = T M \ {0} the tangent space with zero section removed.If (0, +), we define h : T M T M by h(x, y) = (x,y) and wecall h the dilatation of ratio . The set of all dilatations {h, (0, +)}constitutes a one-parameter group. The vector field that has this group as aone-parameter group is the Liouville vector field and in local coordinates ithas the expression C = yi(/yi).

Definition 1.5.1 A function f : T M R that is differentiable on T M andcontinuous only on the null section 0 : M T M is called homogeneous oforder r (r Z) on the fibres of T M (or r-homogeneous with respect to yi) if:

f ha = arf, a R+.

The following Euler theorem holds true:

Theorem 1.5.1 A function f F(T M) differentiable on T M and continu-ous only on the null sections is homogeneous of order r if and only if

LCf = yi fy i

= rf.(1.16)

The following properties hold true:

1) If f1, f2 are r-homogeneous functions, then the function 1f1 + 2f2,1, 2 R is r -homogeneous, too.

2) Iff1 is r-homogeneous and f2 is s-homogeneous, then the function f1 f2is (r + s)-homogeneous.

Definition 1.5.2 A vector field X (T M) is r-homogeneous if

X ha = ar1h,a X, a R+.

An equivalent Euler-type theorem holds true for vector fields:

Theorem 1.5.2 A vector field X (T M) is r-homogeneous if and only if

LCX = [C, X] = (r

1)X.(1.17)

The following properties hold true:

1) The vector fields /xi, /yi are 1 and 0-homogeneous, respectively.

2) If f F(T M) is s-homogeneous and X (T M) is r-homogeneousthen f X is (s + r)-homogeneous.

3) A vector field on T M, X = Xi(/xi) + Yi(/yi) is r-homogeneousif and only if Xi are homogeneous functions of order (r 1) and Yi arehomogeneous functions of order r.


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1.5. Homogeneity 17

4) If X

(T M) is an r-homogeneous vector field and f

F(T M) is an

s-homogeneous function, then Xf F(T M) is an (r + s 1)-homogeneousfunction.

5) The Liouville vector field C is 1-homogeneous.6) Iff F(T M) is an arbitrary s-homogeneous function, then f/yi are

(s 1)-homogeneous functions.7) If X (T M) is an r-homogeneous vector field and Y (T M) is an

s-homogeneous vector field then [X, Y] is an (r + s 1)-homogeneous vectorfield. Consequently, we have that the set of 1-homogeneous vector fields is aLie subalgebra of (T M).

In the case ofq-forms the definition of homogeneity can be stated as follows:

Definition 1.5.3 A q-form q

(T M) is s-homogeneous if

ha = as, a R+.(1.18)We have also an Euler type theorem for q-forms:

Theorem 1.5.3 A q-form q(T M) is s-homogeneous if and only ifLC = s.(1.19)

The following properties hold true:1) If q(T M) is s-homogeneous and q(T M) is s-homogeneous,

then

is (s + s)-homogeneous.

2) If q(T M) is s-homogeneous and X1,...,Xq are r-homogeneousvector fields then (X1,...,Xq) is an (s + r 1)-homogeneous function.

3) dxi (i = 1,...,n) are 0-homogeneous 1-forms, dyi (i = 1,...,n) are 1-homogeneous 1-forms.

4) An 1-form on T M, = idxi + idy

i is r-homogeneous if and only ifi are homogeneous functions of order r and i are homogeneous functions oforder (r 1).

More generally, a vector field T of (1, s)-type is homogeneous of order r ifand only ifLCT = (r1)T. As an example we have that the tangent structureJ is a (1, 1)-type tensor field homogeneous of order 0. In order to prove this wehave to show that

LCJ =

J, which is equivalent to [C, JX]

J[C, X] =

JX,

X (T M). This can be done by taking X {/xi,/yi}.Now, we give some examples of homogeneous functions, vectors and 1-

forms.1) The vertical lift fv and fc of a function f (M) are homogeneous

functions, the first one is homogeneous of order zero, while the second one ishomogeneous of order one.

2) For a vector field X (M), its vertical lift Xv is homogeneous of orderzero and the complete lift Xc is homogeneous of order one.


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3) For an 1-form

1(M), its vertical lift v is homogeneous of order

zero and the complete lift c is homogeneous of order one.Let us explain now why do we have to consider that a homogeneous object

is differentiable only on T M, the tangent bundle with zero section removed,and not on the entire tangent bundle T M.

If a function f F(T M) is differentiable on T M and it is homogeneousof order r, then there exist the functions (i1i2is(x)), such that f(x, y) =i1i2is(x)(y

i1)1 (yis)s , 1 + 2 + + s = r, so f is a polynomial oforder r with respect to y. The homogeneous functions, vectors and 1-forms wepresented in the example above are differentiable on whole T M, consequentlywe have for example that fc is homogeneous of order one and then fc = iy

i,where i = f/x

i. If we want to avoid this particular cases, we have to

assume that the function f is of C-class on T M and only continuous on thenull section. A similar remark works also for tensor fields. Next, if we arereferring to a homogeneous object this will be supposed to be of C-class onT M and only continuous on the null section.


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Chapter 2

Nonlinear Connections

In the geometry of the tangent bundle, and particularly in Finsler and La-grange geometry, an important role is played by the notion of horizontal dis-tribution and its associated concept of nonlinear connection. Despite the factthat a nonlinear connection lives on the total space of the tangent bundle ofa manifold, it can be introduced in the same manner as a linear connection,using the concept of parallel transport. We shall see in this chapter that aparallel transport defines always a connection, which in general is nonlinear.If additional conditions are required for the parallel transport, we obtain ho-mogeneous connections and linear connections.

By studying geodesics of such connections, we obtain that the geometry ofconnections is the same with the geometry of systems of second order differ-ential equations.

If we lift such connections to the tangent bundle of a manifold, we obtainwhat is usually called a nonlinear connection on the tangent bundle. As wehave seen in the first chapter, the vertical distribution V T M is a regular, n-dimensional, integrable distribution on the tangent space. Then it is naturallyto look for complementary distributions of the vertical one in T M. Such dis-tributions that will be called horizontal distributions are induced by nonlinearconnections. In this chapter we introduce the notion of a nonlinear connec-tion on the manifold T M and some geometric structures whose existence is

equivalent to the existence of a nonlinear connection. We also study the in-tegrability of a nonlinear connection. Then, we determine the necessary andsufficient conditions for the integrability of a nonlinear connection in terms ofthe integrability of some induced geometric structures.

The existence of a horizontal distribution together with the vertical onedetermine a decomposition of the tangent bundle T T M into a Whitney sumT T M = HT M V T M. If we express geometric objects using an adaptedbasis to this decomposition, their components are easier to handle. These


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20 Chapter 2. Nonlinear Connections

components behave in a similar manner with corresponding objects from the

base manifold.In this chapter we shall see that a nonlinear connection can be defined

directly on the total space of the tangent bundle of a manifold and it definesa parallel transport on the base manifold. A nonlinear connection is an im-portant tool in the geometry of systems of second order differential equations.Also, the presence of a nonlinear connection on T M will allow us to extendsome results and geometric objects from the vertical subbundle V T M to thewhole tangent bundle T T M.

Equivalent definitions for a nonlinear connection on T M are given anda study of the main geometric objects induced by it is presented. As weintend to apply this theory to dynamical systems, we pay a special attention

to the autoparallel curves of a nonlinear connection and their symmetries. Theparticular cases when the connection is either homogeneous or linear are alsostudied.

2.1 Nonlinear connections on a manifold

When one defines a linear connection using the covariant derivative inducedby a parallel transport, one usually makes the assumption that the map thatassigns to each direction the total derivative in that direction is linear, [65],[111], [113]. If we remove such an assumption the connection we obtain is calleda nonlinear connection. In this chapter we study such nonlinear connectioninduced by a parallel transport. For the introduction of parallel transport andabsolute derivative we follow the book of M. Crampin and F.A.E. Pirani, [65],but we do not make the assumption that the linear transport is linear withrespect to the direction.

Consider M a real, n-dimensional manifold ofC-class. As we have seen inthe previous chapter, there is no canonical isomorphism between two tangentspaces TpM and TqM at two points p and q to M. Consequently, there is nocanonical way of deciding if vectors that live on different tangent spaces areparallel or not.

In order to define a notion of parallelism on a manifold one should beable to identify the tangent spaces at any two points if a curve joining thetwo points is given. This identification has to preserve the linear structureof tangent spaces. Consequently, a parallel transport on a manifold can bedefined as a collection of linear isomorphisms cq,p : TpM TqM, where

p, q M and c is a smooth curve on M joining p and q such that

c1r,q c2q,p = c1c2r,p .

From this we have that cp,p = IdTpM for any closed curve c at p M and


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2.1. Nonlinear connections on a manifold 21

(cp,q)1 = c

1

q,p , where c1 is the reverse curve from q to p.

A vector field X along a curve c is called a parallel vector field along thecurve if Xc(t) =

cc(t),p0

Xp0 , t, where Xp0 is a vector at some point p0 = c(t0)on the curve. We also say that the vector field X is parallel-transported alongthe curve. The definition does not depend on the particular Xp0 we choose.This means that if Xp1 is any other vector on the curve c at p1 = c(t1) andYc(t) =

cc(t),p1

Xp1 is the vector field constructed by parallel transporting Xp1along the curve, then X and Y coincide.

The parallel transport is a collection of maps that preserve the linear struc-ture of tangent spaces. Consider now X and Y two parallel vector fields alonga curve c obtained by parallel transport of Xp and Yp. Then, for any two realnumbers a and b, the vector field aX + bY is a parallel vector field obtained

by parallel transport of aXp + bYp.We shall use parallel transport to define an absolute derivative such that

if a vector is parallel along a curve, then its absolute derivative vanishes.Let X(t) = Xc(t) be a vector field along a curve c. For > 0, we considerX(t+)|| =

cc(t),c(t+)X(t+), the vector at c(t) obtained by parallel transport

of X(t + ). Then, the absolute derivative, along the curve c with parametert, of X at c(t) is defined as

DX

dt= lim

0

1

(X(t + )|| X(t)) =

d

d{cc(t),c(t+)X(t + )}|=0.(2.1)

From this definition, we have that a vector field X is parallel along a curve cwith parameter t if and only if DX/dt = 0. We remark here that at this level,the concepts of parallelism and absolute derivative depend on the parameter-ization of the curve. The following is an assumption we shall use sometimes:

(A1) The parallel transport is independent on the parameterization of thecurve. Under this assumption we have that the absolute derivative satisfies

DX

dt=

DX

ds

ds

dt,(2.2)

where s and t are two parameters along curve c. In his case we say that theabsolute derivative is homogeneous.

Using the definition (2.1) of the absolute derivative, we obtain the followingimmediate properties:

D

dt(X1 + X2) =

DX1dt

+DX2

dt,

D

dt(f X) = f

DX

dt+

df

dtX,

(2.3)

where X1, X2 and X are vector fields and f is a function along a curve c.


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The absolute derivative at a point of a vector field on a curve should depend

on the direction of the curve at that point and not on the particular curve withthat direction. From now we assume that the absolute derivative along a curvedepends at a point only on the tangent vector to the curve and not on theparticular curve that defines the vector. According to this assumption, we canassociate to each vector Y TpM and each vector field near p an elementYX TpM such that

YX = DXdt

t=0

,(2.4)

where the absolute derivative is taken along a curve c such that c(0) = pand (dc/dt)

|t=0 = Y. We call

YX the covariant derivative of X in the

direction Y. Using the covariant derivative, we can construct for two vectorfields X, Y near a point p another local vector field YX whose value at p is(YX)p = YpX. We require also that the rule of association (Y, X) YXis smooth.

The map : (X, Y) XY is called a connection and it has, accordingto (2.3), the following properties:

C1) X(Y + Z) = XY + XZ,C2) X(f Y) = fXY + X(f)Y, where X, Y and Z are local vector fields

and f is a local function.

If we do not require any other properties of linearity or homogeneity for with respect to the first argument, then we refer to

as a nonlinear connec-

tion.

If we require for the absolute derivative to be homogeneous, which meansthat assumption (A1) is satisfied, then the covariant derivative (2.4) is calledhomogeneous and the induced connection is called a homogeneous connec-tion. In such a case, connection satisfies also the property:

C3) fXY = fXY.In order to obtain what is usually called a linear connection we have to

make an additional assumption for the covariant derivative:

(A2) For a fixed vector field X, the map Y TpM YX TpM is alinear map. If assumption (A2) is satisfied, then is called a linear connection.A linear connection satisfies C1), C2), C3) and

C4) X+YZ = XZ+ YZ.A linear connection is also a homogeneous connection.

One can extend the connection to act on 1-forms through

(X)(Y) + (XY) = X((Y)).(2.5)

Then, for this, properties that are similar to C1) and C2) hold true.


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If we consider two coordinate charts, then according to expression (2.8), the

corresponding coefficients of a linear connection are related by

ijk xjxp xkxl = xixm mpl + 2xixpxl .(2.10)For a covector field = idx

i, we have according to expression (2.5) that

X = X(i)dxi + iXdxi. If we use again expression (2.5), we have

(Xdxi)

xj

= dxi

X

xj

= Nij (x, X).(2.11)

From the above expression (2.11) we have that

X =

{X(j)

Nij (x, X)i

}dxj = j|Xdx

j.

A curve c : t I R c(t) = (xi(t)) M is said to be a geodesic for aconnection if the tangent field dc/dt = (dxi/dt) is parallel along the curvec. This implies the following equations:

D

dt

dxi

dt

=

d2xi

dt2+ Nij

x,

dx

dt

dxj

dt= 0.(2.12)

Next, we shall see that by lifting the connection to the tangent bundle,geodesics (2.12) are autoparallel curves given by equations (2.39) for the non-linear connection.

2.3 Nonlinear connections on the tangent bundle

As we have seen in the second section of the first chapter, the vertical distribu-tion V T M is a regular, n-dimensional, integrable distribution on the tangentbundle of a manifold M. It is naturally then to look for a supplementary dis-tribution of the vertical one in T T M. Such a distribution that will be calleda horizontal distribution is induced by a nonlinear connection, [130]. In thissection we introduce the notion of a nonlinear connection on the manifold T Mand some geometric structures whose existence is equivalent to the existenceof a nonlinear connection. We shall study also the integrability of a nonlinearconnection. Then we determine necessary and sufficient conditions for the

integrability of a nonlinear connection in terms of the integrability of someinduced geometric structures.

The vertical distribution V T M and the tangent structure J determine thefollowing exact sequence:

0 V T M T T M J V T M 0.(2.13)Using this exact sequence we can define supplementary distributions for thevertical distribution.


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2.3. Nonlinear connections on the tangent bundle 25

Definition 2.3.1 A nonlinear connection on the tangent bundle T M of a

manifold M is a left splitting of the exact sequence (2.13).

Therefore, a nonlinear connection on T M is a vector bundle morphismv : T T M V T M, with the property that v = IdV T M.

The kernel of the morphism v is a vector subbundle of the tangent bundle(T T M , , T M), denoted by HT M and called the horizontal subbundle. Itsfibres HuT M determine a regular n-dimensional distribution u T M HuT M TuT M, which is supplementary to the vertical distribution u T M VuT M TuT M. Therefore, a nonlinear connection on T M inducesthe following decomposition for the tangent space TuT M, u T M:

TuT M = HuT M

VuT M.(2.14)

The reciprocal of the above stated property holds true. So, we can formulate:

Theorem 2.3.1 A nonlinear connection on T M is characterized by the exis-tence of a subbundle HT M of the tangent bundle T T M such that the decom-position (2.14) holds true.

Next, we refer to a nonlinear connection by N, while we use the notationHT M for the associated horizontal subbundle or for the horizontal distribu-tion.

The restriction of the morphism J : T T M V T M to HT M is an iso-morphism of vector bundles. Using the inverse of this isomorphism J

|HT M

we can define a morphism of vector bundles : V T M T T M , such thatJ = Id|V T M. In other words, is a right splitting of the exact sequence(2.13). One can easily see that the bundle Im coincides with the horizontalsubbundle HT M. The tangent bundle T T M will decompose then as a Whit-ney sum of the horizontal and the vertical subbundle. We can define now themorphism v : T T M V T M on fibres as being the identity on vertical vectorsand zero on the horizontal vectors. It follows that v is a left splitting of theexact sequence (2.13). Moreover, the mappings v and satisfy the relation

v + J = IdT T M.Theorem 2.3.2

A nonlinear connection on the tangent bundle (T M , , M )is characterized by a right splitting of the exact sequence (2.13), : V T M T T M, such that J = Id|V T M.

If a nonlinear connection is given on the tangent bundle T M, then thesequence (2.13) can be represented as follows:

0 V T M

v

T T MJ

V T M 0.(2.15)


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For a nonlinear connection N we denote by h and v the horizontal and the

vertical projectors that correspond to decomposition (2.14), respectively.A vector field X (T M) is called horizontal if h(X) = X and vertical ifv(X) = X. We denote by h(T M) the F(T M)-module of horizontal vectorfields.

Since ,u : TuT M T(u)M is an epimorphism, from (2.14) we can seethat the restriction of,u to HuT M from HuT M to T(u)M is an isomorphismof linear spaces. We denote by lh,u : T(u)M HuT M the inverse map ofthe above mentioned isomorphism. We call lh,u the horizontal lift induced bythe given nonlinear connection. The horizontal lift lh can be viewed also asan F(M)-linear map between (M) and (T M) and it is defined as follows:if X = Xi(/xi)

(M) we define

lh(X)(u) = lh,u(X(u)) = Xi((u))lh,u

xi

(u)

.

The horizontal lift of a vector field X (M) is also denoted by Xh (T M).The horizontal lift lh induced by a nonlinear connection N and the vertical

lift lv are related by:J lh = lv,(2.16)

that is the following diagram is commutative:

TuT M

,u %%KK

KK

KK

KK

K

Ju // TuT M

T(u)M

lv,u

OO

One can prove that if lh : (M) (T M) is an F(M)-linear map suchthat (2.16) holds, then H : u T M HuT M = lh,u(T(u)M) is a horizontaldistribution on T M.

Denote by /xi|u = lh,u(/xi|(u)). We have that {/xi|u}i=1,n is abasis ofHuT M, u T M and under a change of coordinates (1.1) on T M wehave that

xi

=xj

xi

xj

.(2.17)

As ,u(/xi|u) = /xi|(u), u T M, then with respect to the natural

basis {/xi|u,/yi|u} of TuT M, /xi|u has the following expression:

xi

u

=

xi

u

Nji (u)

yj

u

.(2.18)

The set of functions (Nij ) are defined on domains of induced local charts andthey are called the local coefficients of the nonlinear connection.


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2.4. Characterizations of nonlinear connections 27

Proposition 2.3.1 To give a nonlinear connection N on the tangent bundle

T M it is equivalent to give a set of functions Nij (x, y) on every domain ofinduced local chart such that on intersections of such domains, they are relatedby

xj

xkNki =

Njk xkxi + yjxi .(2.19)Proof. The if part is a consequence of (2.17) and the action of the pseudo-group of coordinate transformations (1.1).

For the only if part we suppose that on every domain of induced localchart we have a set of functions Nij such that on the intersection of any two

domains the corresponding functions

Nij and N

jk are related by (2.19). Then

we may define /xi

|u as in (2.18). It is a straight forward calculation to checkthat (2.17) is true and then {/xi|u} span a n-dimensional subspace HuT Mof TuT M. As {/xi|u,/yi|u} are linearly independent, then HuT M andVuT M satisfy (2.14). q.e.d.

Formula (2.19) is the lift to the tangent bundle of formula (2.8). Thereforea nonlinear connection presented in this section as a horizontal subbundle isthe lift of a connection induced by a parallel transport.

Examples Let ijk (x) be the local coefficients of a symmetric linear connection on the base manifold M. Under a change of local coordinates on M we havethat

i

jk =

xi

xl l

pq

xp

xjxq

xk 2xi

xpxqxp

xjxq

xk .

If we denote by Nij (x, y) = i

jk (x)yk and take into account the above law

of transformation, we find that Nij (x, y) satisfy (2.19), so they are the localcoefficients of a nonlinear connection.

2.4 Characterizations of nonlinear connections

Due to its importance in the geometry of the tangent bundle we study neces-sary and sufficient conditions for the existence of a nonlinear connection usingassociated almost product structures, almost complex structure, etc. These

structures were studied also in [76], [80], [130].For a given nonlinear connection N, we have a basis {/xi|u,/yi|u}

of TuT M adapted to the decomposition (2.14). We call it the Berwald basisof the nonlinear connection. The adapted dual basis of this basis, or theadapted cobasis, is given by {dxi, yi = dyi + Nij (x, y)dxj}. Consequently, thehorizontal space HuT M at each point u T M is given by

HuT M = {Xu TuTM,yi(Xu) = 0, i {1,...,n}}.


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A nonlinear connection N on T M induces also a regular, n-dimensional dis-

tribution (the horizontal codistribution) H : u T M HuT M Tu T Msuch that

Tu T M = HuT M Vu T M.(2.20)

Here HuT M = {u : TuT M R, u is R-linear and u(/xi) = 0}. Ahorizontal 1-form can be expressed locally as = iy

i, while a vertical 1-form has the expression = idx

i.The horizontal and the vertical projectors of the nonlinear connection can

be expressed with respect to the Berwald basis as follows:

h =

xi dxi and v =

y i yi.(2.21)

The horizontal and vertical projectors that correspond to decomposition (2.20)are

h = yi y i

and v = dxi xi

.(2.22)

Next we shall refer to both projectors h and h by h and we shall see fromthe context to which one we refer to. A similar remark applies to projectorsv and v.

From expression (1.8) we can see that the tangent structure J acts on theBerwald basis as follows:

J xi = y i and J y i = 0, that is J = y i dxi.Then, for u T M, the restriction of Ju to HuT M, Ju : HuT M VuT M,is an isomorphism. The inverse map of this isomorphism is denoted by u :VuT M HuT M. We can extend the structure u to the whole TuT M bytaking u := u vu. This is equivalent to

=

xi yi or

y i

=

xi,

xi

= 0.

(2.23)

We call the morphism the adjoint structure. It has the following properties:

1) 2 = 0, Im = Ker = HT M;

2) J = h, J = v and consequently Id = J + J .These two global properties uniquely determine a nonlinear connection withrespect to which is locally given by expression (2.23). This can be seen fromthe following proposition:


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2.4. Characterizations of nonlinear connections 29

Proposition 2.4.1 An

F(T M)-linear morphism : (T M)

(T M) such

that 2 = 0 and I d = J + J determines a nonlinear connectionHT M = Ker.

Proof. With respect to the natural basis of the tangent space TuT M, the linearmap u can be expressed as follows: (/x

i) = Aji (/xj) + Bji (/y

j) and

(/yi) = Cji (/xj )+Dji (/y

j ). Using the second property J(/xi)+J (/xi) = /xi we have that Cji (/xj) + Dji (/yj) + Aji (/yj) =/xi, so Cji =

ji and A

ji = Dji . From the second property 0 = 2(/xi)

we get that Bki = Aji Akj . Denote by Nji = Aji . Under a change of coordinates(1.1) on T M the set of functions Nji obey the transformation rule (2.19), sothey are the local coefficients of a nonlinear connection N. We have also that

(/xi Nji /yj ) = 0 and (/yi) = /xi Nji (/yj), which meanthat is locally given by expression (2.23) and the statement is proved. q.e.d.

Proposition 2.4.2 To give a nonlinear connection N on the tangent bundleT M it is equivalent to give for every u T M a linear map Ku : TuT M T(u)M such that Ku Ju = ,u.Proof. If we have a nonlinear connection N, then we consider the structure. For each u T M we define the linear map Ku : TuT M T(u)M, byKu = ,u u. Using the fact that u Ju = hu and ,u hu = ,u, weobtain that the linear Ku satisfies Ku Ju = ,u and therefore we proveddirect part of the proposition.

Conversely, let Ku : TuT M T(u)M be a linear map such that Ku Ju =,u. Since ,u is an epimorphism then Ku is also an epimorphism, u T M.If we denote by HuT M = KerKu we have an n-dimensional distribution onT M. The vertical distribution VuT M = KerJu is n-dimensional, too and fromKu Ju = ,u we have that HuT MVuT M = {0} and then (2.14) is satisfied.q.e.d.

The map K we used in the above proposition is called the connection map.This map was introduced first by P. Dombrowski [76] for the particular caseof a linear connection.

All structures Ju, u, ,u, lh,u, lv,u, Ku discussed above are related through

the following diagram:

HuT MJu //

,u

##HHHHHHHHHHHHHHHHHHH

VuT Mu

oo

Ku

T(u)M

lv,u

OO

lh,u

ccHHHHHHHHHHHHHHHHHHH


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Next, we present two other geometric structures, the almost product struc-

ture and the almost complex structure whose existence is equivalent to theexistence of a nonlinear connection.

Proposition 2.4.3 To give a nonlinear connection N on the tangent bundleT M it is equivalent to give anF(T M)-linear morphismP : (T M) (T M)such that

J P = J, P J = J.(2.24)Proof. If a nonlinear connection N is given, we define the F(T M)-linearmorphism P : (T M) (T M) as

PXi xi + Yi y i = Xi xi Yi y i .Then, J P = J and P J = J, which means that P satisfies the formulae(2.24).

Conversely, let P : (T M) (T M) be an F(T M)-linear morphism suchthat (2.24) is true. Therefore, in the natural basis, the morphism P has theform

P

xi

=

xi 2Nji

yjand P

y i

=

y i.

It can be shown that under a change of induced local coordinates (1.1) onT M, the functions Nij satisfy the formula (2.19) and, consequently, they are

the local coefficients of a nonlinear connection N. q.e.d.The morphism P defined on the if part of the above proof satisfies also

P2 = Id, and consequently it is called the almost product structure of the non-linear connection. It has the property that the distribution of eigenspacescorresponding to +1 is the horizontal distribution and the distribution ofeigenspaces corresponding to 1 is the vertical distribution. With respect tothe Berwald basis of the nonlinear connection, the almost product structure

P has the expression

P =

xi dxi

y i yi = h v.(2.25)

Proposition 2.4.4 To give a nonlinear connection N on the tangent bundleT M it is equivalent to give an F(T M)-morphismF : (T M) (T M), suchthat

F2 = Id, andF J + J F = Id.(2.26)Proof. If we have a nonlinear connection N, we consider the adjoint structure and define F = J. Then F2 = 2JJ + J2 = (h + v) = Id.Also, we have that F J + J F = J + J = h + v = Id.


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2.5. d-tensor fields 31

Conversely, consider an

F(T M)-linear morphism F : (T M)

(T M)

such that (2.26) are true. If we define = J + F, we have that 2 = 0and J + J = Id. According to Proposition 2.4.1, HT M = Ker is anonlinear connection on T M. q.e.d.

Structure F is called the almost complex structure of the nonlinear connec-tion and it has the following expression with respect to the Berwald basis:

F =

xi yi

y i dxi.(2.27)

From the above formula, we see that the almost complex structure has constantcoefficients with respect to Berwald basis and cobasis. The integrability of thisstructure is then equivalent to the fact that /xi is a holonomic frame, whichis the same to the fact that yi are exact 1-forms and we shall see next sectionthat this is equivalent to the integrability of the nonlinear connection.

2.5 d-tensor fields

A tensor field on the total space of the tangent bundle of a manifold is a quitecomplicated structure, while tensor fields defined over the vertical subbundleare very similar to tensor fields over the base manifold. These are the so-called d-tensor fields. The algebra of d-tensor fields on the tangent bundle ofa manifold is studied in [130].

A tensor field T of (r, s)-type on T M is said to be a distinguished tensorfield (or a d-tensor field for short) if under a change of local coordinates (1.1)on T M, its local components change as the local components of a ( r, s)-typetensor field on the base manifold, which means that they satisfy a formulathat is similar to (1.3).

More precisely, we have the following characterization for a d-tensor field.Let T be a tensor field of (r, s)-type on T M, so T is an F(T M)-linear morphism

T : 1(T M) 1(T M) rtimes

(T M) (T M) stimes

F(T M).

Every 1-form

1(T M) and every vector field X

(T M) can be de-composed into a horizontal and a vertical component = h + v andX = hX + vX. If T is a tensor field of (r, s)-type on T M, then T(h1 +v1,...,hr + vr, hX1 + vX1,...,hXs + vXs) is a sum of 2

r+s terms. Each ofthese 2r+s terms is a d-tensor field on T M. Then we have that T is a d-tensorfield of (r,s)-type if and only if it reduces to only one term from those 2r+s

possible terms. This means that T is a d-tensor field if and only if

T(1,...,r, X1,...,Xs) = T(11,...,r

r, 1X1,...,sXs),


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for some choice of 1,...,r, 1,...,s

{h, v

}.

For example a (0, 2)-type tensor field g = gij (x, y)dxi dxj is a d-tensorfield since under a change of coordinates (1.1), if we make use of (1.6), thecomponents gij transform as the components of a (0, 2)-type tensor field onthe base manifold, i.e.:

gij =xk

xixl

xjgkl.

If such a d-tensor field is symmetric and rank(gij) = n, we shall refer to it asa metric d-tensor field (or generalized Lagrange metric). If a metric d-tensorfield g = gijdx

i dxj is given, then one can consider also gv = gijyi yjand gd = gijdx

i yj. Both gv and gd are d-tensor fields on T M. Accordingto the characterization we gave above, this is true also because

g(hX+ vX,hY + vY) = g(hX,hY)

Lagrange Space

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