Initiation to Global Finslerian Geometry

Initiation to Global Finslenan Geometry North-Holland Mathematical Library Board of Honorary Editors: M. Artin, H.Bas...

Author: Hassan Akbar-Zadeh Doctorat d Etat en MathÃ©matiques Pures June 1961 La Sorbonne Paris.

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Initiation to Global Finslenan Geometry

North-Holland Mathematical Library Board of Honorary Editors: M. Artin, H.Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. Hormander, J.H.B. Kemperman, W.A.J. Luxemburg, F. Peterson, I.M. Singer and A.C. Zaanen

Board of Advisory Editors: A. Bjorner, R.H. Dijkgraaf, A. Dimca, A.S. Dow, J.J. Duistermaat, E. Looijenga, J.P. May, I. Moerdijk, S.M. Mori, J.P. Palis, A. Schrijver, J. Sjostrand, J.H.M. Steenbrink, F. Takens and J. van Mill

VOLUME 68

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Initiation to Global Finslerian Geometry

H. Akbar-Zadeh Director of Research at C.N.R.S. Paris France

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First edition 2006

Library of Congress Cataloging in Publication Data A catalog record is available from the Library of Congress. British Library Cataloguing in Publication Data A catalogue record is available from the British Library.

ISBN-13: 978-0-444-52106-4 ISBN-10: 0-444-52106-2 Series ISSN: 0924-6509 © The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.

v PREFACE This book is an initiation to global methods in the study of the differential geometry of Finsler manifolds. It contains my research on the subject during the last twenty years. Most of it has been published in articles in different journals. I have brought it all together in a streamlined manner to offer a coherent vision to global differential Finslerian geometry. The first three chapters form the foundation of Finslerian geometry. They contain the basic notions of global Finslerian geometry and lay the groundwork for the rest of the book. The treatment is deliberately kept transparent and simple so as to highlight the differences between Riemannian geometry and Finslerian geometry. At the start it is best to note that Finslerian geometry is the most natural generalization of Riemannian geometry. Consequently the results established in the book are the generalizations of the Riemannian case. Our particular interest bears on complete or compact manifolds. The book contains detailed proofs of a certain number results that I have published in the Comptes Rendus de l'Academie des Sciences de Paris. The book has eight chapters. Each chapter begins with a resume of the results contained in it. The number of publications on Finslerian geometry is high. I indicate only a few of them in the bibliography. I want to thank here my friend Dr. Cyrille de Souza who has helped me in the preparation of this book. My sincere thanks go to Mr. Sevenster, editor of Elsevier, for his kindness, patience and advice during the preparation of the manuscript. I am also thankful to Ms. Andy Deelen, Administrative Editor, for her suggestions for the final preparation of the text. Finally, I am grateful to Elsevier for its interest in the progress of this branch of differential geometry.

VI vi

Introduction Finslerian geometry is the most natural generalization of Riemannian geometry. In his dissertation (1854[1]) Riemann already imagined a generalization of his metric. Then P.Finsler in his thesis (1918[17]) generalized a certain number of theorems of classical differential geometry. Berwald contributed to the progress of this geometry [1928 [8]). But the connection introduced by him is not Euclidean and deprives the Finslerian geometry the simplicity and elegance of Riemannian geometry as Elie Cartan remarks in his book (1933, [13]). Unfortunately some scholars of Finslerain geometry have not paid heed to Cartan's observation. In his work Cartan studied the geometry of Finslerian manifolds in the framework of metric manifolds with the help of a Euclidean connection. He introduced the notion of the manifold of line elements ([13],[14]) that is formed by a set of points and the direction starting from those points. The parallel transport defined by him preserves the length of vectors. With the introduction of Euclidean connection in the neighbourhood of each linear element, the manifold enjoys all the properties of a Euclidean manifold. In other words the manifold is locally Euclidean. The different infinitesimal connections introduced by Cartan (linear, projective and conformal) can be dealt with in the same geometric framework: C. Ehresmann published an article on this topic with the title "infinitesimal connections in a differentiable fibre bundle" (1950, pp. 29-55 [16]) in the context of any Lie group provided a general framework for the introduction of connections. A clear treatment of the subject was given by Lichnerowicz in his "Theorie Globale des connexions et des groupes d'holonomie"(1954 [27]). Thus the modern foundations of Finslerian geometry are best laid in the framework of fibre bundles as done in this book. This book falls naturally into three parts:

Introduction Introduction I. II. III.

vii

Basics of Finselrian Geometry (Chapters I and II) Classification of Finslerian manifolds (Chapters IV, V, VI) Isometries, Projective and Conformal Transformations (Chapters III, VII, VIII)

viii

CONTENTS Preface

v

Introduction

vi

Chapter I Linear Connections on a Space of Linear Elements Abstract I.

Regular Linear Connections

1. 2. 3. 4. 5.

Fibre Bundles V(M) and W(M) Frames and Co-frames Tensors and Tensor forms Linear connections Absolute differential in a linear connection. Regular linear connection 6. Exterior differential forms

1 2 4 5 6 9

II. Curvature and Torsion of a regular linear connection 7. Torsion and curvature tensors of a general linear connection 1. Torsion tensors 2. Curvature tensors 8. Particular case of a linear connection of directions. Conditions of reduction 9. Ricci identities 10. Bianchi identities 11. Torsion and Curvature defined by a covariant derivation

13 14 17 18 20 21

Contents

ix

Chapter II Finslerian Manifolds Abstract 1. 2. 3. 4. 5. 6. 7. 8. 9.

Metric manifolds Euclidean connections The system of generators on W. Special connections Case of orthonormal frames and local coordinates for the class of special connections Finslerian manifolds Finslerian connections Curvature tensors of the Finslerian connection Almost Euclidean connections

23 24 27 30 32 33 36 40 44

Chapter III Isometries and affine vector fields on the unitary tangent fibre bundle Abstract 1. Local group of 1-parameter local transformations and Lie derivative 2. Local invariant sections 3. Introduction of a regular linear connection 4. The Lie derivative of a tensor in the large sense 5. The Lie derivative of the coefficients of a regular linear connection 6. Fundamental formula 7. Divergence formulas 8. Infinitesimal isometries, the compact case 9. Ricci curvatures and Infinitesimal isometries 10. Infinitesimal affine transformations 11. Affine infinitesimal transformations and Covariant Derivations

49 53 54 57 58 61 64 67 70 75 76

x

Contents

12. The group Kz(L) 13. Transitive algebra of affine infinitesimal transformations 14. The Lie Algebra L 15. The case of Finslerian manifolds 16. Case of infinitesimal isometries

78 79 81 84 86

Chapter IV Geometry of generalized Einstein manifolds Abstract I . Comparison theorem 1. The Laplacian defined on the unitary tangent fibre bundle and the Finslerian curvature 2. Case of a manifold with constant sectional curvature

89 95

II. Deformation of the Finslerian metric. Generalized Einstein manifolds 1. 2. 3. 4. 5.

Fundamental lemma, Compact case Variations of scalar curvatures Generalized Einstein manifolds Second variationals of the integral I(gt) Case of a conformal infinitesimal deformation

98 101 105 112 118

Chapter V I.

Properties of compact Finslerian manifolds of non-negative curvature

Abstract 1. Landsberg manifolds 2. Finslerian manifolds with minima fibration

123 125

Contents 3. Case of isotropic manifolds 4. Calculation of (δ A)2 when (M, g) is a manifold with minima fibration. 5. Case when (M, g) is a Landsberg manifold. The calculation of ∇iAj2 6. Case of compact Berwald manifolds 7. Finslerian manifolds whose fibres are totally geodesic or minima

xi 127 129 131 132 135

II. Compact Finslerian manifolds whose indicatrix is an Einstein manifold 1. The first variational of I(gt) 2. Second variational

138 141

Chapter VI Finslerian manifolds of constant sectional curvatures Abstract I.

Isotropic Finslerian manifolds. Notations and recalls

1. 2. 3. 4.

Finslerian manifolds Indicatrices Isotropic manifolds Properties of curvature tensors in the isotropic case

144 147 150 151

II. Finslerian manifolds with constant sectional curvatures 1. Generalization of Schur’s theorem A. Case of Berwald connection B. Case of Finslerian connection 2. Necessary and sufficient conditions for an isotropic Finslerian manifold to be of constant sectional curvature

153 156 157

xii

Contents

3. Locally Minkowskian manifolds 4. Compact isotropic manifolds with strictly negative curvature

162 164

III. Complete manifolds with constant sectional curvatures 1. Operator D1- the isotropic case 2. Complete manifolds with strictly negative constant sectional curvature 3. Complete manifolds with strictly positive constant sectional curvature 4. Complete manifolds with zero sectional curvature

167 169 171 173

IV. The plane axioms in Finslerian geometry 1. 2. 3. 4.

Finslerian submanifolds Induced and intrinsic connections of Berwald Totally geodesic submanifolds The plane axioms

175 179 181 182

Chapter VII Projective vector fields on the unitary tangent fibre bundle Abstract 1. Infinitesimal projective transformations 2. Other characterizations of infinitesimal projective transformations 3. Curvature and infinitesimal projective transformations 4. Restricted projective vector fields 5. Projective invariants 6. Case when Ricci directional curvature satisfies certain conditions

191 194 197 201 205 210

Contents

xiii

7. The complete case 8. Case where the Ricci directional curvature is a strictly positive constant. 9. The second variational of the length 10. Homeomorphie to the sphere

211 214 216 218

Chapter VIII Conformal vector fields on the unitary tangent fibre bundle Abstract 1. The Co-differential of a 2-form 2. A Lemma 3. A characterization of conformal infinitesimal transformations 4. Curvature and Infinitesimal Transformation in the compact case 5. Case when M is compact with scalar curvature H is constant 6. Case when X = Xi (z) dxi is semi-closed

223 225

References

243

Index

246

227 229 234 239

This Page is Intentionally Left Blank

1 CHAPTER I Linear Connections on a Space of Linear Elements (abstract) Let M be a differentiable manifold of dimension n of class C<x. Let p: V(M) -> M be the fibre bundle of non-zero tangent vectors to M , with fibre type Rn-{0} with structure group GL(n, R) the general linear group in n real variables. We denote by n: W(M) -> M the fibre bundle of oriented directions tangent to M. Let E(M) be the linear fibre bundle of frames on M and p"'E(M) the induced fibre bundle of E(M) by p. An infinitesimal connection on p"'E(M) is called a linear connection of vectors ([1]). The study of this connection leads us to single out a condition of regularity (§5). In this case, independent tensor forms can be introduced on V(M). To a regular linear connection of vectors are associated canonically two torsion tensors S and T as well as three curvature tensors R, P and Q; we find expressions for them in (§7). In view of obtaining the formulas of the habitual linear connections we establish a reduction theorem (§8). With the help of covariant derivations of two types V and v ' we form three Ricci identities for a vector field in the large sense (§9). In §10 we show that there exist between the two torsion tensors S and T as well as among the three curvature tensors R, P and Q of a general regular connection five identities called Bianchi identities. We then give explicit formulas for them.

I.

Regular Linear Connections

1. Fibre Bundles V(M) and W(M) over M. a. Let M be a differentiable manifold of dimension n of class C°°. The space V(M) of non-zero tangent vectors to M can be given the structure of differentiable fibre bundle over M of dimension 2n with structure group the linear group of n real variables GL(n, R), the fibre being isomorphic to the vector space Rn minus the origin. In the following a point of V(M) will be denoted by z = (x, v). We denote by p the canonical projection of each vector z of V(M) to its origin xe V(M): pz = x. b. We call oriented tangent direction at a point xeM the equivalence class defined on the non-zero vectors of the origin x by the positive collinearity: two non-zero vectors z\ and Z2 eV(M) have the same direction if there existe a scalar X>0 such that

2

Initiation to Global Finslerian Geometry

Z2=X,zi.The quotient space of V(M) by the equivalence relation defined by the positive collinearity will be called the space of oriented tangent directions to V(M) and will be denoted by W(M).The space V(M) is fibred over W(M) with the group of positive homothecies as the structure group.We denote a point of W(M) by y;and the canonical map of V(M) over W(M) by "n.We have r|z=y .It is clear that W(M) is endowed with the structure of a differentiable fibre bundle over M, whose dimension is (2n-l) and whose fibres are homeomorphic to the ball Sn-1 and whose structure group is O(n). If n is the corresponding canonical projection, we have 71 y = X

Between the maps p, r\ and n we have the relation 7 1 . Tl = p

2. Frames and Co-frames. a. Let E(M, F, G) be a differentiable fibre bundle over the base space M of fibre type F and of structure group G. If f denotes a differentiable mapping of a manifold M' to the manifold M we know (Steenrod [23], p.47) that we can, starting with E(M, F, G), construct, a differentiable bundle E'(M', F, G) with base M' and of the same fibre types and the same structure group as E, this space is the induced fibre bundle from E by f and will be denoted by f"'E. If one denotes respectively by g and g' the canonical maps of E —> M and of E' —» M' we have the following commutative diagram:

f >

M

where h is the induced map from £" —» £ and f.g'— g.h

Linear Connections on a Space of Linear Elements

3

b. Let E(M) be the space of principal fibre bundles of linear frames over M of fibre type and structure group the linear group Gl(n,R). The fibre bundle, induced from E(M) by TT: W(M) -> M, (resp. by p:V(M)—>M) is then a principal fibre bundle ^"^(M) (resp. p"'E(M)) with base W(M) (resp. V(M)) of fibre type and structure group identical to GL(n, R) and will be called the fibre bundle over W (resp. over V) of linear frames. It is clear that p"1E(M) is none other than the fibre bundle induced from rc'^M) by: r|:V(M) -> W(M). To construct these spaces one can also proceed with the usual method of coverings. c. Let us consider a covering of M by open and connected neighbourhoods (U,V,...). At each point z e p~'(U) we consider a frame Rzu, depending differentiably on z: it is a basis of Tpz, that is to say an ordered set of n linearly independent vectors of Tpz,. If U and V are two neighbourhoods of the covering, let Rzu and Rzy be the frames attached toz e p"'(UnV), then there exists a regular matrix element of the group GL(n, R) such that Rzv = Rzu A u v (z), z e p'CUnV), To such a frame of Tpz corresponds in T*pz a dual basis or co-frame a u z : it is an ordered set of n linear forms (a 1 ,a 2 , ...a n ) linearly independent in T*pz. If a z and a z are dual co-frames of Rzu and Rzy we have, in matrix notation, the relation a u z = A u v a v z where z e p'^UnV), Let Qz be the set of frames attached to z e p"'(UnV), over the union (2.1)

uzeVQz

one can define a topology and a natural structure of a differentiable manifold: to every system of local coordinates (x1) of M we let correspond a system of local coordinates for (2.1), where the coordinates of an element of Rz e uZ£V Qz are defined by the

4

Initiation to Global Finslerian Geometry

coordinates (x1, v1) of its origin z, the (v1) being the components of the vector z with respect to the natural frame of the origin x, and by the matrix A defining Rz (with respect to the same natural frame). The projection which makes correspond to every element Rz e u zeV Qz its origin z defines u zeV Qz as a principal fibre bundle over V(M) with GL(n,R) as the structure group. This principal fibre bundle will be identified with p"'E(M). 3.Tensors and Tensor Forms. a. Tensor fields in the large sense. We call an affine tensor field in the large sense a map which lets correspond to every z e V(M) an element of the affine tensor algebra constructed over Tpz. By abuse of language we call a tensor in the large sense instead of a tensor field in the large sense. b. Tensor fields in the restricted sense. Let t be tensor field in the large sense. To two points z\ and Z2 belonging to r\'l(y) correspond two values t(zi) and t(z2) in general not proportional. Let us suppose that for Z2 = A. zj. We have necessarily t (Z2) = f (X) t(zi), (X > 0); where f(A.) is a continuous function of A,. Let us write this relation for three points of r\'l(y) say z\, Z2 = X z\ , Z3 = fj, Z2 we obtain

Since we have Z3 = \xX z\ we also have

It follows that f must satisfy

The only continuous solution of this functional equation is of the form f(X) =

Linear Connections on a Space of Linear Elements

5

where p e R. We call a tensor field in the restricted sense (by abuse of language a restricted tensor) a tensor field t such that if z, and z2e T( '(y), we have for z2=A,Zi (3.2)

t(z2) = Xp t(z.)

p will be called the degree of the tensor t. In particular, a tensor field of degree zero in the restricted sense is nothing else but a tensor field over W(M). C. By a tensorial r-form in the large sense over V(M) of definite type (k, 1) we understand a map which lets corrrespond to every element z e V(M) an element of T^® T*'pz ®A(r)pzwhere A(r)pz is the space at z of r-forms with scalar values of V(M). Such a form is a tensorial r-form of the usual type of the fibre bundle p"'E(M), A, being a given real constant > 0. Let \i\ be the transformation of V defined by \ix: z —> Xz a tensorial rform T in the restricted sense of homogeneous degree p is a tensorial r-form over V(M) such that for every A, we have

4. Linear Connections. Let p"'E(V(M)) be the principal fibre bundle of linear frames on V(M). An infinitesimal connection of p~'E(M) will be called a linear connection of vectors [1], [lc]. Given a covering of V(M) by neighbourhoods with local sections of p"'E(M) a linear connection of vectors can be given by the data in each U of a 1-differential form co'u over p"'(U) with values in the Lie algebra of the group GL(n, R) and such that for z e p"'(UnV), we have RZV=RZU Auv(z),

AeGL(n,R)

they satisfy the conditions of coherence

6

Initiation to Global Finslerian Geometry

(4.1) (o z v=A- lu v co z uA U v + A- |U vdA U v where in each U endowed with frames the form wzu is represented by an (nxn) matrix whose elements are the 1-differential forms on p'^U): the elements of the matrices cozu and cozy will be denoted by cozu = (co'j),

cozv =(cok'h)

To the linear connection thus envisaged corresponds a one form over V(M) with values in the Lie algebra GL(n,R). An infinitesimal connection of 7T~'E(M) will be called a linear connection of directions. Such a connection can be defined in a manner similar to the preceding. 5. Absolute differential in a linear connection. Regular linear connection. a. Given a covering of M by neighbourhoods with frames, let us consider a vector field W in the large sense in V (M). For z e p"'(U), it can be defined by one row matrix of its components W u . The absolute differential of W relative to the linear connection of vectors is (5.1) where VWU defines a linear differential form over V(M) with vector values. Similarly, the absolute differential of a covariant vector Pu is defined by (5.2)

Vp u = d(3 u -p u co u

There exists a canonical vector field in the large sense: one which to every z of V(M) lets correspond the vector Tpz defined by z. This canonical vector field will be denoted in the following by v. From the consideration of its absolute differential we deduce the local nforms

Connections on a Space of Linear Elements Linear Connections (5.3)

7

91 = Vv1 = dv1 + coy

which define a one form 8 of vector type. The set (a1, dv 1 ) form a co-frame of the vector space 0 z tangent to V(M) at z ; in order that the set (a1, 61) forms a co-frame of 0 z it is necessary and sufficient that the system of n-forms to which are reduced the 91 for a'=0, (i=l, 2,.., n) be linearly independent. Let us suppose that the linear connection of vectors is defined with respect to a local section of E(M) by: (5.4)

co'j= r V d ^ + CVdv".

Using the coherence condition we see that C'jk are the components of a tensor of type (1, 2); if we denote by \i the restrictions of 0'to the fibre p~'(x) of (5.3) and (5.4) it follows that (5.5)

yij = dvj+ v j C' k dv k = (8'k + V C^) dvk

We are thus led to study the system in dv defined by |a'=0; in order that this system admits only the zero solution it is necessary and sufficient that the matrix (5.6)

(8V + v1 C jk), 5'k being the Kronecker symbol

be regular ; in this case the set (a1, 91) forms effectively a co-frame of 0 Z . It is clear that this matrix defines a tensor of type (1,1). Definition 1.5.-^4 linear connection of vectors is called regular if the set (a1, 6' = Vv') forms a co-frame of the vector space tangent to V(M)atze V(M). b. In the following we suppose that the linear connection of vectors is regular; with respect to the co-frame (a1, 91) the matrix co'j is written (5.7)

a>ij=Yijk

8

Global Finslerian Geometry Initiation to Global

co being regular we deduce by putting (5.7) in (5.3) that the matrix L\=h\-JClfr

(5.8)

is regular. In the following we will denote by M the inverse ofL L.M = M.L = I where I is the identity matrix; thanks to the condition of coherence (4.1) the coefficients of two kinds of linear connection of vectors get transformed according to the formulas (5.9)

y k ' hT = A^AVAVy'k + Ak'r SrAV,

(5.10)

C k ' hT = Ak', AVA k r C J jk + Ak'r d- ,-AV

where <9p and 5* y denote respectively the Pfaffian derivatives with respect to a and 6 1 . If we consider a covering of M endowed with the local sections of E(M) the Arh depend only on x therefore df AV = 0, the relation (5.10) becomes (5.11)

C k ' hT = Ak',AVA1VCijk

Thus it follows that relative to the co-frames of such a covering, the Cjk are the components of a tensor T of type (1,2). Thus with respect to the coframe (dx1, 91) the matirx a>) of the linear connection of vectors can be written (5.12)

(D i j=r i j k

(5.12) is identical to (5.4); in virtue of (5.3), we have

Linear Connections on a Space of Linear Elements f Vdx^fjke^rjkdx'+c'jhte 11 -v r ( f

h

9 rkdx

k

+Thrker)]

hence (5.13)

rijk = r 1 J k - c i j h v r r h r k

(5.14)

T i j k =C i j k -C i j h v r T h r k

It is clear that for the co-frames (dx1, 0'), the L'k = 8'k- v* Tj k are the components of a tensor of type (1,1); it is the same for the matrix M, the inverse of L defined by (5.6). 6. Exterior Differential Forms. a. Given a regular linear connection of vectors, we have shown in the previous paragraph that the set (a1, 0 ') constitutes a basis for the 1-forms over V(M). Thus any 1-form over V(M) can be written (6.1)

7t = a i a i + b i e i

For 7t to be the inverse image over V(M) of a 1 -form overW(M) by the canonical map r\ : V(M) —>• W(M) it is necessary and sufficient that whatever be z\ and z2 E rf'(y) one must have z2 = Xz\, (X > 0) (6.2)

7t(z2) = 7t(z,)

From V2 = X\\ we obtain

Putting this relation in (6.2) and on identifying the two parts we have (6.3)

ai(z2) = ail b'vj =(

10 10

Initiation to Global Finslerian Geometry

In the following, the operation of multiplication contracted by v will be denoted by the index 0; thus the last relation of (6.3) can be written bo = O Similarly given a differential 2-form on V(M) (6.4)

- aijaW +by a'A6J + - Cy 9' A9 J

In order that it be the inverse image on V(M) of a differential 2form on W(M) by r\ (V -> W) it is necessary and sufficient that we should have for z2 = Xz\ (X>0) (6.5)

a,j (z 2 ) = aij(zi)

(z2) = X-\ bij (zi) bio = 0 Cu(z2) = X-2C jj (zi), Loj = 0

In a general manner, for an r-form on V(M) to be the inverse image of an r-form on W(M) by the canonical map r/: V(M)—> W(M) it is necessary and sufficient that the coefficient of the mixed term containing d, p times and 6\ q times, p+ q = r be restricted of degree -q satisfying the condition of homogeneity of multplication contracted by v. In this case this r-form will be identified with an r-form on W(M). The preceding result applies to the connection form: For a regular linear connection co on V(M) to be the inverse image of a linear connection on W(M) by r\:V(M) —» W(M) it is necessary and sufficient that for z\ and z2 = Xz\, (X > 0), € r\-\ (y) we have (6.6)

yiik(z2) =YV C| jk (z 2 )=r 1 C' jk (z 1 ) C jo = 0

Linear Connections on a Space of Linear Elements

11

and we will identify co with a linear connection of directions. It is clear that the condition of homogeneity of the coefficients of the second kind of the connection is invariant by a change of frames. Let t be a tensor in the large sense on V(M) of type (1, 1), the absolute differential of t relative to the linear connection of vectors is Vtj = dtjj + ©Vtj - t'r corj.

In what follows we denote by Vk and V% respectively the covariant derivatives with respect to the co-frames a k and 9k; thus in virtue of (5.7), the above relation gives us

V* kt'j = 3'kt j + C'rktrj - t'r Cjk .

In particular we have V k v j = 0 ; Vjv^S'k In earliers sections we have introduced the Pfaffian derivatives with respect to the co-frames (a k , 9 ). In the following, we are thus led to reason in terms of the co-frame (ock, dvk). Denoting by 8k and b\ the Pfaffian derivatives relative to (cck, dvk) we propose to establish the relations that exist among these different derivations. To do this we substitute the expression for co j defined by (5.7) in (5.3), (6.7)

e k =dv k + (y k l h a h + C k i h e h ) v 1

Let O be a function with real values in the large sense on V(M) at a point ze V(M), its differential can be written indifferently dO = S k Oa k + 5 k O6 k - 8 k Oa k + 8'k 0>dv k

12 12

Initiation to Global Finslerian Geometry

In virtue of (6.7), the above relation becomes 8k
8\ = 8-k - Crok 8v

Thus the derivation operators d and d' are expressed by means of 5 and 8' and the coefficients of the connection by (6.8) and (6.9), the linear connection in question being regular the preceding formulas are invertible and we have (6.10)

8k = 3

(6.11)

6-k = M r k 5v

b. Let us denote by TV(M) the fibre bundle of tangent vectors to V(M), and by p the canonical linear map TV(M)^p"'T(M). This means that if X e TV(M), we have p(X) = (z, p*(X)), z e V(M) et p* is the linear tangent map. Let Vz be the set of vertical vectors at z eV(M), that is to say, vectors that are tangent to the fibre passing through zeV(M) (Vz =ker p* where p*: TV(M) ->• T(M)). Let V be a covariant derivation in the fibre bundle p"'TM —> V(M). We now define a linear map u,:T z V(M)->T p z (M) where X e TV(M) Now we show that if the connection is regular the linear map \iz defines an isomorphism of Vz on Tpz.for each zeV(M). Taking a vertical basis Si — -&— of V z, we have SvJ

Linear Connections on aa Space of Linear Elements ^ = V ^

13 13

(v i 5 i ) = (5iJ + v r C' r j )§ 1

Now the matrix in the bracket of the right hand side is regular. So \xz defines an isomorphism of Vz onto Tpz. Let us suppose that the connection is regular and let Hz c TZV(M) be the kernel of fiz. Then at each point z e V(M) we have the following decomposition TZV(M) = Hz 0V Z ,

Hz = ker \iz,

Hz nV z = 0.

where Hz (respectively Vz) is the horizontal (respectively vertical) space defined by V. Every tangent vector X eTV(M) decomposes into

i =HI+vl,

HXeHz,VXeVz

II. —Curvature and torsion of a regular linear connection 7. Torsion and curvature tensors of a general linear connection. i. Torsion tensors. Let © be a regular linear connection of vectors represented in each U by the matrix co'j defined by (5.7) and satisfying the condition of coherence. It is clear that the one row matrices whose elements are (7.1)

S j = da1 + co'jAoc'

define on V(M) a 2-form with vector values. We call this vector 2form the torsion form associated to the linear connection. Since the Pfaffian system a1 = 0, is completely integrable, the elements da1 can be written (7.2)

doc' = - bjjk a j Aa k + aV aJA9k

14 14

Initiation to Global Finslerian Geometry

where the bjk are skew-symmetric with respect to the lower indices; taking into account this relation, the torsion form can be put in the form (7.3)

S1 = da 1 + co'j A a j = - Sjjk a j Aak + r j k e W

from which we have

s V + s'kj = o where the Sjk and Tjk are the components of two tensors of type (1, 2) which will be called torsion tensors of the two kinds of the connection. On substituting (7.2) in (7.3) and on identifying the two sides, we obtain (7.4)

S'jk = b'k - (Y^k - Vkj)

(7.5)

Tjk = Cjk - a'jk

ii.

Curvature tensors.

a. To the regular linear connection co on V(M) we can associate a 2-tensorial form Q of type adj g"1 over V(M) having for components (7.6)

Qjj = <Wj + coV A ©rj

We call this 2-form the curvature form of the connection. With respect to the co-frames, the Q'j can be written (7.7)

Q j j = | R'jkia'Aa'+P'jk^Ae'+lQ'jkie^e 1 ;

the tensors R, P and Q will be called the curvature tensors of the connection, the first and the last being skew-symmetric with respect to the indices k and 1. In order to evaluate these tensors it is necessary and sufficient to develop the right hand side of (7.6) and to identify it to (7.7). To do this we obtain by differentiation

Linear Connections on a Space of Linear Elements (7.8)

15

<Wj=! (dky'ji -dr/jOaW-Ka kCV a; y ^ a V e 1 + - (d'k c'ji- a; cijk)ekAe1+yijIdar+c Vder

In the above relation we see appear the differentials of the coframes; da r and d9r the first being calculated from (7.3); as to the second we have by (5.3) (7.9)

d9r = dcors vs - cors Advs = 9s A cors+ Qr0

On the other hand, the last term of the right hand side of (7.6) becomes (7.10)

(oUfflr^ (Y'rkYVyVjOaW +(YirkCrjl-yrjkCiri)akAe1

+|(c i rk c r jl -c i rl c r Jk )e k Ae 1 Adding (7.8) to (7.10) and identifying the expression thus obtained to (7.7) we have (7.11)

R^i = ^ jki + CV Rrokl

(7.12)

P V ^ H i + C'jrP'oki

(7.13)

Q V ^ g j k i +C'jrQroki

where we have put (7.14) (7.15)

R ijki=(5kyiJi-%1Jk)+(y'rkyrji-yiriyrjk)+Y jr(Srki+yrki-yrik) P'jki^SkCV-^y'jk) + (yVkC^YjkCV, -/ikC^+Yjr (C'kl - Trkl)

(7.16)

Qsjkl = (d'k C'j, -d] Cjjk) + (C' rk Crj, - Cjrl Crjk) + Cjr (Crkl - Cr|k)

16 16

Initiation to Global Finslerian Geometry

Multiplying the two sides of (7.11) by v j we get * = Ss - C o) R okl kl = R R okl

The expression within the brackets is none other than the matrix L's defined by (5.8); the linear connection involved being regular, it follows, on multiplying the two sides by M'r that (7.17)

Rioid = M i r i r o k l

Thus (7.11) become (7.18)

R i jkl = J R i jkl + C i jr M r a i? a o k ,

In the same way, for the other two tensors we obtain (7.19)

Pijki=i>jki + C i j r M r a P a ok ,

(7.20)

Qijki=0ijki + C i j r M r a e a o k i

where R*, P*, Q* are defined by (7.14), (7.15) and (7.16). Thus we see that three curvature tensors of a regular linear connection are expressed in terms of the coefficients of the connection and their first derivatives as well as torsion tensors by the formulas (7.18), (7.19) and (7.20). b. It is often convenient to reason in terms of the co-frames 1 (dx ', 8 ) of local coordinates. In this case the matrix of the linear connection is represented by (5.12), by (7.2) it follows that (7.21)

bjjk = 0,

a j jk =0

From (7.4) we obtain (7.22)

S!jk= -(fV-r'kj)

Connections on aaSpace of Linear Elements Linear Connections

17 17

and the coefficients of the second kind Cj k coincide with the torsion tensor Tj k . The formulas (7.14), (7.15) and (7.16) become in this case (i ox\ y/.z,j)

J?l = P\ r 1 P\ r 1 -i-T^1 F r F 1 Fr i A jk] C k l ji - C| l jk ' i rk i jl " rl * jk

(7.25)

Qjjkl = V-kT', - V, T'jk + Tjr, T r j k - T ! r k Trj,

and the curvature tensors become (7.26)

RVi = R V, + fjr M r a Raokl

V ' ••^ ' /

^ jkl

-* j k l ~ *

A

jriV1 a

r

okl *

*

*

*

By the preceding formulas it is clear that R, P, and Q define in this case three tensors. 8. Particular case of a linear connection of directions. Conditions of reduction. a. In the case of regular linear connection of directions we know that the coefficients of co'j satisfy (6.6), the preceding formulas giving the torsion and curvature tensors of a general linear connection are in particular valid for such a connection, and we have in addition Co n

V"- 1 /

T ' = A pi C\\ = O'-. = f) = n i jo v/, r j k o vy, v j o l Vjko "

b. Let to be a regular linear connection on p" E(M); let us suppose that it is the inverse image of linear connection on E(M) by the canonical map of p"'E(M) on E(M) with respect to the coframe (dx',dv') the matrix of this can be written

18 18 (8.2)

Initiation to to Global Global Finslerian Finslerian Geometry Geometry Initiation coiJ = r i j k (x)dx k ,

It follows , by (5.13) and (5.14)

r'j^rvcx),

rjk=o

In virtue of (7.27), (7.24) and (6.9), we have Pjki= / » > = - 3 ; r ' j k = - ^ r v = o Thus the tensors T and P are zero. Conversely, let us suppose that the tensors T and P of the regular linear connection on p"'E(M) are zero; from the relations (5.13), (5.14), (6.9), (7.27) and * (7.24) it follows that C'jk is identically zero and the T jk = F j k do not depend on the direction, and the matrix of the linear connection is of the form (8.2). Thus we have: Theorem. In order that a regular linear connection a> on p'lE(M) be the reciprocal image of a linear connection on E(M) by the canonical map of p'lE (M) over E(M) it is necessary and sufficient that the tensors T= 0 and P = 0. If it is so, Q = 0 and the connection can be identified with a linear connection on E(M). 9. Ricci Identities. -Take a covering of M by neighbourhoods (U) endowed with a natural frame of local coordinates. Let X be a vector field in the large sense; in each U, X defines a 0-form with vector values XZU = (X1); the relation ddX' = 0 becomes - (dkdi-d,dk)XidxkAdx1 + (dkd , - d•i5k)XidxkA0l

+ - (e k8 -1 - d \d ox'e^e 1 + er x w = o

Linear Connections on a Space of Linear Elements

19

where 9k and 9 k the Pfaffian derivatives with respect to the local co-frame (dxk, 0k). Taking into account (7.9) the above relation gives us (9.1)

(dkd\ - dA)X[ + 9vX1Rrokl = 0

(9.2)

(9k9 , - WX1

(9.3)

(d\d\ - 9 -,9 'k)Xj + 9'rXi(Qrokl + Trkl - Trlk) - 0

+ &TX\?TM - f

r

lk)

=0

On the other hand the covariant derivation of type Vi of X is

A second derivation of the same type gives us 1

9 k x s r j s l +x s 9 k f js, 9,x r r i rk +x s f r sl f i rk -v r x 1 rrrl k

From this we deduce the identity, on taking into account (9.1), (7.22) and (7.23), (VkV, - ViVOX1 = XrR'rkl -V.X'R'ok, - VrXJSrkl

(9.4)

In an analogous manner we obtain (9.5)

(VkV,-V1Vk)Xi=XrP1rkl-VrX1Prokl+VrXiTrkl

(9.6)

(V k V, - V,Vk)X' = XrQVkl - WX'Q'ok,

More generally, let t''"'a 7l...^ be a tensor field in the large sense; we obtain (9.7)

(VkVLV.VkK'•J'h...Jf = Yd P

,

^JMkl^1

,

RW'"'"-V;,

" Ji~r-J/i -R r ok|V'r^ l ' t t y 1 ..^-S r k lV r ^ 1 ''" 7 l ... 7 / ,

20 (9.8)

Initiation to Global Finslerian Geometry (V k V, -V.Vk)'* •'" ,,.,, P

Z (9.9)

nr

/'

ijukl'

'«

Dr

V7- fh-'a

7 1 ./-.j / i -r 0 klVr/

4_Tr

T7 /'i---'«

i.-.j/,+ 1 k l V r r "7l...7/,

( V - k V W , V k ) ? " •'-„...;,= J QiYrki/'•'•••'" v,, / ( r=\ P . . " X Q^Hkl^ " h-r-Jn - QroklV'r /''•••'or y,...^

To the identities of (9.7). (9.8) and (9.9) we give the name the Ricci identities [1]. 10. Bianchi Identities [1]. We have associated to the regular linear connection on V(M) three curvature tensors and two torsion tensors; among these different tensors and their covariant derivatives there exist relations which we now are going to establish. By exterior differentiation from the formulas (7.1), and (7.6) we obtain (10.1) (10.2)

^ '

'

The relations (10.1) and (10.2) are called the Bianchi identities. On identifying in (10.1) the terms in a k A a A a m we obtain (10.3)

S R'mk, - S TVR'ok, = S VmSVi + S S 1 ^ ,

where S denotes the sum of terms obtained on permuting cyclically the indices (k, 1, m), the coefficients of the terms in cck A a1 A 9m and in a k A 91 A 9m vanish in (10.1). Similarly on identifying in (10.2) the terms in a k A a1 A a m , a k A a1 A 6m , a k A 61 A 9m , 9k A 91 A em we obtain successively

Linear Connections Connections on on aa Space of of Linear Elements Elements (10.4) (10.5)

21 21

S V m R V S S'WR'J™- SPjjmr Rroki = 0 Vfc R'jkl+T'kmRjrl+T^ R'jkr+VkP'jim-ViP j k m +S r klPjrm-(P jkrProlm-PjIrProkm)+Q1jrmRrokI = 0

(10.6)

V* P'ju- V;P' jkm + VkQ'j,™ + Pjjrl T r km - P' jrm T r kl +Q jrm P'okl " Qjri P'okm "Pjkr Q'olm = 0

(10.7)

S V^QVSQ'jrmQ'ok^O

11. Torsion and curvature defined by a covariant derivation. Let us suppose that the general linear connection is regular. We denote by V the corresponding covariant derivation in the fibre bundle p'TM -+ V(M). Let X, Y, Z be three vector fields on V(M) over X, Y, Z belonging to T pz. Then we can express the torsion and the curvature of the connection by the following formulas: (11.1) (11.2)

T(X, Y)= Vx Y-VfX-p[X,Y] Q(X, Y)Z= VxVf Z- V^ ViZ- V ^ Z

t and Q being respectively the torsion and the curvature of V. They determine two torsion tensors, denoted by S and T, and three curvature tensors R, P, and Q. following the decomposition of vector fields into horizontal and vertical parts. (11.3) x ( H I , H 7 ) = S(X,Y), T ( V X , H 7 ) = T(X,Y)

22

Initiation to toGlobal Finslerian Geometry

(11.4) Q(Hi,H7) = R(X,Y), Q(Hl,V7)

where X = ji(V X), Y = \x(V 7 ) On deriving the equations (11.1) and (11.2) and on using the Jacobi identity for the A

A

A

three vector fields X, Y, Z on V(M) we obtain (11.5) S Q ( 1 , 7 ) Z = S V f S ( l , Y) +

Sx(Z,[XY])

(11.6) S Q ( l , f ) Z + S Q ( Z , [ l i > ] ) = 0 where S denotes the sum of the terms obtained by permuting cyclically X, Y, and Z . The equations (11.5) and (11.6) are called Bianchi identities. On decomposing the vector fields into horizontal and vertical components of (11.5), we obtain an identity (see (10.3), and of (11.6) we obtain four identities (see (10.4), (10.5), (10.6) and (10.7)). A

A

A

23

CHAPTER II FINSLERIAN MANIFOLDS (Abstract) A metric manifold is defined by the data of a tensor field gy in the restricted sense of degree zero on W(M). To this tensor field is associated a scalar of degree two in the restricted sense F2 = gy (x, v) vV where F > 0 is by definition the length of tangent vectors v to M at x e M. With the help of g we can define the scalar product of two vectors of T^, ye W(M), consequently an orthonormal frame on T^,. Let us denote by E(W, g) the principal fibre bundle on W(M) of orthonormal frames. An infinitesimal connection on E(W, g) is called a Euclidean connection. We give the necessary and sufficient conditions in order that a linear connection on W(M) is naturally associated to a Euclidean connection of directions. We say that the datum of a function F > 0 homogeneous of degree one on V(M) defines a Finslerian metric if it leads to a regular problem of the calculus of variations. The following result is the fundamental theorem of Finslerian Geometry: Given a Finslerian manifold there exists a regular Euclidean connection such that its torsion tensor S vanishes and the tensor T satisfies a condition of symmetry. Such a characterization of the Finslerian connection leads us naturally to Cartan's Euclidean connection ([lc], [13]). Using the results of chapter I we establish the fundamental formulas of Finslerian geometry the three curvature tensors, five Bianchi identities are completely made explicit[§ 8]. §9 is devoted to the semi-metric connection and we give a characterization of Berwald connection. We show that there exists an infinity of torsion-free connections of directions attached to F that define the same splitting of the tangent bundle as the Finslerian connection. These connections, differ from Berwald or Cartan connections by a homogeneous tensor t'j^ of degree zero satisfying t'oj = 0 = tV0-, and have the same flag curvature as Berwald and Cartan connections.

l. Metric Manifolds. Let gjj be a tensor field in the restricted sense of degree zero, symmetric and positive definite. To the tensor field gy- we can associate a scalar in the restricted sense of degree 2 such that

24 (1.1)

Initiation to Global Finslerian Geometry 2L = F 2 = g l J (x,v)vV

(i,j = l,...n)

where F (> 0) is, by definition, the length of the vector v tangent to M at x. Then we say that the data of g makes M a metric manifold, gij is called the metric tensor, L the fundamental function of the metric manifold. With the help of the tensor g we can put a norm on the tangent vector space Tny (x - Try). An orthonormal frame at y e W is by definition an orthonormal base of the Euclidean vector space T%y; it is thus an ordered set of n unitary vectors (ei; Q2, ...en) of Tny such that their scalar products two by two are given by

where 8jj is the Kronecker symbol. We denote by E(W, g) the principal fibre bundle of orthonormal frames over W. This space admits a fibre and a structural group identical to the orthogonal group O(n). Let us consider a covering of M by neighbourhoods (U) endowed with orthonormal frames ; let R(y, an orthonormal frame at y e n'\\J) if y e 7t"'(UnV) we have R> = R>, Cl! (y),

where the matrix C is an element of the group O(n); with respect to such a covering the metric of the space becomes (1.2)

ds2 = 2L(x, dx) = £ (a ) 2

2. Euclidean Connections. An infinitesimal connection on E(W, g) is called a Euclidean connection of directions. Let us consider a covering of M by neighbourhoods endowed with local sections of E(W, g); a Euclidean connection of directions is defined here by the data in each U of a 1-form coyv (y e TT'^U)) over 7t"'(U) with values in the Lie algebra of the orthogonal group

Finslerian Manifolds

25

0(n). Such a form is represented by an (nxn) skew-symmetric matrix which we denote again by (Oyv = (fflij),

(oOij + COji = 0 )

its elements are differential 1-forms of y. have R> = R»Cvv
(2.1)

If y 6 7i"'(UnV) we

yen'CUnV)

where co^ must satisfy the conditions of coherence. (2.2)

<

= {Ci;(y))"1 <

C'J(y) + ( C ? ( y ) ) " ^ 7 ( y ) , Ce O(n)

We now show that to every Euclidean connection of directions is naturally associated a linear connection. Let us consider the connection defined with respect to a covering of M, endowed with orthonormal frames by the matrices n*, such that

The linear connection thus defined is evidently independent of the covering chosen. It is called the linear connection associated to the Euclidean connection the absolute differential of the tensor g,j(y) in this connection is given by (2.3)

Vgjj = dgij - cohi ghj - cohj gih

The frames being orthonormal we have gjj = 8jj The relation (2.3) then becomes (2.4)

Vgij= - (coji + coy) = 0

It thus follows that the absolute differential of the metric tensor is zero. Conversely let 7i be a linear connection such that

26

Initiation to Global Finslerian Geometry

the absolute differential of the tensor metric in this connection is zero ; by (2.3) and (2.4), it is clear that for a covering of M by the neighborhoods endowed with local sections of E(W, g) the matrices of the linear connection are skew-symmetric; on the other hand if 7i"' (UnV) we have

> ClJ (y), where C" is an element of the group O(n); for the linear connection envisaged we have the condition of coherence

It thus follows that the matrices (oy{! = nl define a 1-form of connection with values in the Lie algebra of O(n), that is to say a Euclidean connection. Thus we have the Theorem. In order that a linear connection on W be naturally associated to a Euclidean connection of directions it is necessary and sufficient that absolute differential of the tensor metric gy (y) of the Euclidean connection be zero. The Euclidean connection is called regular if the associated linear connection is regular. This will be identified with the Euclidean connection in question. Remarks on the Curvature.—Let co be a regular linear connection. Let us consider a covering of M by the neighbourhoods endowed with orthonormal frames; we can therefore put all the indices in lower positions and the curvature form is written Qjj =dC0jj +C0ir ACOrj

(Qij =gih & j)

Finslerian Manifolds

27

With respect to the orthonormal frames the symmetric; hence

COJJ

are skew-

Qij +Qji = 0

This relation, valid in orthonormal frames, is also valid in arbitrary frames. So it follows that the three curvature tensors of a regular Euclidean connection are skew-symmetric with respect to the first two indices. 3. The System of Generators on W. — a. To the canonical vector field one can make correspond the vector

It is evidently of degree zero, therefore it defines canonically a unitary vector field of degree 0, that is to say a vector field over W. Let co be a regular Euclidean connection on W, at a point x e M, if the vector tangent space to p"'(x) is referred to the co-frame (\il), the vector space tangent to the 7i~'(x) is defined by the equation liUj = 0

(3.1)

The absolute differential of 1 in the regular Euclidean connection is (3.2)

pj= Vlj = ¥-\Ql -l j dF),

1 being of unit length we have (3.3) whence (3.4) Thus (3.2) becomes

1, (3j = F"1 (liB1 - dF) = 0 5hF = 0, 8i F = lh.

28 (3.5)

Initiation to Global Finslerian Geometry pi = F - i ( e i - i i i h e h ) , v k r = o,

(3.6) Vi l^F^S'k-l'lk) By (3.5) it is clear that the B1 define on V(M) a 1-form with vector values. This 1-form being of degree zero and satisfying (3.1) can be identified to a 1-form on W with vector values. If we denote by y1 the restrictions of the B1 to the fibre 7i~'(x), they satisfy (3.7)

li y j = 0

Thus every linear combination of the y1 is a 1-form with values in the vector space tangent to 7t"'(x) at y. Conversely the y' form a system of generators for these 1-forms , in other words among the y ' there does not exist any non trivial relation distinct from (3.7). In fact, let (3.8)

ary^O

where the a; do not all vanish; so in virtue of (3.5) it becomes

let (a 1 -a h l h l,)u' = the Euclidean connection being regular the u1 are thus linearly independent; we have ai = (ahlh)l, and the relation (3.8) is not different from (3.7). It thus follows that for a regular Euclidean connection the set (d, ft') forms a system of generators for the 1-forms of the vector space tangent to W(M). b. Let 7i be a linear form on W, its inverse image on V(M) by r\ :V —> W, which we continue to denote by n, can be put in a unique way under the form

Finslerian Manifolds (3.9)

29 7C

i

i

where the Q and di are restricted vectors of degree 0 and -1 respectively and the d{ satisfy the condition of homogeneity (chap. I §6) (3.10)

do=0

If we refer n to the system of generators (a1, (31) it becomes

(3.11)

7C = a i a i +b i |3 i .

On substituting in (3.9) the expression for 0'drawn from (3.5), taking into account (3.10) we obtain (3.12)

TC-Qa' + Fdip 1 .

On identifying (3.11) to (3.12), we have b ~ Fd(. a. = q , Thus referred to the system of generators (a1, (31) every 1-form on W can be written under the form (3.11) where the a, and b; are restricted vectors of degree 0 and the b; satisfy (3.13)

b o =0,

Let us prove the uniqueness of the expression for n defined by (3.11). In fact, suppose that n = 0. The restriction of the fibre 7C~'(x) gives, on taking into account the fact that b0 = 0,

The \i are linearly independent, whence

30

Initiation to Global Finslerian Geometry

From the fact that n — aj a1, = 0 we have a\ — 0. This proves the uniqueness. If we relate the tangent vector space to W to the system of generators (a 1 , P') the 1-form of the regular Euclidean connection of directions is written in a unique way under the form (3.14)

co'j

where the B j k satsfy (3.15)

B' J O =0.

The Yjk and B j k are restricted quantities of degree zero. On substituting in (3.14) the expression for P' defined by (3.5) we then have with the notations of chapter I (§5) (3.16)

B i jk (x,v) = F(x,v)C 1 jk (x,v)

4. Special Connections [lc]. Let co be a regular Euclidean connection of directions ; the absolute differential of the metric tensor in this connection is zero (4.1)

dgij - cohighj + cohjgih

We put (4.2)

Yjik= gjhYhik,

CJ)k = gjhChlk

Thus (4.1) becomes dgij = (Yijk+ Yj,k)a k + (C ijk + C j i k )9 k

whence (4.3)

Yijk + Yjik = 3kgij

where 5k and d\ are defined by the formulas 6.8 and 6.9, Chap I In virtue of (6.6) of the chapter I the d\ gy satisfy

Finslerian Manifolds

31 d'ogij = 0

Let us suppose that the torsion tensors, associated to co, satisfy (4.5)

Sijk = 0

(Sijk= gir Srjk),

(4.6)

Tijk = Tjik,

(Tijk = girTrjk)

From the formulas (7.4) and (7.5) of chapter I, it follows that there exist, in addition, the following relations among the coefficients of the Euclidean connection in question: (4.7) (4.8)

Yijk-Yikj = b i j k Qjk - Cjik = ajjk - ajik

(bi jk = girbrjk) (ayk = gir a r j k )

On adding the equations (4.4) to (4.8), we get (4.9)

Cj jk = -

a-kgij + - (a ijk - ajik)

On the other hand from the relations (4.3) and (4.7) it follows (4.10)

yjik + yikj = 3 k gy - bijk

On permuting cyclically the indices i, j , k we obtain (4.11)

Ykji + Yjik = 5ig jk - b j k i ,

(4.12)

Yikj +Ykji = 5j g ki - b k i j .

On redoing (4.12) the sum of (4.10) and (4.11) we have, after a change of indices, (4.13)

Yijk = - (dkgy +5jg ik - 5ig kj ) - - ( b i k j + b j i k -b kji )

32

Initiation to Global Finslerian Geometry

Conversely, it is easy to verify that the quantities yyk and Cijk defined by (4.9) and (4.13)) satisfy the equations (4.3), (4.4) (4.7) and (4.8), whence Definition. [ 1 c] We call a special connection any regular Euclidean connection such that the corresponding torsion tensors satisfy (4.5) and (4.6). The coefficients of such a connection can be expressed by the formulas (4.9) and (4.13). 5. Case of Orthonormal Frames and Local Coordinates for the Class of Special Connections [lc]. a. Let us consider a covering of M by neighborhoods endowed with orthonormal frames and let (ei) be such a frame, we then have ejej=5 ij

gij

in this case, the formulas (4..9) and (4.13) become (5 •1) (5 .2)

Qjk= ^(a,jk -ajik)

(Cijo = 0)

1 k + bjkj -bkji) •I(bji

b. If M is covered by neighborhoods (U) endowed with natural frames of local coordinates the quantities a and b vanish, the coefficients of the two kind of the special connection become (5-3)

T l j k =-a-kgij

(5.4)

T ijk = - (dkgij + Sjgik - Sjgkj)

(Tijo = 0)

1 where dk and b\ are the Pfaffian derivatives with respect to the local covering (dx\ 9k = Vvk )

Finslerian Manifolds

33

6.Finslerian Manifolds. a. In paragraph 1 we have defined the length of the canonical vector vx tangent to M at x by a positive restricted function F(x, vx). On substituting dx for v in F and putting (6.1)

ds = F(x,dx),

F is by definition the arc element. Let l(xo,xi) a path in M with origin at xo and the extremity at xi ; in virtue of (6.1) the length l(xo, xi) is defined by (6.2)

s(x o ,xi)= f

F(x, x)du

(*=|T)

F being restricted of degree 1 the integral of the right hand side is independent of the parametric representation chosen for the path l(xo, Xi). The first variation of this integral with non fixed extremities l(xo, xi) is given by (6.3)

8 = 71,-710- f •»(-to.*

Sx

where n is a restricted linear differential form of degree 0 defined by

(6.4)

n = | £ 8x'

and TTO and n\ correspond to the points xo and x\. We call the extremal of the problem of calculus of variations attached to F(x,x). It is a solution of the differential system of the second order.

34 (6.5)

Initiation to Global Finslerian Geometry A^_|i=o du ox ox

x=dx/du,

On choosing for the parametric representation of the path l(xo, x the length of the arc s(xo, xi) = s the system (6.5) becomes =0

f f

0 = l.2.-"U*-dx/d.)

The system admits the first integral F(x, x ) = l

(x=dx/ds)

It thus follows, on multiplying the two sides of (6.5)' by F, we obtain the differential system of the second order ^%-§l ds ox'

ax

= 0 1

( i = l , 2 , ...n). v

'

where we have put F2(x, v) = 2L(x, v). Let, on developing, (6.6)

5"ijL x j + 5-ijLxj-5iL = 0 (i,j= 1, 2, ...n). The problem of the calculus of variations attached to the

function F(x, x) is called regular if the coefficient of x in (6.6) is a non degenerate quadratic form., that is to say if (6.7)

det(<%L)*0

This problem is called positively regular if <%L is a positive definite quadratic form. It is clear that for a covering of M by neighborhoods endowed with natural frames of local coordinates the<%L are the components of symmetric tensor of order 2 restricted of degree 0 ; we put (6.8)

gij(x, Vx) = Sij L(x, Vx).

Finslerian Manifolds

35

This tensor defines on M the structure of a metric manifold. Let us multiply the two sides of (6.8) by v1 and v* successively; in virtue of the homogeneity of F, we have gij (x,

v

x

) v V = ^ 8jF2 = 2L = F 2 (x, vx).

We are thus led to the following definition : Definition . Let M be a differentiable manifold and V(M) the space of non-zero tangent vectors to M. The structure of a Finslerian manifold on M is defined by the data of a function F(x, Vx), positive, positively homogeneous of degree 1 on V(M) leading to a regular problem of the calculus of variationsL In the following we suppose that F leads to a positively regular problem. b. In the previous sections we have seen that a structure of a Finslerian manifold determines on M a structure of a metric manifold by the tensor gy defined by (6.8); we propose here to study the converse problem. Let us consider a metric structure defined by the data of tensor field gy(x, v) symmetric, restricted of degree zero, and definite positive. To this tensorial field we associate a scalar 2L = gyv1 v* defining the square of the norm of the tangent vector to M at x. Under what condition is it the metric tensor of a Finslerian manifold ? In other words, under what condition gy can be deduced from (F2 = 2L) by (6.9)

gy = 5"y L

(6.10)

2L = gy(x,v)vV?

It is clear that for a covering of M by neighborhoods endowed with natural frames of local coordinates the right hand

36

Initiation to Global Finslerian Geometry

side of (6.9) is a tensor if for such a covering gjj is defined by (6.9) we obtain by derivation S'k gij = SiJk L

whence S'kgij = 5"i gjk = 5 j gki,

It thus follows (6.11)

v i 8- k g u = 0.

Conversely, this condition is sufficient: in fact, if the relation (6.11) is satisfied, from the expression 2L we deduce by derivation 5"k L = - (5"k gy) vV + gkj v1 = gkjv1 A second derivation gives us 5"ki L = (8"i gkj) v1 + gk, = gkl Theorem [lc]. - In order that the metric tensor gy be the metric tensor of a Finslerian manifold it is necessary and sufficient that we have (6.11.) 7. Finslerian Connectionflc] The above analysis helps us establish that there exists for the structure of a Finslerian manifold one and only one special linear connection corresponding to the Finslerian metric. The terminology Finslerian connection was introduced in [lc] in analogy with the Riemannian connection. Let gy be the metric tensor of a Finslerian manifold defined by (6.8) Let co j be the matrix of the special Euclidean connection associated to gjj. (7.1)

co'j = r' jk dxk + C'jk dvk = f

j

jk dx

k

+ T j jk 9k

Finslerian Manifolds

37 37

where the T jk and F jk are defined by (5.3) and (5.4). In virtue of (5.3) and (6.9) of the chapter I we have in local coordinates (7.2)

Tijk = - d% gy = - (§'k gij - Trok5v gjj)

The metric tensor gy of the Finslerian manifold satisfies (6.11); multiplying the two sides of (7.2) by v1 we obtain v1 Tijk = - v1 8 k gij = 0 T being symmetric with respect the first two indices , we have

(7.3)

r0k = o.

Conversely if the relation(7.3) satisfies after (6.9) of chapter I, the Pfaffian derivatives of the two types d\, dj; ( local coordinates ) coincide and on multiplying the two sides of (5.3) by v1 we find the relation (6.11). Thus (7.3) is equivalent to (6.11) and the torsion tensor of the special connection associated to gjj becomes (7.4)

Tijk= - 8 k gij = VT. 5"kij L,

(Tojk = Tiok = Tijo = 0),

the T and C being related by the formula (5.14) of Chapter I; due to the homogeneity of T they coincide and we have (7.5) We have thus evaluated the torsion tensor with the help of g and its first derivatives with respect to v. Before calculating the coefficients F and F we remark that in virtue of the homogeneity of the tensor T, the linear connection (7.1) is necessarily regular,

38

Initiation to Global Finslerian Geometry

For the matrix (5.6) of chapter I reduces the identity matrix. the formula (5.13) of chapter I, we have

By

fijk = r i j r -T i j r r r o k

(7.6)

due to the homogeneity of T we have (7.7)

f'ok-r'ok

On the other hand in virtue of (6.8) of chapter I and (7.4), we have * 5kgij = 8 k gij-2Tij r r r ok

(7.8)

* whence the coefficients T jjk , defined by (5.4) become (7.9) r

ijk

= - (5 kgij + Sjgik - 5igkj) - vr(Tijs r

s rk

+ Tiks T srj - Tkjs T s n )

Let us multiply this relation by v*, taking into account the homogeneity of the tensor T we have (7.10) Let us multiply the two sides by v : (7.11) vk v> T ijk = -(8-ikL + v'Sjg.k - 5i
2Gr(x,v)=rrjk(x,v)vJvk

whence (7.13)

2girGr(x, v) = T ^ v V = vk §-,kL - 5jL.

Finslerian Manifolds

39 39

The above relation determines G as a function of L and its derivatives,; we notice, in addition, that last term of this relation figure in the differential system of extrema (6.6); a derivation with respect to v gives us (7.14)

gis5v Gs = - (8-ir L+ vk 5 kgir - 8-ri L) - 2TirsGs = v" F ijr,

whence 6- k G r =r r o k

(7.15)

On substituting (7.15) in (7.9) we obtain (7.16) 1 (S

S

5 )

(T5Gs

Tjks5-jGs - Tkjs 8'jGs).

To obtain the expression of F rjk it suffices to multiply the two sides of (7.16) by gir. * Thus the coefficients F j k are completely determined by the metric tensor of the Finslerian manifold and its first derivatives. As to the coefficients F j k we have by (7.6), (7.17)

Fljk= - (8kglJ +5 jgik - 5igkj) + Tkjs8-iGs - Tiks 8jGs

the F and C determine then a unique regular linear connection on W. Conversely it is easy to show that the quantities F and C defined by (7.17) and (7.4) determine canonically a special linear connection associated to the tensor g, whence : Theorem [lc], [13]. Given a Finslerian manifold M there exists a special Euclidean connection and only one which corresponds to the Finslerian metric. I call this connection the Finslerian connection.

40

Initiation to Global Finslerian Geometry 8. Curvature Tensors of the Finslerian Connection.

a. The formulas of the paragraph 7, chapter I giving the curvature tensors of a general regular linear connection can be applied in particular to the Finslerian connection. We reason in local coordinates. We recall that in virtue of homogeneity of the torsion tensor T of the Finslerian connection the matrix L [(5.8), chap. I], as well as its inverse M, reduce to the identity matrix (local coordinates ). After (7.23), (7.26), and (6.8) of chapter I, we then have *

*

(8.1)

Rjki = R jki + T j r RrOki

(8.2)

R)a =(5 k r i jl -S- s r 1 J i5- k G s )

- (5, fjjk - 8-s fjjk 5-, Gs) + ( f jrk f rj, - fjrl f r jk ). We now establish the following proposition : The curvature tensor P of the Finslerian connection is expressed in terms of the torsion tensor T and its covariant derivative of the type Vf, and the curvature tensor Q is deduced algebraically from the tensor T In fact, after (7.24) and (7.27) of chapter I, we have (8.3)

P'jkl^jkl + T'jrKkl

(8.4)

Pjk^VkTye-irv.

On the other hand Sr ( r S J k ) = sr (g s ,r i J k ) = gsi Sr ( r i J k ) + 2T s i l r j J k .

Finslerian Manifolds Manifolds Finslerian

41

On evaluating the left hand of this relation using (7.16), it becomes (8.5)

gsi Sr ( r

j

jk)

= (5 k T sjl + 5j T skl - 8S Tkjl) - 2T srl F rjk

- (Sr Tsjr 5 k G r + Sr T skr 8j Gr- Sr Tkjr 5-sGr) -(TSjr Sff G r + T skr S-ji Gr - Tkjr Ssl G r ). On multiplying the two sides by v J we obtain

(8.6)

v i 5-,f i jk = Vor kl .,

On deriving with respect vJ the relation (7.15) we have (8.7)

<%-G r =f r lk + V 0 T r kl

Taking into account this relation , (8.5) becomes (8.8) 5-, f 'Jk = Vk TV + VjTV. - V'Tkj, + TrkjVo TV, - TjkrV0 Trj, - T'jrV0Trkl whence (8.9)

P'jkl = V1 Tkj, - Vj fk, + TVr Vo Trj, - Trkj Vo T1,,.

We deduce from it (8.10) Pijoi = Pjko = O As to the curvature tensor Q, we have by (7.25) and (7.28) of chapter I (8.11)

QVi= g'jki + T V Q W

(8.12)

g'jki = (4'T'j, - ^ T V ) + (T'j.TVk - fr, Trjk)

42

Global Finslerian Geometry Initiation to Global

On the other hand <% TV = Sg (gir Trjl) = - 2T'rk Trj, + gir S[ Trjl On substituting this relation in (8.12), we obtain (8.13) It satisfies (8.14)

Qijki=eijki = T i ri T r j k -T i rk T r j 1 Qojk, = Qioki = Qijoi = Quko = 0.

Thus we have established the proposition. b. In paragraph 10 of chapter I we have defined the complete list of Bianchi identities for a general regular linear connection. These five identities become in the case of a Finslerian connection [1]. (8.15)

SRUi-ST^R^O

(8.16)

SV m R i jk i-SP' jmr R r oki = 0

(8.17)

(8.18)

(8.19)

V^R'k, + T r km RVi + Trlm Rijkr+ Vk P ^ - V, Pjjkm - (P jlr V0 T fkm - P'jkr V0 TV) + Qjrm R'okl = 0 V m Pjjkl - y , PVm + Vk Qjjlm + P'jrl T rkm - P ^ Trk, + Q'jrlV0Trkm - Q'jrm V0 Trkl = 0 SVmQV^O

where we have denoted by Vm and V m the two covariant derivations in the Finslerian connection and by S the sum of terms obtained by permuting cyclically the indices (k, 1, m). In virtue of (8.1), the identity (8.15) becomes (8.20)

S0)k>i)l?ijki = 0

Finslerian Manifolds

43 43

Let us multiply the two sides by v' * (8.21)

S(j,k,l) i?ojkl = 0

Let us multiply this relation by vk * RO)o\

(8.22)

=

* Ro\oy

The tensor Ryki being skew symmetric with respect to the indices i and j , k and 1, we have *

*

*

(8.23)

R yki = - i?jiki - 2 T jjr R Oki •

(8.24)

* * = i? yki - tfjiik

2 R ljkl

Taking into account (8.23), the identity (8.20) becomes * (8.25)

*

*

R jikl - R kijl

=

*

*

*

" R lijk " 2(T jjr R Okl + T kir R olj + T |ir R ojk)

On changing in the above relation the indices j , i, k, 1 into k, 1, j , i respectively, and on adding the relation thus obtained to (8.25), and taking into account (8.24) we have (8.26)

j

*

*

*

*

— -2Rijjk - 2[T jjr R Ok| + Tkir R olj + Tiki--/? oji + Tj] r R oik]

Similarly, on changing in (8.26) the indices 1, i, j , k into j , k, 1, i respectively and on putting in (8.26) the relation thus obtained we have * (8.27) R|jjk - Rjkli

=

*

*

*

Tjjr R 0\k + Tkir R ojl + T]kr R oij + T j | r R oki

44

Initiation to Global Finslerian Geometry

9. Almost Euclidean Connections Let c : [ 0 , l ] - > M a differentiate path in M. For t o, 11 e [0, 1], the length of the arc joining x o = c(t o) to x i = c(t i) is defined with the help of the function F by

(9.1)

s=JF(xiii)dt

On choosing the parametric presentation of C(x o, X | ) the arc length s = s(x o, xi), as in §6 the extremal of (9.1) is a solution of the system of a second order differential equation, defined by (6.6) which we write,

(9-2)

^ ds

+ g ir (5 kgrj - \ 6 rgjk ) xJ x k = 0 2.

where gir is the inverse of g rj. Let us look for a torsion-free connection of directions TIT such that the geodesic of rn coincides with the extremal of (9.1). If the T'rk are the coefficients of this connection, the system of differential equations representing the geodesies of this equation can be put under the form (9.3)

^ ds

+ f 1 j k (x, x J ) i J i k = 0 ,

If D k is the covariant derivation associated to w, from (9.2) and (9.3) it follows: (9.4) where x = v. The relation (9.4) determines a necessary and sufficient condition in order that the geodesic of the connection m of directions without torsion coincides with the extremal of (9.1)^

Finslerian Manifolds

45

This being the case, a torsion-free connection of directions is necessarily regular. For the map \i becomes the identity map. On the other hand its curvature 2-form on W(M) decomposes into a 2-form of the type (2, 0), that is to say two times horizontal and a 2-form of the type (1,1), one time horizontal and one time vertical, which we denote by P . We then have Theorem. Let (M, g) be a Finslerian manifold. There exists a unique torsion-free connection D of directions such that 1) the geodesies of D coincide with the extremal of the variational problem corresponding to F. 2) the second curvature tensor of this connection satisfies (9.5)

?(X, F)v = 0

The coefficients of the connection mare defined by : (9.6)

f

j jk

= - gir(5k gjr + 5j grk - dt gjk) + D oTV

where T is the torsion tensor of the Finslerian connection defined by (7.4). Proof. On deriving vertically (9.4) and on taking into account (9.5) we obtain (9.7) v k (D k g rj + Djgrk- D r g j k ) = 0 (9.8)

vk D kgjr= 0,

vk Djgrk = vk Dr gjk

Deriving once again vertically the relation (9.8), in virtue of (9.5) we get (9.9) Dlgli+ Do(8)m)=0 Thus D i gy is completely symmetric. From the fact that D is torsion-free from (9.9), we obtain, on permuting cyclically the

46

Initiation to Global Finslerian Geometry

indices i, j , and k, the three relations on subtracting the third one from the sum of the other two, we obtain the formula (9.6). This is called the Berwald connection. We denote by H and G the corresponding curvature tensors. It is clear that if g is Riemannian, the condition (9.5) is automatically satisfied, and D is none other than the Riemannian connection. The connection w defined by (9.6) can be obtained ([8]) on putting in 2G1 the homogeneous expression of second degree in x in the differential equation (9.2) and on deriving twice vertically Sj G1 = G'J; 5$ G1 = G jk = Y jk. [this method was suggested to Berwald by Emmy Noether (see E. Cartan Oeuvres Completes [14] p. 1393)]. However, if we abandon the hypothesis (9.5) P (X, Y)\ *• 0 and we suppose P (v, F)v = 0, we still have the relation (9.8). In this case G jk is related to Y jk by Gjk = F jk + t'jk where tjk is a symmetric tensor, homogeneous of degree zero in v and satisfies t'ok = t'ko = 0- The choice of this tensor determines the connection F . We note that this connection defines the same splitting of the tangent bundle as the Berwald connection and that the expression of the curvature H 'ojo of this connection, called flag curvature, is identical to that of Berwald and Cartan. In the following we denote by V and D respectively the connections of Finsler and Berwald (D = D) associated to g and we put:

Remark. D. Bao and Z. Shen in their work on Finslerian Geometry use constantly a semi-metric connection which they call Chern connection. Now S. S. Chern tried to obtain by the method of local equivalence a set of semi-Euclidean connections for the Finslerian metric (E. Engel Zur Flachen theorie. I Leopz Ber. 1901pp 4004412, E. CARTAN sur un probleme d'equivalence et la theorie des espaces metriques generalisees, Mathematica t.4 1930pp 114-136). He establishes the following theorem

Finslerian Manifolds

47

Theorem. There is a unique set of local 1-forms co'j on TM/{0} such that dco1 = coJ AW j dgij = g kj cok, + g lk cokj + 2 Cijk « n + k

^ '

V

Unfortunately, the expressions defining co j are none other than the horizontal part of the Cartan connection denoted by

f> (x, v) dxk (see E. Cartan C.R. Academie .Set 198 (1933) pp 582-586. and 'Les espaees de Finsler Paris Hermann (1934).

48

CHAPTER III ISOMETRIES AND AFFINE VECTOR FIELDS ON THE UNITARY TANGENT FIBRE BUNDLE (Abstract). This chapter is devoted to the study of infinitesimal isometries of a compact Finslerian manifold without boundary, and affine infinitesimal transformation of a regular linear connection of directions.([1], [la], [lb]). We recall the calculus rules of Lie derivatives of a tensor field in the large sense and of a form of a regular linear connection of vectors. Let L be the Lie algebra of infinitesimal transformations of M. To an X e L is associated a certain endomorphism A x of Tpz whose expression contains the torsion tensor T. Let Az(L) be the Lie algebra of endomorphisms of Tpz corresponding to the elements of L. We establish then a relation between A[x, Y], A X , A Y and the curvature of the linear connection, generalizing from the Riemannian case due to B. Kostant ([26], [1]). For the study of compact Finslerian manifolds we establish the divergence formulas for the horizontal 1-forms and for the vertical 1-forms on W(M) ([1],[2]). Next we study the 1-parameter group of infinitesimal transformations that leave invariant the splitting of the tangent bundle defined by Finslerian connection. We give a local characterization of isometries. In case M is compact and without boundary we prove the largest connected group of transformations that leave invariant the splitting defined by the Finslerian connection coincides with the largest group of isometries. We establish a formula linking the square of the vertical part of the lift of an isometry X on V(M) and the integral involving an expression of the flag curvature (R(X,u)u,X). If this form is negative definite the isometry group is finite. Finally, to every infinitesimal isometry X of a Finslerian manifold is associated an antisymmetric endomorphism whose square of the module modulo a divergence puts in evidence a quadratic form cp depending on two Ricci tensor Ry and Pij.[lb]. We determine the conditions on them so that the isometry group of the manifold is finite. We study the particular case of Py = 0 In paragraph §10 we give a characterization of affine infinitesimal transformation (respectively partial) of regular linear connections of vectors. In paragraph § 11 we show that the Lie derivative L(X) commutes with the covariant derivatives of two types V and V* (respectively of type V) when X defines an affine infinitesimal transformation (respectively partial), and conversely. Let L be the Lie algebra of affine transformations of a generalized linear connection, and L its lift on V(M). The Lie algebra Az(L), corresponding to L is the Lie algebra of a connected group Kz(L) of linear transformations of Tpz. The study of this group is the objective of paragraphs §

Isometries and Affine Vector Fields

49

12, 13 and 14. In the case when the Lie algebra L is transitive on V(M) we have a relation of inclusion az c Kz(L)
1.Local Group of 1-parameter local transformations and Lie derivative a A vector field X on M generates a local 1-parameter group of local transformations of M by the integration of the differential system (1.1)

^=Xx(u)

(ueR)

starting from an initial point x(0) = x. From the existence theorem for the solutions of the differential system it follows that it is possible to find neighbourhoods U( x ) and positive numbers s( x ) for every x e M such that (1.1) admits a solution denoted by (1.2)

x(u) = exp (uX) x

defined for |u| < e( x ) starting from a point x e U( x ) and satisfying the group relation (1.3)

exp [(u+u')X] x = exp (uX) exp (u'X)x

provided that the two sides are defined. In general, the numbers s( x ) depend on x ; if we can choose s( x ) independent of x then exp (uX) can be defined for every x e M, and for |u| < oo, in this case X generates a 1 -parameter group of global transformations of

50

Initiation totoGlobal Geometry Initiation GlobalFinslerian Finslerian Geometry

M. If the manifold M is compact every vector field on M generates a 1-parameter group of global transformations of M. b. The differentiate map exp (uX) is defined over U(x) for every |u| < s( x ) and it admits a linear tangent map denoted exp (uX)' which defines an isomorphism of Tx with Tx(u). To every element z of V(M) above U( x ) corresponds thus an element z(u). From the map it follows that for each U(x) and s(x) a differentiable map is defined in p"1(U( x )) (1.4)

exp(ul):z-»z(u)

exp ( u l ) defines a local 1-parameter group of local transformations of V(M) which we call the extended group. By its definition we have (1.5)

p. exp (uX)z = exp (uX). pz,

we conclude from it, on deriving with respect to u that the vector field X over V(M) is projectable by p (1.6)

pI=X

From (1.5) it follows that (1.7)

p'. e x p ( u i ) ' = exp(uX)'.p',

X is called a lift of X to V(M). Let (a1, 8j) be a co-frame at z e V(M), where the 61 are supposed arbitrary ; if we denote by i the operator of the interior product, we put (1.8)

Vector Fields Isometries and Affine Vector

51

Thus at the point z e V(M) the vector X has for components with respect to the frames in question X1 and X'; in particular for the local co-frame (dx1, dv1) at z e V(M) the components of X are dx'ju) |

*• ^ ~ ^ ~ '

u=0

~

dv(u) |

'

~w~

u=0

_j*Yi_«.Yi

~

j

~ °

We remark that in a change of frame the quantities 80X1 do not transform themselves according to the components of a vector. c. Let O be an r-form on V(M) with values in a vector space N. We call the infinitesimal transformation of O by X at z the rform L( X )O [by abuse of notation L(X)O] at z with values in N defined by (1.10)

L(X)O = i(X)dO + d i ( X ) O

It is clear that L(X) is a derivation of degree 0 which commutes with d. If f is a function with real values on V(M) by (1.10) we have L(X)f=i(X)df=Xkdkf+

Xkd-kf,

where Xk and X k are the components of X with respect to the coframe (a 1 ,6'); relative to the co-frame (dx1, dv1) this relation can be written L(X)f=X k 5 k f+ 80 Xk 8% f, L(X) commutes with the operations 5k and 5% of ordinary derivations. Given a tensorial field in the large sense ( or a connection form) on V(M), for it to be invariant by the extended group exp (u X), it is necessary and sufficient that its Lie derivative with respect to X

52

Initiation to Global Finslerian Geometry

is 0. Besides, it is clear that the canonical vector v is invariant by the extended group exp(uX), in other terms, exp u(X) does not change the differentiable structure of the manifold V(M) To a frame Rz, attached to z, corresponds, by the action of the group exp ( u l ) , a frame Rz(u) at z(u), the differentiable map exp(ul):Rz->Rz(u) defines a local 1 -parameter group of local transformations of the fibre bundle p" E(M) generated by the vector field X which is a lift of X. We will define in the same way as in classical differential geometry [28] the infinitesimal transformation of a form on p"1E(M). Given a form A on p"'E(M) with values in a vector space N, the infinitesimal transformation of A by X is by definition the Lie derivative by X and will be denoted by L(X)A. One proves that [28] 1. If A is a tensorial q-form of type R(G) (linear representation of G in a vector space ), L(X)A is also a tensorial q-form of the same type. 2. If co is a 1-form of connection with values in the Lie algebra GL(n, R) of adjoint type, L(X)co is tensorial 1-form of adjoint type. Let Y be a vector field in the large sense, and A a semi-basic 1-form on V(M), that is to say it depends only on the 1-form of the base. Since L(X) satisfies the Leibniz rule for the tensor product and commutes with trace operator we have (1.11)

L(X) i(Y)A = i(Y) L(X)A + i (L(X) Y)A .

Isometries and andAffine Affine Vector Fields

53

2.Local Invariant Sections. a. Let X be a vector field on V(M), a point z of V(M) is called ordinary if it is not a zero of X; z is a zero of the first kind if every neighbourhood U of z contains ordinary points ; z is a zero of the second kind if there exists a neighbourhood U of z such that every point of U is a zero of X in this case exp(uX) defines the identity on U and the infinitesimal transformation of a tensor in the large sense or of a connection at the point z is zero. Concerning the zeros of the first kind, if t is a tensorial field in the large sense (connection form); if we evaluate [L(X)t]z at a ordinary point z e V(M) we can obtain [L(X)t]zo at a zero zo of the first kind by passing to the limit. b. Given an ordinary point z e V(M) there exists a neighbourhood U <= V(M) of z consisting of ordinary points ; if Rz is the frame attached to z the points of p"*E(M) defined by Rz(u) = exp(uX) Rz belong to the trajectory of X starting from Rz. We are thus led to the following definition: Definition. Given a neighbourhood U a V(M) consisting of ordinary points we call a local section ju of p~lE(M) generated by the trajectories of X will be called a local invariant section ofX over U . On U the frames of the local section define a vector field ej(z), invariant by X; if the n vectors e;(z) determine the frame at z, Rz = uz, we have (2.1)

L(X)ei(z) = 0

On the other hand, we have the equality between the scalars i(ej) a j = 8) The Lie derivative of the right hand side vanishes. So, taking into account (1.11), we have L(X) i(ej)<xj = i(L(X)ej)aj + i(ej) L(X)a j = 0

54 54

to Global Global Finslerian Finslerian Geometry Geometry Initiation to For (2.1) to be satisfied it is necessary and sufficient that i(ej)L(X)a j = 0

whatever be ej, whence, L(X)aJ = 0

(2.2)

c. Let A be a form on p"]E(M) with values in a vector space N. Let A the form induced on the local section of p'1E(M) over U . On U there exist a form ®?7 with values in N such that u* A = O;y For the local section in view, we obtain (2.3)

u* L( X) A = L(X) O(7 = [i( X) d + di( X)]
This formula helps us evaluate L(X) A. relation (2.2) becomes

In virtue (2.3), the

[L(X)a] i = L(X)a i = 0

(2.4) 3.

Introduction of a regular linear connection

a. Let us consider a covering of M by neighborhoods U, let co be a regular linear connection on V(M) represented in each p"'(U) by the matrices co'j of 1-forms defined by [(5.7), chap I]. Let us introduce the connection cosmd to be associated to co defined by the matrices a> j such that (3.1)

a ij = coj

where S is the first torsion tensor of co. If V denotes the absolute differential in the associated connection, from (3.1) it follows

Isometries and andAffine Affine Vector VectorFields Fields Isometries (3.2)

55 55

O'l= Vv 1 = 9 i + S i 0 k a k

After (3.1) it is clear that the associated connection is regular. Writing down the two connections co and W with respect to the coframes (dx, G) and (dx, 6 ) respectively, the relation (3.1) becomes

rjjkdxk + Tjjk ek = r' jk dx k + T'Jkek - (r j j k - r k i k = r 1 k1 kjJ dx dx k + +T Tijki kfe where Y and T are the coefficients of the connection co ; on substituting in this relation the 9 k, defined by (3.2), and on identifying the two sides we obtain (3.3) (3.4)

~* * r jk = r 'kj - T jh s ok T)k=T)k

If S denotes the torsion tensor of a>, from (3.3) we have (3.5)

Sji^T'jhS

o k -T' k hS

oj-Sj k

Thus the torsion tensors of To are defined by (3.4) and (3.5). Let Y be a vector field in the large sense, its absolute differential with respect to 15 can be put under the form (3.6)

VY j = V k Y i a k + V.-Y1 <9k = D k Y i a k . + A - Y j e k

where DkY' and Di Y1 are two tensors in the large sense of the type (1.1). On putting in this relation the 6 k defined by (3.2), we obtain (3.7)

D k Y j = V k Y j + V^Y'S^k

56

Initiation to to Global Global Finslerian Finslerian Geometry Geometry Initiation A'Y^V.-Y1

(3.8)

b. Let U be a neighborhood of V(M) consisting of ordinary points . Let us consider a local section invariant by X over U . We have shown that for such a section we have (2.4), namely [L(X)a]j = L(X) a j = i ( l ) d a j + di(X)a j = 0

(3.9)

Taking into account [7.3 chap I], this relation becomes (3.10) dX1 +(co'j + S j jk a k )X J -ay\(X)ak + T i t h i h a k - X h T i h k 9 k = 0 Let (3.11) VX i + T i k h ^ h a k = fflik(X)ak + X h T 1 h k e k In virtue of (3.6), for the frames of a local section in question we deduce (3.12) (3.13)

DkKi + TikhXh = (o'\(X) DkX^xT'hk

c. It is now easy to evaluate the Lie derivative of a tensor in the large sense or of a regular linear connection. First let us calculate the infinitesimal transformation of a vector field Y in the large sense. Given a neighborhood U consisting of ordinary points, let us consider a local section invariant of X over U . With respect to the frames of this section, Y defines a 0-form with vector values Yjj — (Y1) of (2.3), we then have for these frames [L(X)Y)]j= L(X) Y j = i(X) dY1 = VY'(X) - Yk co'k ( 1 ) Taking into account (3.12), it becomes

and Affine Vector Vector Fields Isometries and (3.14)

57 57

[L(X)Y)]j = VY1 ( 1 ) - Yk(Dk Xj + T'kh Xh)

the two sides of this formula being vectors in the large sense on V(M) the equality is valid for arbitrary frames and at the ordinary points of V(M). Passing to the limit it is valid at all the points. In order to evaluate the quantities X that figure in the right hand side of (3.14), it suffices to remark the Lie derivative of the canonical vector field v is zero, whence X ^ D o X ' + T'ohX11

(3.15) Let (3.16)

X h ( 6 ' h - T 1 o h ) = D0X1

On the other hand, the formula (3.14) can be put under the form j

= Xk Vk Y1 - Yk Dk X1 + X \ Vj Y1 - Yh

j

where the X are defined by (3.16). In a local co-frame (dx, 6) the term within the parenthesis in the right hand side of the formula is none other than the Pfaffian derivative of Y' with respect to 6k; on taking into account [(6.9), chap. I] and (3.16), we obtain (3.17)

[L(X)Y)]1 - Xk Vk Yj - Yk Dk X1 + Si Yj Do Xs.

The right hand is independent of the connection introduced. Let us suppose that the coefficients of o with respect to the co-frame (dx, dv) are zero. Then it will be so also for the associated connection, and the relation (3.17) is reduced to (3.18)

[L(X) Y)]1 = Xk 5k Yj - Yk 5k Xj + St Y] 50Xs.

4. The Lie Derivative of a Tensor in the large sense. We will now apply the above method to calculate the Lie derivative of a tensor in the large sense. Let t'jk be a tensor in the

58

Initiation to Global Finslerian Geometry

large sense on V(M); in the frames of a local section, invariant by X, we have

= VtVC X) + tVkCorj( X) + t > r k ( X) - trjkcoV( X) In virtue of (3.12) we obtain (4.i)

[L(X)t]'Jk = v t ' J k ( i )

j

h

tjjr(Dkxr + Trkh xh) - t ^ x ' + r r h xh)

where X is defined by (3.16). This formula is valid in arbitrary frames and at all points. By the same reasoning as above, we have in local co-frame (dx1, 91) (4.2) and (4.3)

[L(X)t)]jjk = Xr Vr t'jk + trrk Dj X r +1 jjrDk Xr -t r j k D r X i +&-t i j k D 0 X s [L(X)t]jjk = Xr5r tjjk+ tVk5jXr + t V 5k X r -1 rjk5r X' + &-tj k 5 0 X s 5. The Lie derivative of the coefficients of a regular linear connection.

Let co be a regular linear connection on V(M). We know that the Lie derivative of co is a tensorial 1-form on p"IE(M). of the type adj. Let U be a neighborhood consisting of ordinary points. Let us consider a local section invariant under X over U . After (2.3), in terms of these local frames we have (5.1)

[L(X) co] jj - L(X) cojj = i ( l ) dcojj + di(!)© i j

The operator i is an anti-derivation from [(7.6), chap I]; so we obtain

Isometries and Affine Vector Fields

(5.2)

i ( l

i

l

i

i

l

59

l

In virtue of [(7.7), chap I], the first term of the right hand side becomes

With respect to the frames in question we have, after (3.12), (5.4)

di(X)d)) + (oir(orj (X) - (D\ (l)co r j = V(Dj X 1 + T\

Taking into account (5.2), (5.3) and (5.4), the formula (5.1) becomes (5.5)

[L(X) co]) = L(X)co) = V(Dj X1 + T jjh X h) + (Xh R jjhk - Xh P 'Jkh)ak + (XhP jjhk + X hQ 'Jhk)e k

where X is defined by (3.16). Relative to the above frames, the Lie derivative of 0r becomes (5.6)

[L(X) 6] r = L(X) 6 r = L(X)(dvr + v s cors) = [L(X)co]r0

Taking into account this relation, the formula (5.5) becomes in local co-frame (dx, 0), (5.7)

[L(X) co]jj = L(X)f jjk dx k + L(X)T jjke k + T'jr[L(X)co]r0

whence, on multiplying the two sides by v1 [L(X) coj'o = L(X) f From it, we have

j ok

dx k + L(X) T jok 6 k + T ^

60 (5.8)

Initiation Global Finslerian Finslerian Geometry Geometry Initiation to Global [L(X) co]r0 = M r s (L(X)f

s

ok dx

where M is the inverse of the matrix L. (5.7) and taking into account (5.5) we get

k

+ L(X) T s ok 9 k )

On substituting (5.8) in

(5.9) [L(X)co]'j=(L(X)f ijk+T'jrlVfsLCX)f s ok )dx k +(L(X)T i jk + T' jr M r s L(X)T s ok )9 k ^(DjX'+T'jh X h )+(X h R' jhk - X h P' jk h)dx k +(X h P i jhk + X h Q' jhk )e k In virtue of (5.6), the formula (5.5) gives us (5.10) L(X)f 1 j k dx k + L(X)T 1 jk O k = V(DjX" + T ' J h X h ) +(X h RVk - X h P ' k h ) d x k

+ ( x V ^ + x "Q^e" - rjrjrvv s [ v(Dsxr+Trsh xh) +(X h R r shk - X h P r skh )dx k +(X h P r shk + X h Q r shk )e k ] where X is defined by (3.16). The two sides of the formulas (5.9) and (5.10) being tensor 1-forms of type adj, these relations are valid in arbitrary frames and at all points, whence (5.11) L(X)f

j jk

= V k (D j X i + T i j h X h ) + (X h R i j h k -

Xh?)kh)

- T 'Jrvs [V k (D s X r + T rsh X h) + (XhRrshk - X h P r skh ] and (5.12) L(X) T jjk = V* (DjX j + T jjh Xh) + (X h P j jhk + Xh

Q)hk)

- T > s [ Vj (D s X r + T rsh Xh) + (X h P r shk + X h Q r shk )] In virtue of (7.26), (7.27) of Chapter I, the formula (5.11) becomes

Isometries Affine Vector Fields Isome tries and andAffine

(5.13)

fi

61 h

i

As to (5.12), on developing the right hand side we re-find the formula giving us the Lie derivative of the tensor T (4.2). 6. Fundamental Formula. -In the following we suppose that V(M) is endowed with a regular linear connection and we reason in local co-frames (dx1, 91). Let g the vector space over R of infinitesimal transformations defined by the vector fields on M. To every pair (X, Y) e g, we let correspond the usual bracket (6.1)

[X,Y] = L(X)Y;

With this composition law g becomes a Lie algebra, which will be called the Lie algebra of infinitesimal transformations of M, and will be denoted by g. Let us consider the map which lets correspond to every X e g the tensor of the type adj defined by (6.2) where (6.3)

Xh(5rh-Troh) =

In local co-frames (dx, 9) the expression between parentheses in (6.3) is none other than the matrix Lrh defined by [(5.8), chap.I], the linear connection in question being regular; this matrix has therefore an inverse which we have denoted by M; from (6.3) we obtain (6.4)

X h = MhrD0Xr

On substituting (6.4) in (6.2), we have 'j = -(DjX1+T1jhMhrD0Xr) Thus to every X e g we let correspond the endomorphism Az(X) of Tnz We denote by Az(g) the vector space of endomorphisms

62

Initiation to Global Finslerian Geometry

corresponding to all the elements of g. Az(g) can thus given the structure of a Lie algebra by the composition law defined by the usual bracket: if AX and AY are two elements of Az(g), their bracket is defined by (6.5)

[AX, AY] - AXAY - AYAX.

We will denote this algebra by A. Let X be an element of g and AX the corresponding endomorphism: if t j k is a tensor field in the large sense, we put (6.6)

(AXt)'jk = (AX)1 r t r jk - (AX)rj 4 - (AX)rk t' r

With these notations, the formula (4.1), giving the Lie derivative of tjk by X can be written in a simple manner (6.7)

[L(X)t]'jk = V t'jk( X ) + (AXt)'jk

Let X and Y be two elements of g and AX and AY the corresponding elements of A; to the bracket [X, Y] corresponds the element A[X,Y] of A ; we propose to evaluate A[X, Y] as a function of AX and AY. If we put

we have by (6.2) (6.8) with (6.9)

Z h = D o Z h + Thok Z k

From (6.7) it follows [L(X)AY]iJ = V {AY))(X)

+ [AX, AY]jj

Taking into account the definition of AY, this relation can be written

Isometries and Affine Vector Fields

63

[AX, AY]jj + V(AY)jj( X) = - [L(X)Dj Yj + L(X)Tjjh Yb + TjjhL(X) Yh]

(6.10)

where (6.11)

7 h = D0Yh + Thok Yk.

On subtracting (6.10) from (6.8), we get (6.12) (AZ)' J -[AX,AY] i j -V(AY) 1 J (X) = L(X)Dj Y1 - Dj L(X) Y1 + T jh(L(X) Yh - Z h ) + L(X) Tj h Yh We are going to evaluate the right hand side of this relation; first of all, in virtue of (3.7), we have DjY'= V J Y 1 + Y m T 1 mh S h Oj = 5 j Y ' + Y k f i jk whence (6.13)

L(X) DjY1 - Dj L(X) Y1 = Yk L(X) f

j jk

On the other hand, on taking the Lie derivative with respect to X of the two sides of (6.11) we have L(X) Yh = L(X)D0 Yh + L(X) Thok Yk+ Thok L(X) Yk On substituting in this relation the expression of Z h defined by (6.9) and on taking into account (6.13), we obtain L(X) Y h - Z h = T hok(L(X) Y k - Z k)+Y k L(X) f whence

h ok

+ Y k L(X)Thok

L(X) Y h - Z h = Mhr (Yk L(X) f

r Ok

+ Y k L(X) Trok).

(6.14)

In virtue of (6.13) and (6.14), the relation (6.12) can be written

64

Initiation to to Global Finslerian Geometry (AZ) i J -[AX,AY] i J -V(AY) 1 J (X) <

= Y k (L(X) f 'Jk + T'jh Mhr L(X) f rok) + 7 k (L(X) f jk + r j h M h r L(X)Trok). On evaluating the right hand side of this relation by (5.9), we obtain the fundamental formula (6.15) A[X,Y] = [AX,AY] + VAY(1)-VAX(#)

7. Divergence Formulas When the manifold M is compact we have established in [1] a divergence formula for a horizontal 1-form. Using the same calculus, we establish here an analogous formula for a vertical 1— form (See [2]). Let u: M ->• W(M) a unitary vector field. We denote by u = Uj dx1 the 1-form associated to u. Now du is of rank 2(n-l) everywhere on W(M). We denote by (du)11'1 the (n-l) th exterior power of du, by x\ the volume element of W(M):

•

^

*

*-M1»)-

N=*tii

Let (x, ei, e 2, .. .e n -i, u) be an ortho-normal frame of Tny (y e W), (u = e n ) Then we have U = U j e j = e n =>Ua = 0 ,

Un = 1

Let p i = VUJ ((3.2) chap II); then we have Pn=COnn = 0,

p \ = C0^n ,

(A, = 1, .. .n-1)

Relative to this frame the volume element r\ becomes

r\ = P where

Isometries and Affine Vector Fields

65 65

(3 = Pi A...AP n -l, tfl = G>iA...A©n

with (COJ) is the dual co-frame of (e j) Let b be a vertical 1-form on W

(7.1)

b = ] T bx$x

(bn = 0)

X=\

The adjoint of b relative to r\ is (7.2)

*b=]T (-l^'b.PiA^^A

Ap n ,Aft)

X=\

where the sign /? signifies that the corresponding term should be omitted in the exterior product under consideration. Hence we have

(7.3) d(.t>)=£ (-l)X"1dKAp,A...A^^A Apn.,A(O x=\ X=\

f

2

E

(-l^'b^p.A^A

APn.,AdC0

X=\

Therefore we should calculate the terms dp^ and do. First of all we have i = Ph ACOih + ^ F^RjoklCOk ACOi + Pjokl

whence, modulo the terms containing the co k (7.4)

dp i = P hA0) ih

(mod co k)

66

Initiation to Global Finslerian Geometry

Similarly n

d o = 2_j ( - 1 ) ' dWj AM] A ...A<5 j A...ACO n

Now dco i = Oji ACOJ = F Tjjh (3 h ACOJ

Thus (7.5)

dco = F T i i h p h ACO

On substituting (7.4) and (7.5) in (7.3) we obtain

Where V} denotes the covariant derivative with respect to the coframe Vv Now, u j Vrb n = 0 => un Va bn = 0 = V% bn So (7.6) 5b = -F(Vrbi + bjT Hj ) where 8 is the co-differential (see [1], [2]) Let us suppose M to be compact without boundary. We then have (7.7)

J Sbt| = - | W(M)

F( V; b'+ b1 Ti)n = 0

W(M)

To this relation we give the name the divergence formula for a vertical 1-form (7.1). Similarly, if (7.8)

a = ai(z)dx j ,

zeV(M)

is a horizontal 1-form on W(M) we obtain (see [1] pp 68-70) (7.9)

5 a = -(V i a 1 -a i V 0 T 1 )

and Affine Vector Vector Fields Isometries and

67 67

In case M is compact without boundary, we have

(7.10)

J 5 a r| = - J (Via1 - a,V0 r)Ti = 0 W(M)

W(M)

This is the divergence formula for a horizontal 1-form. 8. Infinitesimal Isometries [la], [lb] A. Let X be an infinitesimal transformation on M and Hz the horizontal sub-space of TZV(M) defined by the Finslerian connection. The extended group exp(u X) maps Hz to a subspace of TZ(U)(V(M)). An infinitesimal transformation X leaves Hz invariant if the image of Hz by exp(uX) is a horizontal subspace ofT z(u) . Theorem 1. In order that the infinitesimal transformation X leaves the horizontal distribution Hz invariant it is necessary and sufficient that it satisfies one of the three following equivalent conditions: (1) (2) (3) vector

L(X) commutes with the projection H The splitting of the 1-form Vv is invariant byX The bracket [X,v] is zero, where v is the horizontal field above v (p*v - v)

(1) is evident. To show that (1) implies (2) we calculate the Lie derivative of Vv. On identifying w in (3.1) with the Finslerian connection (S = 0) and on multiplying the two sides of (5.11) by v1 we obtain (8.1)

L(X) Vv(HF) = VH- V0(X) + R(X, Y)v + (V,T) (V,X, Y)

Now [ X, H Y ] is, by hypothesis, horizontal. This implies that the right hand side vanishes.

68

Initiation to Global Finslerian Geometry

(2) => (3): it suffices to take Y= v in (8.1) and to remark that [ X, v ] is always vertical from (8.1) the theorem follows. (8.2)

V;V,X + R(X,v)v = O « [ X ,v] = 0

* (3)=> (2) In fact we have noted for 2 G' the expression F jkvVk ( see 7.12 chapter II). Now the left hand side of (8.2) is none other than the Lie derivative of 2G ' (see 5.11) So, on deriving vertically we obtain (2). B. Let X be an infinitesimal transformation on M. Then X is an infinitesimal isometry if it leaves invariant the metric tensor g: (8.3) L(X)g = 0 Let us decompose L(X): where A^ is defined by

(8.4)

4

Theorem 2. X is an infinitesimal isometry if it satisfies one of following three equivalent conditions: (1)

g(4Y,Z)

(2)

g(V^X,v)

(3)

g(V,X,v) = 0

for all Y, Z Gp"1T(M) where g ( , ) denotes the local scalar product. (1) signifies that the Lie derivative of g is zero. (1) => (2) => (3) is evident. To prove the opposite implication we must derive vertically

Isometries and Affine Vector Fields

69

The compact case Theorem 3. For a compact manifold without boundary M the largest connected group of transformations which leave invariant the splitting defined by the Finslerian connection coincides with the largest connected group of isometries([la], flbj) Proof. If X is an infinitesimal isometry, it leaves invariant the 1form of the splitting V. Hence (8.2). Conversely for X satisfying (8.2) we obtain (8.5)

V, [g(X, u) g( A, u u)] + g( A, u, u) 2 = 0

where v = L u (L = ||vj|) and v = Lu. M being compact without boundary and on integrating (8.5) on W(M) the first term is a divergence. So, after the formula(7.10), we obtain u, u) 2 r| = W(M)

whence g( A% u, u) = 0 This is the condition 3 of theorem 2. C. Let X be an infinitesimal isometry and f = i g(X, X). On using (8.2) we obtain u u (f) = g (V, X, V, X) - g(R(X, u)u, X) Since M is compact and without boundary, the left hand side is a divergence. So integrating over W(M):

(8.6)

<( V, X, V, X> = J (R(X, u)u, X) r\ W(M)

where < , > and ( , ) denote respectively the global and local scalar product on W(M).

70

Global Finslerian Geometry Initiation to Global

Let us suppose that the integral of the right hand side of (8.6) is non-positive. We then have Vri X= 0. From it we obtain, by vertical derivation, and the fact that X is an isometry, X is of covariant derivation of horizontal type zero. If the quadratic form (R(X, u)u, X) is negative definite the group reduces to the identity. Theorem 4. Let (M, g) be a compact Finslerian manifold without boundary and X an infinitesimal isometry. If the integral of (R(X,u)u,X) on W(M) is non-positive, X is of covariant derivation of horizontal type zero. If this form is negative definite then the group of isometry of(M, g) is a finite group ([la], [lb]). 9. Ricci Curvatures and Infinitesimal Isometries In the preceding section we have shown the influence of the sign of the sectional curvature (R(X, u)u, X) (the flag curvature) on the existence of a non-trivial isometry group. Here we will highlight the significance of the Ricci curvatures Ry and Py of the Finslerian connection. After theorem 2 of the previous paragraph, to every infinitesimal isometry X is associated an anti-symmetric endomorphism AX of Tpz defined by (9.1) To this endomorphism is associated a 2-form (Ax) (9.2)

A x = 1 (Vj Xj - Vj Xj) dx * AdxJ

X being an isometry, the Finslerian connection is invariant under X by (5.11). So we have (9.3) whence

Vk(Ax) ) = Xh Rjjhk - P'jkhVo Xh

(9.4)

-VJ (Vj Xj - V, Xj) = 2 (Rij XJ + Py V0XJ)

Isometries Affine Vector Fields Isome tries and andAffine

71

where we have put Rrjrj = Rjj, and Prirj = Pij Then by (8.9 chap II) (9.5)

Py = Vr Trij - V, Tj + Tr Vo Trij - T rsi Vo T srj

from (9.4) and using the divergence formula (7.10) we obtain (9.6)

18(i(X)Ax)= \ (A x , Ax) -Ry X'XJ - PijXV0XJ + lx1(ViXj-VJXi)V0Tj

where i(X) is the inner product with X. We are going to calculate the last two terms of the right hand side when X is an isometry. Let us put the result in a quadratic form in X modulo divergences. After (9.5) we have, on taking into account the divergence formulas (7.7) and (7.9): the term: -T rsi XV 0 T srj V0Xj = Div +[V0(TrsiV0 T s rj )+1 (VrV0 Try - Vo Trij V0Tr)]XjXJ the term Vr Try Xj Vo X j = Div - ^ (V0Vr T rij)X'Xj the term Tr Vo Trjj X1 Vo X J = Div - 1 V0(Tr Vo T rij)XiXj the term :-ViTjXV 0 X j (9.7) = Vo Vj Tj XjXj + Vj Tj Vo XjXj + div on W(M) where div = divergence as in the rest of the book. Take the term in the expression: VjTj =D jTj + Tr Vo T rjj where V is the covariant derivation in the Finslerian connection and D is the covariant derivation in the Berwald connection. So we get (9.8)

ViTjVoX'XJ - DjTj Vo XlX> + 1 TrV0TrijV0(XiXJ) 'VJ = Div on W(M) - 1 V 0(TrV0Trij) XjXJ + D,Tj Vo X'X

But the last term of the right hand side is

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Initiation to Global Finslerian Geometry

DjTj Vo X'XJ = glk D o Xk dr D0Tj XJ - glk D 0 X k D 0 Or Tj Xj = g l k # [D 0 X k D 0 TjX J ] - glk (DiXk + D o df Xk)D0Tj XJ - D o dr Tj X J D 0 X' Now D 0 X 0 - 0 since X is an isometry. The first term of the right hand side is a divergence. So we get DjTj Vo XjXj = div on W(M) + gik DiXkD0 Tj Xj + 1 D0(D05r Tj)XjXJ = -D0(X, T)D 0 Tj X J + 1 [D08r D o Tj - D0V,Tj + Vo (TrV0 Trij)] XjXJ+div on W(M) On taking into account this relation, (9.8) becomes (9.9)

ViTjV0X'Xj = 1 ( D o 5? Do Tj - D o VjTj) X1 XJ D o (X, T) D o Tj Xj + Div on W(M)

We calculate the last term of the right hand side in another manner: X being an isometry we have -ViV0TjXiXj =-Vi(V0TJXiXj) + VjTjViX'X-1 + V J j X V i XJ = XjV0TjV, Xj - XjV0Tj X'V0Ti - VoTjXj (VjXj + 2 ThijVoXh) +Div on W(M) = - T'V0 Xj XJ V0Tj - XjV0 Tj X1 V0Ti - 1 Vo TJ Vj X1 X, - Vo T r Trij V0(XjXJ)+ Div on W(M) = - XJV0TjV0 (X, T) - 1 Vj (Vo TJ Xj X, ) + 1 Vj Vo Tj X % + Vo (Vo Tr Trij )X'XJ + Div on W(M)

Isometries and Affine Vector Isometries and Affine Vector Fields Fields = Div on W(M) - XjV0TjV0 (X, T ) - l v

73 o

TJ VoTjX1 X*

+ 1 Vj V o T j X j X, + V o (V o T r Trij )X i X j = DivonW(M) +[V0(V0TrTrij ) + \ (V rV 0 T r -V 0 T r V 0 T r ) gijJX'XJ - Xj V 0 T jV 0 (X,T) Let us carry the expression - XJ V 0 T jV 0 (X, T), drawn from the preceding relation into (9.9): J

= 1 (D 0 5r D 0 Tj - D 0 V j Tj)X'XJ - ViV0Tj XjXJ

-[Vo (V o T r T r ij)+1 ( V rV 0 T r -V 0 T r V 0 T r )gy]XiXJ + Div on W(M) This is the last term of the right hand side of (9.7). On substituting it in (9.7) we get: - V i T j X i V 0 X J = Div + {\\ V o Vj Tj - V, V o Tj + 1 Vodf V0Tj - V 0 (T, jh V 0 T h ) + l g i j [V 0 T r V 0 T r - V r V 0 T r ]}X j X j the term 1 X j (VjXj- VjX,)V0 TJ = D i v + 1 [gij(V0TrV0Tr- V r V 0 T r ) + V 0 (T l j h V 0 T h )]X i X j Let us put i|/(X, X) = { 1 [(Vo Vr Try + V0V0(TrTrij) + Vr V o Trij - Vr TrijV0Tr V o a? VjyVoViTj] - V 0 (T r si V o T s rj ) +V 0 (T ljh V 0 T h ) + V,V0Tj }XjXj

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Initiation to Global Finslerian Geometry

We have (9.10)

-PijXVoX*

+^X 1 (V i Xj-V J X i )V 0 T j = Div on W(M) + \|/(X, X)

On substituting this relation in (9.6) we obtain (9.11)

\ 5(i(X)Ax) = \ (Ax, A x ) -
whence (9.12)

M being compact without boundary, on integrating on W(M) we get (9.13)

l<(Ax,Ax)>= f

By a reasoning analogous to the Riemannian case we show that the isometry group of a compact Finslerian manifold is compact since it is the isometry group of the manifold W(M) with the Riemannian metric of the fibre bundle associated to the Finslerian metric. Let us suppose that (p (X, X) in (9.13) is negative definite. It then follows that X vanishes so that the dimension of the isometry group is zero. Therefore it is a finite group. If cp(X, X) in (9.13) is negative semi-definite it follows that X is of covariant derivation of horizontal type zero. Theorem 5. If the quadratic form cp(X, X) is negative definite on W(M) then the isometry group of the compact Finslerian manifold without boundary is finite. If the form is negative semi-definite then X is of zero horizontal type covariant derivation [IbJ. If the Ricci tensor Py vanishes everywhere then by (9.10) \|/(X, X) is a divergence. Hence the formula (9.13) reduces to

Isometries and Affine Vector Fields

75

Corollary. Let (M, g) be a compact Finslerian manifold without boundary such that the second Ricci tensor Py vanishes everywhere on W(M). IfRjjX'X' is negative definite everywhere on W(M) then the isometry group of this manifold is finite. If this form is negative semi-definite then X is of zero horizontal covariant derivation [lb].. 10. Infinitesimal Affine Transformations^] Let exp(uX) be the local 1-parameter group of local transformations of M, generated by X and its extended group exp(uX). Let co be a regular linear connection V(M). We call infinitesimal affine transformation an infinitesimal transformation X on M such that its extended group exp(uX) leaves the linear connection co invariant. For it to be so it is necessary and sufficient that (10.1)

L(X)co = 0

In virtue of (5.7) the above relation gives (10.2) (10.3)

L(X)f> = 0 L(X)T' jk = 0

Conversely, if the relations (10.2) and (10.3) are satisfied for all X, then after (5.8) and (5.9), we have (10.1) and X defines an infinitesimal affine transformation. For an infinitesimal transformation to define an affine infinitesimal transformation it is necessary and sufficient that it leaves invariant the coefficients of the connection. After (5.7) the relations (10.2) and (10.3) imply

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Initiation to Global Finslerian Geometry

(10.4) V(AX)'j = i(X)Q) This is equivalent to (10.5)

Vk (AXVj = X h R V - i

h

(10.6)

V; (AX)1) = X h P i j h k + i h Q i j h k

PVh

•

where AX and X are defined respectively by (6.2) and (6.4). Conversely let us suppose that (10.4) is satisfied for all X, we then have, after (5.11) and (5.12), the relations (10.2) and (10.3) so that X is an affine infinitesimal transformation. We are thus, in the following, led to envisage the infinitesimal transformations that leave invariant the coefficients of the first kind F/* of the connection. Such infinitesimal transformation will be called a partial affine infinitesimal transformation. In virtue of (5.9) and (5.10) , a partial infinitesimal transformation can be characterized by the system (10.5). Thus we have the Theorem. In order that an infinitesimal transformation X defines an affine infinitesimal transformation (respectively partial) for the connection co, it is necessary and sufficient that it satisfies (10.4)[respectively (10.5)].[l] It is clear that a partial affine infinitesimal transformation leaves invariant the geodesies of the linear connection co. If X defines an affine infinitesimal transformation we have (10.7)

L(X)R i jk , = 0 ,

L(X)Pijkl = 0, L(X)Q!jkl = 0.

11. Affine Infinitesimal Transformations and Covariant Derivations Let t be a tensor of type R(G) with values in a vector space N and co a regular linear connection on V(M). We will denote by R (G) the representation of the Lie algebra G of G induced by R .

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77

Let U be a neighbourhood of V(M) consisting of ordinary points and a local section invariant under X over U . The tensor t and the linear connection define on U respectively a 0-form tn with values in N, a 1-form

<%with values in G.

The absolute

differential of t is a 1-form V tn with values in N

whence L(X) V% =dL(X) tn + R(L(X)

con)tn

From this one deduces L(X) V% -VL(X) tn = Hence we have at every point of V(M) (11.1)

L(X) Vt - VL(X)t = R (L(X)ra)t

In particular, for a vector field in the large sense Y, we have L(X) Vk Y1 - Vk L(X)Yj = L(X) f )k YJ - L(X) f rok Sr Y L(X) Vj Yj -VjL(X) Yj = L(X) T"k Yj -L(X) Trok & Y' From the formulas (11.2) and (11.3) we have the following

Theorem. In order that an infinitesimal transformation X defines an affine infinitesimal transformation (respectively partial) for the connection co, it is necessary and sufficient that the covariant derivatives of the two types Vk and V'k (respectively of type Vk) of a tensor in the large sense commutes with its Lie derivative by X [1].

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Initiation to Global Finslerian Geometry 12. The Group Kz(L)

Let L be the Lie algebra of affine infinitesimal transformations of V(M) for the connection co. In the preceding paragraphs we have seen how to every vector field X € L on M, we can let correspond a vector field X (X, X) on V(M) which is a lift of X. We denote by I the lift of L over V(M). On L we can define a Lie algebra structure by the usual bracket operation. The bracket of two elements of L can be projected by the map p. Let Az(L) be the Lie algebra of endomorphisms of T^z corresponding to the elements of L. Az (L) is the Lie algebra of a connected group Kz(L) of linear transformations of Tpz. We know that at a point z e V(M) the Lie algebra of infinitesimal holonomy can be generated by the elements(see [1], §11 chapter I) (12.1)

riJ = V...VR 1 jk ,(Z 1 ...Z ? )u k ,u 1 2

(12.2)

Pj = V... VP'jki(Zi...Z?)uk, v!2

(12.3)

qij =

V .

i

where Z\...Zq, u $xi\, ii $ and «2 (U2, ui) are the local vector fields in the neighbourhood of z e V(M). We denote byCT^Zthe Lie algebra of infinitesimal holonomy at z e V(M), by r'z, r/z, g'z those generated by r, p and q respectively. If X e L, the Lie derivative with respect to X commutes with the covariant derivative and X leaves invariant the three curvature tensors, thus L(X)r, L(X)p, and L(X)q belong respectively to r'z, r/z, Q'Z. On the other hand, in virtue of (6.7), the Lie derivative of r'jby X ( for example) becomes [Az(X),r]iJ The left hand side as well as the first term of the right hand side belong to r'z, whence [A z (X),r'z]e_r'z Similarly [AZ(X), r/z ] c r/z and [AZ(X), ^ z ] e q'z

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79

&z being the sum of r'z, rVz, g'z, [AZ(X), g^z ]
algebra

of

affine

infinitesimal

1°. Let 9z be the tangent space to V(M) at z Definition. The Lie algebra L of affine infinitesimal transformations is said to be transitive on V(M) if at every point z e V(M) the subspace of Oz generated by the X z for X e L coincides with Oz. In this paragraph we will suppose that L is transitive on V(M) . Let z e V(M) and let U and V be two neighborhoods of z. From the definition above it follows that for every point z e U there exists an affine transformation of V on a neighborhood of z that maps z on z . From this we deduce that over U the groups of infinitesimal holonomies have the same dimension, hence z is regular[27] for the infinitesimal holonomy. Thus all the points of V(M) being regular for the infinitesimal holonomy, the group of infinitesimal holonomy &_z at every point of z e V(M) coincides the restricted holonomy group CTZ 2°. Let X and Y be two elements of L, from (10.4) we have VAX(f ) = Q(X, From (6.15) it follows

Y)

Q( X, Y ) = [AX, AY] - A[X, Y] whence, for every pair X, Y e L, we have

80

Initiation to Global Finslerian Geometry Q(X,

F ) e Az(L)

Let Zi e L , from (6.7) we have (13.1) L ( Z , ) n ( A \ F ) = [AZi,Q)] Let us put

[Z h X]

= 4,,

F

[Z,,Y]=TII

Since Zi belongs to L, we have L(Z,) 1 h= Do L(Z,) Xh + Thok L(Zi) X k whence

The right hand side is none other than $, whence

L(Zx)Xh=tf Similarly

L(Zi) P = ffi

Thus the left hand side of (13.3) can be written

Whence, the left hand side is an element of Az(L), from the relation (13.3) it follows that VQ(X, F ) ( Z i ) belongs to Az(L), on repeating this procedure we see therefore that the vector space Cq ( q= 0, 1, 2, ...), generated by elements corresponding to the qth covariant derivation (q= m+n, m-times horizontal derivation and n-times vertical derivation ) of the curvature tensors at z eV(M) belongs to Az(L) (See [1] page 20). Thus the restricted holonomy group az is contained in the connected group Kz(L). Taking into the preceding theorem we have the

Isometries and Affine Vector Fields

81

Theorem. If the Lie algebra L of affine infinitesimal transformations is transitive on V(M) at every point z e V(M) we have ozaKz(L)

^No(oz)

where No(oz) is the connected normaliser of the restricted holonomy group oz in the group of linear transformations of Tpz [!]•

In the following L will be called basic Lie algebra of affine infinitesimal transformations.

14. The Lie Algebra L 1°. Let L be the basic Lie algebra of affine infinitesimal transformations and Az(L) the Lie algebra of endomorphisms of Tpz corresponding to the elements of L. The following lemma follows from the system of linear differential equations (10.4) Lemma. Let X and Ybe two elements of L and AX and AY be the corresponding endomorphisms of Tpz, if at a point xo= p ZQ we have Xx 0 = Y x 0 ,

Azo(X) = Az 0 (Y)

Then X and Y coincide on the entire M After this lemma, the algebra L is finite dimensional. 2°. Let G be the vector space defined by the direct sum (14.1)

G = Az(L) + Tpz

On G we define the following composition law : (14.2) [A,,A 2 ] = A 1 A 2 -A 2 A 1 [A, s] = - [s, A] = As

82

Initiation to Global Finslerian Geometry [si, s2] = - S ( ? i , 's 2) - Q.(J xJ 2)

where Ai, A2 e Az(L), s e Tpz, A eAz(L), and s\,s\ e Tpz and I is as usual defined by the pair ? = (s, s = MDo s) The composition law (14.2) does not in general define on G the structure of a Lie algebra since the Jacobi identity is not satisfied in general. Let us consider the map which to every X e L lets correspond Iz:XeL^-Az(X)-Xx where Az(X) e Az(L) and Xx is the value of X at x = pz. Let G = Iz(L); it is clear that G is a vector subspace of G . We take up the study of the properties of Iz, thus the structure of G. First of all Iz is a linear and bijective map of L on G; in fact the linearity of Iz follows immediately from the definition ; let AX - Xx be an element of G, we say that to it corresponds at most one X e L such that Iz(X) = Az(X) - Xx , x = pz If Y is another element of L to which corresponds Az(X) - Xx , we must have Iz(Y) = Az(X) - Xx

(x = pz)

From the definition of Iz(Y) we then have Az(Y) - Yx = Az(X) - Xx,

(x = pz)

Az(Y) = Az(X),

(x=pz)

whence Yx = Xx

In virtue of the above lemma, X and Y coincide on the entire M. Thus Iz is a linear and bijective map. Let us show that

Isometries and andAffine Affine Vector Fields

83 83

Iz is a homomorphism of L on G. Since it is linear it suffices to show that it preserves the bracket . Let X and Y be two elements of L, we have (14.3)

Iz([X, Y]) = Az [X, Y] - [X, Y]x

(x= pz)

On the other hand, the last term of the right hand side can be easily put in the form [X, Y]x = Az(X)Yx - Az(Y) Xx + £(JT, Y) On putting this expression in (14.3) and on substituting the value of Az[X, Y] taken from (14.2), we obtain , Y]) = [Az(X), Az(Y)] - Az(X)Yx +Az(Y)Xx - I ( X, Y) - Q( X, Y) In conformity with (14.2) we then have Iz([X, Y]) = [Az(X) - Xx, Az(Y) -Yx] = [Iz(X), Iz(Y)], where x = pz Thus Iz defines an isomorphism of L with the Lie-subalgebra G of G . So we have the Theorem. Let L be the basic Lie algebra ofaffine infinitesimal transformations and G the vector space defined by the direct sum G = Az(L) + Tpz where Tpz is the tangent vector space to M at x = pz andAz(L) the Lie algebra of endomorphisms of Tpz corresponding to the elements of L. Let suppose that G is endowed with the composition law defined by (14.2); Then L is isomorphic to the Lie algebra G = Iz(L) czGby the map defined by [I] IZ:XGL->AZ(X)-XX

84

Initiation to Global Finslerian Geometry 15 The case of Finslerian manifolds

In this section we suppose that M is given the structure of a Finslerian manifold. Let g be the metric tensor of this manifold and co the corresponding Finslerian connection. We know that the torsion tensor S of this connection is zero; if in the formula (3.1), co represents the Finslerian connection, then the linear connection W associated to co coincides with co, the matrix L as well its inverse M reduce to identity (local coordinates) and the endomorphism A(X) defined by (6.2) can be written (A(X))iJ = -(V j X 1 + T 1 jh V 0 X h )

(15.1)

where V represents the covariant derivative of the Finslerian connection. The results obtained in the preceding paragraphs relative to affine infinitesimal transformations of a general regular linear connection are valid for the Finslerian connection. Let X be an infinitesimal transformation of M. In virtue of (6.6) and (6.7) the Lie derivative of the metric tensor g by X has the form (15.2) where (15.3) Let us put (15.4)

L(X)gij = -(A(X)ij + A(X)ji) A(X)ij = -(VjXi + T,jhV0Xh) t(X)ij = -(A(X),J + A(X)JI)

For the infinitesimal transformation X we have r jk

+ grj L(X) f

r ik

.5)

Vk t(X)ij = g t L(X) f

.6)

Vj t(X)y = gir L(X) Trjk + grj L(X) T rik

+ 2 T ljr L(X) f

From the above formulas we obtain i h (15.7) L(X) f ''Jkk == l gl i h [Vjt(X) hk + V k t(X) jh - Vht(X)jk] - (T"r L(X) f

r ok

+ T"kr L(X) f

r 0J

- T jkr g ih L(X) f r oh )

r ok

Vector Fields Isometries and Affine Vector (15.8)

85 85

L(X) T'jk = 1 glh [ V; t(X)hk + Vj t(X)jh - V% t(X)jk]

If X defines an affine infinitesimal transformation (respectively partial) for the Finslerian connection , after (15.5) and (15.6) it is clear that the covariant derivative of two types Vk and Vj (respectively of _ the type Vk) of the tensor t(X) is zero. Conversely let us suppose that covariant derivative of two types Vk and V\ (respectively of type Vk) vanishes, we show that infinitesimal transformation X defines an affine infinitesimal transformation (respectively partial) for the Finslerian connection. In fact, if the covariant derivative of type Vk of the tensor t(X) vanishes, the relation (15.7) becomes .9)

L(X) f "jk = - (Tjr L(X) f

r ok+

Tjkr L(X) f

r oj

-T j k r g i h L(X)f r o h ) On multiplying the two sides by v" and taking into account the homogeneity of the torsion tensor we get (15.10)

L(X)r i 0 k --T 1 k r L(X)f r oo

On multiplying this relation by vk we get (15.11) L(X)f' o o =0 From (15.10) we get L(X)f J o k =0 Thus the relation (15.9) gives us L(X)r'jk=0 where X is a partial affine infinitesimal transformation. If, in addition, the covariant derivative of type Vj of the tensor t(X) vanishes from (15.8) we then obtain

86

Initiation to Global Finslerian Geometry L(X) T'jk= 0

Thus X defines an affine infinitesimal transformation for the Finslerian connection. So get the theorem Theorem. Given a Finslerian manifold M, in order that an infinitesimal transformation X on M defines an affine infinitesimal transformation (respectively partial) it is necessary and sufficient that the covariant derivative of two types Vk and W'k (respectively of the type V/J of the tensor t(X)tj defined by (15.4) vanishes [1]. If X defines a partial affine infinitesimal transformation for the Finslerian connection of [(8.6), chap II] we obtain (15.12)

L(X)Pjoki = - ^ ^ L ( X ) f

j jk

=-L(X)Vorkl =0

16. Case of Infinitesimal Isometrics An infinitesimal isometry is evidently an affine infinitesimal transformation for the Finslerian connection. So the results of the preceding paragraphs apply to the Lie algebras of infinitesimal isometries. Let L be the basic Lie algebra of infinitesimal isometries. Let L be its lift to V(M). To every vector field X e L we can associate a skew-symmetric endomorphism A(X) of the Euclidean vector space Tpz. For the usual bracket operation these endomophisms define a Lie algebra Az(L) which is the Lie algebra of connected group Kz(L) of rotations of Tpz. On Az(L) let us introduce the scalar product defined by (a(z), P(z)) = ( 1 )<Xij P1J,

a(z), p(z) e Az(L)

Let az be the Lie algebra of the restricted holonomy group at z and Bz the ortho-complement of
Isometries and Affine Vector Fields

87

U , Sy(z) belongs to o(z)(U ), ( see theorem 1 §11 chapter I [1]), so also V Sy( Z) for all local vector fields Z (Z, Z ) on U . Consequently if for all z e U , Py e BzU , by differentiation of •A- Sjj p'-"= 0. From this we have ivSij(Z)piJ +isiJvp1-i(Z) = o the first term being zero, the second terms must be so whatever be Sy therefore according Vp lj (Z) belongs to Bz. Let X be an infinitesimal isometry and Az(X) the corresponding endomorphism. Az(X) e Az(L) can be decomposed according as (16.1)

Az(X) = S(z) + p(z), (S(z) e gz,, p(z) e Bz)

On the other hand after (10;4) we have VA(X) jl (Z) = Q u From (16.3) it follows

The left hand side as well as the first term of the right hand side belong to Bz, whence whatever be Z we have VPij(Z) = 0 Hence it follows that the covariant derivative of the form p is zero. So it is invariant under the holonomy group v|/z, in particular we have [g(z),P(z)] = 0 Thus P(z) belongs to connected centralizer Co(crz) of the restricted holonomy group az in the group of rotations of Tpz. After (16.3)

88 88

to Global Global Finslerian Geometry Initiation to

it follows that Kz(L) is contained in the connected normalizer No(az) of the restricted holonomy group az in the group of rotations of Tpz. If the Finslerian manifold does not admit 2-form with zero covariant derivative, the group Kz(L) is contained in az ; if in addition the Lie algebra of infinitesimal isometries is transitive on V(M) then the group Kz(L) coincides with az, so we get the theorem. Theorem. Let M be a Finslerian manifold and V(M) the manifold of non-zero tangent vectors to M Let L be the Lie algebra on Mof infinitesimal isometries and L its lift on V(M). If the Finslerian manifold does not have 2-forms with vanishing covariant derivative , the group Kz(L) is contained in the restricted holonomy group az; if in addition L is transitive on V(M), Kz(L) coincides with az [I].

89

CHAPTER IV GEOMETRY OF GENERALIZED EINSTEIN MANIFOLDS (abstract). Let (M, g) be a compact Finslerian manifold of dimension n, and let W(M) be the fibre bundle of unitary tangent vectors to M. Let cp(x, v) be a function on W (M). We now introduce the Laplacian Acp = Acp(x, v) of cp(x ,v). Now Acp decomposes into two parts, Acp = A (p + A cp. We call A cp the horizontal Laplacian, and A cp the vertical Laplacian. To the function cp(x, v) on W(M) we associate a symmetric 2-tensor A,j((p), and we establish a formula linking the square of A^cp) to A cp and a quadratic form O(cp*, 0. In case M is simply connected and X= n.k, then (M, g) is homeomorphic to an n-sphere. This is a generalization of a theorem of Lichnerowicz-Obata in the Riemannian case. The rest of the work is devoted to the deformation of a Finslerian metric. Let F°(M, gt) be a deformation of (M, g) preserving the volume of W(M). We prove that the critical points g0 e F°(gt) of the integral l(gt) on W(M) of a certain Finslerian scalar curvature define a Generalized Einstein manifold [6]. We evaluate the second variational of I(gt) at the critical point g0 and show that in certain cases and for an infinitesimal conformal deformation, we have I"(go) > 0.

I. Comparison Theorem 1. The Laplacian defined on the Unitary Tangent Fibre Bundle and the Finslerian Curvature A. Let cp be a differentiable function on W(M). With respect to a local chart of W(M) its derivative is (1.1)

dcp = Di(pdxj+ d ;( p. Dv',

Dj

- T £,. 5;q> = q>i

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Initiation to Global Finslerian Geometry

where Dj is the covariant derivation in the Berwald connection. Then by (7.6), (7.9) ch. III)the Laplacian of cp on W(M) decomposes (1.2) (1.3)

Acp= Acp+ Acp Acp^DiDjip,

Acp = - F 2 g 1 J 5 ; a * c p ,

(3)=—y)

ov We call A is the horizontal Laplacian and A the vertical Laplacian. Let us consider the symmetric tensor Ay defined by 2 Aij((p) = Di(pJ + Dj(pi+-pg ij , n

(1.4)

(
We choose p in such a way that the trace of A vanishes. g1JAij = 2(g1J Dj cpj + p) = 0,

(1.5)

p = A cp

Let us put

Lemma 1. Let (M, g) be a Finslerian manifold of dimension n; we have .6)-(A,A) = ( 1 - - )(Acp, Acp)-O(cp* #>) + DivonW(M) 2 n where ®(cp* (p) is a quadratic form in q>j and

Geometry of Generalized Einstein Manifolds

91

Proof. We have (1.9)

(A, A) = | Aij Ay = g ' V Di cpj Akl = glkD, (Akl (p1) - glk(p'DiAkl -2D0 T/VAki

Now (1.10)

DiAki = D,Dk(p, + DjD, cpk + - D i P g k l - - p D0Tik, n n

whence (1.11)

- (p'gik DiAki = -
First, the first term of the right hand side becomes (1.12)

-(p'glk Dii (Dkkq>,-D,<j>k)= - g ^ D . t p C p ^ ) ] +g lk g jl D i (p J (D k (p,-D,(p k ) = -

Now the last term of the right hand side takes the form (1.14)

- 25;cp k Hroll(p'gik = - 2 D k
Finally let us denote

92

Initiation to Global Finslerian Geometry

(1.15)
On taking into account (1.15), (1.14), (1.13) and (1.12) the right hand side of (1.11) is completely calculated, and on putting the result thus obtained in (1.9) and on dividing by 2, we obtain the lemma. B. Let us consider a 1-form C, = Q(x) dx1 on W(M). We have Lemma 2. Let (M, g) be a Finslerian manifold of dimension n. On W(M) we have (1.16)

H *(C, Q = n ( H( ;, u)u, Q + Div on W(M)

Proof. First let us write a Bianchi identity containing the vertical derivation of H ( see chapter I formula (10.5) adapted to the Berwald connection) (1.17)

a ; H'jk, + D, G\km - DkGVm = 0

where H and G are the two curvature tensors of the Berwald connection . Whence (1.18)

V^HV^O

Let us consider the 1-form belonging to Ai(W) defined by its components

whence, on taking into account (1.18), (1.19) F glk d\ Zs = - 2 Tjmk Cn£r Hroji glk + H.(£, Q - (H(C, u)u, Q

Geometry of Generalized Einstein Manifolds

93

Now the first term of the right hand side vanishes, in virtue of the property of the symmetry of the torsion tensor T. On the other hand, let us put 7

= 7 -u-^-

7 = V7

Now the Z i are the components of a vertical 1-form on W(M). We thus obtain, after (1.19) the formula (1.16). Lemma 3. Let f be a differentiable function on M; let us put f t= Df. We have g1J fjfj - ntfu1)2 = Div on W(M)

(1.20)

This lemma is proved easily, by using the formula (7.6 chap.III). C. In this paragraph We suppose that the eigenvalue of the Laplacian A is zero, cp is therefore independent of the direction. The quadratic form O(cp*,cz>) becomes H*(cp*, cp*) and on identifying C, with dcp and on using the lemma 2 we obtain (1.21) - (A, A) = (1 - J-) (Acp, Acp) - n(H(cp*, u) u, cp*) + Div on W(M) where cp* is the gradient associated to dcp. If A = 0, after (1.21) it follows then is positive. We suppose in what follows that (H(cp*u)u, cp*) is positive. Moreover there exists a constant k > 0 such that F- 2 H' 0 J 0 >kh;.

(1.22) where (h', = 8 ' J

- u'u,, k - constant> 0 and 8 the Kronecker J

J

symbol.). Since HJOJO is symmetric ([1] as a consequence of the first Bianchi identity), and positive, we then have

94 (1.23)

Initiation to Global Finslerian Geometry n(H(cp*, u)u, cp*) > n.k^Vi - {^)2],

(q* = D.cp)

On taking lemma 3 into account we have : (1.24) n (H(q>*, u)u, cp*) > (n-l)k cpVi + Div on W(M) Now the last term becomes: kcp'cpi - kg1J Dj(p (pi = k. ( A(p, cp) + Div on W(M) On putting it in (1.24) we obtain n(H(q>*, u) u, (n-l)k ( A < [> -nk >] 2 n Acp being zero, (p is a function on M, A (p depends on v in general. We put A cp =A, cp where A, is a function defined on W. We have 0 < - < ?— 2

n

[

XCk -nk) (p2ri

iV(M)

Now we do not always have (1.25)

X
If A, = nk, then A = 0. That is to say (1.26)

Djcpj + k 9 gy = 0

(k = constant > 0), cpj = DjCp

Geometry of Generalized Einstein Manifolds

95

If, in addition, we suppose that M is simply connected, after a result shown in ([3]) the existence of a function (p on M satisfying a differential equation of the second order (1.26) implies that (M, g) is homeomorphic to an n-sphere. Similarly on supposing that H*(cp*, cp*) defined by (1.8) is positive and satisfies (1.27)

H*(cp*, cp*)>(n-l)k cp*2

with k = constant > 0.

On using the formula (1.21) and on reasoning as we have done above we obtain the following result. Theorem. Let (M, g) a simply connected compact Finslerian manifold, without boundary, of dimension n. We suppose the curvature tensor H of the Berwald conneciton satisfies the inequality (1.22) or the inequality (1.27) where k is positive constant. Moreover we suppose that the vertical Laplacian of

nk. In case of equality, (M, g) is an n-sphere. ( see [28] and [33]) 2.Case of a manifold with constant sectional curvature Let us suppose (M, g) is manifold with positive constant sectional curvature. H'jki = k( 8'kgji - 5'igjk) (k = constant > 0) And to simplify let k — 1. Then we have H*(cp*,cp*) = (n-1) g1J cpi cpj = (n-1) (Acp,
96

Initiation to Global Finslerian Geometry

= - | F 2 gij ((p*, (p) defined by (1.7) becomes ®(cp*; ) = (n-l) (Acp, cp) —(A
u

Since M is compact and without boundary we have : <7(A, u ) cp r\ (M)

First if |o, = 0 and l = n w e are in the hypothesis of the preceding theorem. From the polynomial of the second degree a it follows that if \i > ——-, 6

then a > 0, and if

_ n(n-\) n 2 V7—> <*-(/---)> o z we have thus an estimate for the eigenvalue of the vertical Laplacian and ^

' > (n4)

Geometry of Generalized Einstein Manifolds

97

Thus the study of the quadratic form 0(
98 II

Initiation to Global Finslerian Geometry Deformation of the Finslerian metric. Generalized Einstein Manifolds

1. Fundamental lemma Let t e [-s, E] where e is sufficiently small > 0. By the deformation of a Finslerian metric we mean a 1 -parameter family of this metric gij(x, v, t). For such a metric co = Uj dx1, the volume element as well as the connections and curvatures attached to g depend on t. The derivative of the volume element r\ of W(M) is given by the following lemma proved in ([5]pp. 345-346) Lemma 1. The first variational of the volume element of W(M) defined by (§7.chap III) of the fibre bundle of unitary tangent vectors to a Finslerian manifold (M, g) is given by [5]: (1.1) where the notation' denotes the derivation with respect to t. Proof. First of all, we have co, = (eFt/SvVx1,

co' = (SF'/SvVx1, (Sr = dj)

This derivative commutes with the differentiation d so that from the expression of the volume element (j) defined in (Chap III, §7) we have (*)

4>' = co'A(dco)""1 + (n-l)coA(dco)n"2Adcof

By a simple calculation from (1.1, Chap II) we obtain 5F75V1 = g'ir ur- (F7F) u, co'=g'ir urdx' -(F7F) co, Let us denote by 9=Vv, (3=Vu, we have

(u = F"1 v) where co = Uj dx1

Geometry of Generalized Einstein Manifolds Pj = F 1 ( 8 i k - u j u k ) e k ,

99

(UjPJ) = O

From the above relation we obtain by differentiation dco'= (dV/ Sv'dx1') d x W + F (52F75v'8vi)(3jAdxi where(5/5x', 5/5vJ)denote the Pfaffian derivatives with respect to (dx j , 9") define a basis of TZV(M) at z e V(M). The first term of the right hand side is a 2-form in dx, by putting it in (*) it cancels the second term. The coefficients of the second term are given by F ^ F V S v W ) = g'y - (F7F)g jj - g'jr uru { - g',ruruj + 3 F'/Fu i Uj Taking into account the above relation the derivative of § can be written (**) (|)'= -n (F/F)4> + g'iru'dxUcdco)11"1 + (n-l)coA(dco)n-2Ag'ijpJAdxi where

(F7F)= 1 g'ijuU

To evaluate the last two terms of the right hand side of the relation (**) we take an orthonormal frame (ej) (i= 1, ...n) at x e M such that u= en, we get un = 1, u a = 0, p n = 0, p a = ooan ( a = 1, ...n-1) where coy is the Finslerian connection. Thus the last term of the right hand side of (**) is g la g'i a <> | , and the last but one term is m equal to g g'jn <j), thus their sum is g 'Vij <)>• Dividing the two sides of (**) by (n-1)! we get the lemma. Compact Case Fundamental Lemma 11. Let (M, g) be a Finslerian manifold of dimension n &2. Let Wbe a differentiate function, homogeneous of degree zero in v, defined on W(M) and ty = (gy)' then we have [6]

100 100

Initiation to Global Finslerian Geometry

(1.2) T trace t - n ¥.t(u, u)+

F2

•,

>

(tJ-g1' trace t)d) ' *¥ = Div on W(M)

where t(u, u) =tjj u1 u* and d) =—dvJ Proof Let Y be a co-vector field on W(M) components (1.3)

defined by its

i ^ Y j - U j Y o F - 1 where Yj = F-1 M/toj, (f 0 = v J 7j = 0)

where o denotes the contracted multiplication by v. Y defines a vertical 1-form; after (7.6, chap III) we have (1.4)

- 8 Y = F g11 d'j Yi = \\i trace t- n v|/.t(u, u) + t'o d) \\i

We are going to calculate the last term of the right hand side. For this, let us consider the co-vector field Z defined by (1.5) Z k = Z k - u k Z o F - \ Zk =Ft 1 k d;i|/ We have (1.6) F g"k d'j Zk = t'od] v)/+F2(tjl-trace t g*1) d) } vj/ + F2 g"1 dr (trace t d;\\i) Also, (1.7) FgJ' 5- Z, = F gj' d'j Z, - (n-1) t'o d] v|/ =F2(tjl-gjl trace t)5* } v|/-(n-2)t'o5' \|/ + F2 g"1 5* (trace t &, \|/) The last term of the right hand side is a vertical divergence after (7.6 chap. III). So also is the left hand side. On putting the expression t'05*i|/ taken from (1.7) in (1.4) we find the formula. Let suppose that (M, g) is compact and without boundary, by integration on W(M) we get

Geometry of Generalized Einstein Manifolds (L8)

101

n

= f [MI trace t + (tjl - gjl trace t ) 3 * ; vi/lri iv(M) LY («~2) Let us suppose \j/ = cp a differentiable function on M, independent of the direction. From (1.8) we have (1.9)

f

O trace t TI= n f

O t(u, u) ri

Let us put O = 1 and use (1.1) we find (1.10)

(vol W(M))' = J^ If = —

[trace t - - t(u, U)]TI

wf trace t r\- — I

t(u, u) ri

2. Variations of scalar curvatures Let U(x') be a local chart of M and p"'(U) (xj, v') the local chart induced on p"'(U). A 1-parameter family of Finslerian connections is represented ([1]) (2.1)

oo'jIp-Ku). = f i j k ( x , v , t ) d x k + T' j k (x,v,t)Vv k

where V denotes the Finslerian covariant derivation associated to the 1-parameter family of Finslerian metrics. The coefficients of the Finslerian connection are determined by (5.4 chap II) are written explicitly : (2.2)

and (2.3)

102 102

Initiation to Global Finslerian Geometry

On deriving the relation (2.2) with respect to t we find (2.4)

-(T i j S G l S k + T i k r G i r j -T k j s g i r G l S r )

* where t y- = g'ij, and G j = Y ' o j. Let us multiply the two sides of (2.4) by V , on taking into account of the property of the torsion tensor we have :

(2.5) ( f 'oky = G- v = i (v k t'o + v 0 tjk - v j tok) - 2 r k r G- r Let us multiply this relation by vk :

(2.6)

(f'oo)1 = 2G' i =V o t i o -^V i t o o

Let now n'j be the matrix of 1-form of 1-parameter family of Berwald connections associated to gt defined by : (2.7)

n) I P.,(U). = G j jk (x, v, t) dxk

Let H and G two curvature tensors of this connection. derivative with respect to t the tensor H becomes ( see [5]) (2.8)

H' jki = D k G' ji - Di G' j k + Gj r k G' ri - Gj r | G

The

r k

where D is the co variant derivative with respect the 1 -parmeter family of the Berwald connection. On the other hand the tensor H is related to the curvature tensor R of the Finslerian connection by ([3 p.56]):

Geometry Manifolds Geometry of Generalized Generalized Einstein Einstein Manifolds

103 103

(2.9) Rjki = HVi +TV Rroki+ViVoT'jk -VkVoT'j, +V0T'IrV0Trjk - VoT'krVoT'ji Let us denote by Ry = Rr;rj and Hy = Hrjrj the corresponding Ricci tensors ; from (2.9) it follows, (2.10)

Rij(x, v) v1 v> = Hjj (x, v) v V = H(v, v)

We call the Ricci directional curvature the expression H(u, u) = R(u, u) ( see [5] p. 350) we have Lemma 3. Let (M, gt) be a deformation of a Finslerian manifold we have the formula (2.11) H'(u, u) = H'ij u1 u" = x t(u, u) + div on W(M) with (2.12) x = g i j ( D i D o T J + d ; D o D o T J ) , t(u, u) = tyu1 u> Proof. The derivative of H(v, v) with respect to t is obtained by means of the formula (2.8) (2.13)

H'(u, u) = H'ij u' u j = 2[Vj (F '2 G'')

-F' 2 G' i V o Ti]-V o (F" 2 G' i i) = Div on W(M) + 4 F ~l G" V0Tj where V denotes the covariant derivative in the Finslerian connection. Let us calculate the last term of the right hand side. In virtue of (2.6) and (7.9 chap III), we have

104 104

Initiation to Global Finslerian Geometry

(2.14) 4 F "2 G" V0Ti = 2F"2 (Votjo - | Vjt00)V 0Tj = 2V0 (F-'t'oVoTO^F"2 t'oVoVoT + t(u,u)[V i V 0 T 1 -V 0 T,V 0 T i ] = - 2 F-2 t'oVoVoT, + t(u, u) gij DjDoTj + Div on W(M)

Now the first term of the right hand side becomes, in virtue of (7.6 Chap III): - 2 F"2 t'oVoVoT, = F"2gij 5- (tooVoVoTj) + t(u, u) gij 5- (D0D0Tj) Taking into account this relation and on putting (2.14) in (2.13) one finds the lemma. Let / / j k be the tensor defined : (2.15)

H jk = | ( Hjk + Hkj + vr d) Hkr)

where Hjk is the Ricci tensor of the Berwald connection. We have Lemma 4. Let (M, gt) be a deformation of a Finslerian manifold of dimension n : we have g"k H 'jk = n T. t(u, u) + Div on W(M) where T is defined by (2.12). Proof. With the preceding notations we have H'(v, v) = H'rs vr vs = F2 H'rsur us = F2 H'(u, u), (u = — ) F Whence by vertical derivation d) H'(v, v) = 2 Vj H'(u, u) + F 2 d) H'(u, u) A second vertical derivation gives us :

Geometry of Generalized Einstein Manifolds (2.16)

105

/ / > = - 5v \ H'(v, v) = gjkH'(u, u) + Vj d\ H'(u, u) + vkdvH'(u,u)+iF2dv;H'(u,u)

whence (2.17)

gjk H)k

=nH'(u,u)+|Fg i k a' j ,.[F5;H 1 (u,u)]

The last term of the right hand side is a divergence after (7.6, chapter III). On using the preceding lemma we find the result looked for. 3. Generalized Einstein manifolds A. In the following we suppose that the Finslerian manifold is compact and without boundary. Let F(gt) be a 1 -parameter family of Finslerian metric. We denote by F (gt) the sub-family of Finslerian metrics such that for t e [-s, s] the volume of fibre bundle of unitary tangent vectors corresponding to gt is equal to 1. We look for gt e F° (gt) such that the integral I(gt) is an extremum. (3.1) with (3.2)

1(6)= f

rit=l

iv(M)

where//jk is defined by (2.15), on taking into account lemma 4, the derivative of H t is (3.3)

(Ht)' = (g* / / j k ) ' = - tjk Hik + gJk i/' jk

Thus in virtue of (1.1) the derivative of Ht r\t becomes

106 106

Initiation to to Global Finslerian Geometry

(3.4) ( Ht -n 01 = [- tjk Hjk + Ht tjk gjk + n v|rt(u, u)] ri + Div on W(M) where (3.5)

V|/ = T - —

and T is defined by (2.12) On replacing in (3.4) the expression n \|/ t(u, u) drawn from the formula (1.2) of lemma 11 we obtain finally (3.6)

(Ht r|t ) ' = - A jk tjk TI + Div on W(M)

where we have put: (3.7) ^jk=^jk-gjk(^+ H)

n-2

F g

j k

g S ; ; y F d y n-2

k

y

M being compact and without boundary , on integrating on W(M) we get

A point g0 G F°(gt) (t= 0) is critical when I'(go)= 0. So at the critical point go the tensor ^4jkis L2 orthogonal to tjk. Since the volume of W(M) is supposed to equal to one, after (1.10) we have (3.9) = f

trace t . ri = 0 and = f

t(u, u) r| = 0

Thus the tensors g and u®u are L2 orthogonal to t. If A is a linear combination of g and u®u with constant coefficients at a point g0 e F°(gt) (t= 0), that is to say if

Geometry Geometry of ofGeneralized Generalized Einstein Manifolds (3.9)'.

107 107

Aik — cgjk + buj uk where c and b are constants,

then from (3.8), (3.9) and from the above relation it follows that go is a critical point of I(gt) at t=0. But A is non-decomposable. We now show that b = 0. In fact, on deriving vertically the above relation, and taking into account the expression of A defined by (3.9)', we obtain gik v J d* Aik = - J - F 2 g*kd- \ V = (n-l)b, n-2

(n*2)

Now the second term is a divergence on W(M). On integrating on W(M) we have b = 0. Thus we have (3.10) Aik=Hjk - (y + //)g jk + - i - F2 gjk 4 - - ^ - F 2 dyk v/ = C gjk n-2 n-2 where (3.11)

*=g " a ; ; v .

v=T - y

After (3.10) the vertical derivative of Cgjk is completely symmetric, so also the vertical derivative of Aik that is to say d' A j k . Thus on writing the equality d] A j k = 8* Aki , and on suppressing the common elements of this equality we find (3.12) - 2 V, gjk O + - F 2 gjk d-,O 2vi 5v; VJ/ «-2 «-2 «-2 2 =-5- (v|/+i/)gkl+-i^2vjgk|O+-i^F gk , a- o--^2vjaj ;H/ -d;(i|/+/Ogjk+

Let us multiply the two sides of (3.12) by gJk we obtain (3.13) (\-n)dl(\\i+ H) + 2\lO+—F2 a; O - — ^ - 5} v|/= 0 n-2 «-2

108 108

Initiation to Global Finslerian Geometry

where \\i and H are homogeneous of degree zero in v while O is homogeneous of degree -2 in v; on multiplying (3.13) by v we obtain (3.14)

— ! _ F

n-2

2

O = 0,

O = grs d;;i|/ = 0 , n * 2

Let us multiply the relation O = 0 by F f w e get o = F 2 g r s d;(y a-v|/) - F 2 g rs 5;i|/5; v|/ After (7.6 chap III) the first term of the right hand side is a divergence, thus by integration on W(M) we obtain : (3.15) whence (3.16) So from (3.13) it follows (3.17)

&,

d'H=0

On taking into account the expression \\i defined by (3.5), the relation (3.10) becomes (3-18) On multiplying the above relation by g'k and on dividing by n we obtain (3.19)

- # - ( T + -

n From (3.18) and (3.19) it follows

2

Geometry of Generalized Einstein Manifolds

(3.20)

109

//Jk=-//gjk n

where H is independent of the direction. Definition . A Finslerian manifold is called a generalized Einstein manifold (G.E.M) if the Ricci directional curvature is independent of the direction, that is to say ( [5], [6]) (3.21)

/ / j k ( x , v ) = C(x)g jk (x,v)

where C(x) is a function defined on M. We have proved the following theorem: Theorem. For a compact Finslerian manifold without boundary the Finslerian metric g0 e F°(gt) at the critical point (t = 0, g0 = g(0)) of the integral I(gt) defines a Generalized Einstein Manifold[6]. For a Generalized Einstein manifold His independent of the direction, and after (3.19) it is defined by

(T + C). Let us

2-n suppose that H is constant then x must be constant But x being defined by (2.12) is a divergence on W(M). Consequently x = 0. We have Corollary. If the Finslerian metric go e F°(gt) at t = 0 is critical for the integral l(gt) and defines at this point a manifold with Ricci directional curvature constant, then we have at this point ~

1 ~

~

(Hjk = — Hgjk, H = constant), ,)and n (3.22)

x = gij (DiD0 Tj + 5;D 0 D O TJ) = 0

B. Let us consider now the integral (3.23)

I,(g t )= 1(M)

Ht(u,u)Tit

110 110

Initiation to to Global Finslerian Geometry

With the condition that the volume of W(M) is constant and is equal to one we look for a gt e F°(gt) such that Ii(gt) is an extremum. To derive Ii(gt) with respect to t we have, first of all,: [H, (u, u)]1 = (HyXu1 u1 + H ^ u 1 ) ' ^ + u V ) ' ]

(3.24)

F'

1 where (u )' = u = — t(u, u)u'. In virtue of the lemma 3 the relation (3.24) becomes 1

1

[H, (u, u)]' = [x - H(u, u)] t(u, u) + Div on W(M) whence (3.25) [H(u, u)rit]'=[v|/.t(u, u) + H(u, u) trace t ]r]+ Div on W(M) where we have put (3.26)

v|/ = t - ( ^ + l ) H ( u , u )

Substituting in (3..25) the value of \\i t(u, u) drawn from the formula (1.2) we get:

(3.27)

r,(gO= l m

5 jk t jk r 1 t=2

Thus (3.28) 5 j k = [ H ( u , u ) + - - — ^ — O ] g j k + — ^ — 5v-v|/ n n(n-2) n(n-2) where (3.29)

Hence, by identical reasoning as above, at the critical point t = 0, go = g(o) e F°(gt) the integral Ii(gt), the tensor B ^ becomes

ofGeneralized Einstein Manifolds Geometry of

111 111

(3.30) F2

5 j k = [ H(u, u) + «

— O ]gjk +— — S< \ v|/ = c gjk w(«-2) n{n-2)

In fact, after (3.9), 5 j k is proportional to gjk and w. <E>M^ . As in paragraph A we show that b = 0. Let us multiply the two sides by g*k and contracting, we have

(3.31)

[H(u,ii)+---A-
J2

<> = C

n n(n-2) n (n-2) Let us multiply the two sides by gjk and subtracting from (3.30) the relation thus obtained, we have F2 „ n(n-2)

F2 ^ n (n-2)

whence, on multiplying by v* and v k we obtain (3.32)

0 = 0,

5*lP = 0.

From (3.31) and (3.26) we then have (3.33)

H ( ) C

n

n-2

^(U

with x is independent of the direction. Thus H(u, u) is independent of the direction. And g 0 is a generalized Einstein manifold. If on the other hand we suppose that H(u, u) is constant then x = 0. Thus we obtain the same conclusion. If we write after (3.30) that d] B^ = 5} 5 k | and proceed then in the same manner as the preceding paragraph.

112 112

Initiation to Global Finslerian Geometry

Theorem. The Finslerian metric g0 e F°(g t) at the critical point (t= 0, go=g(0)) of the integral Ii(gt) defines a generalized Einstein manifold. Moreover ifH(u, u) is constant, then T= 0 4. Second variational of the integral I(gt) A. In view of reducing the calculation of the second derivative of I(g t)) we are going to suppose that the trace of the torsion is invariant by deformation and prove some lemmas. Lemma 5. Let T be the derivation with respect to t e [s, e] of the torsion tensor of Finslerian connection and ty = g'y ,the following conditions are equivalent (4.1) (T'jk)T = 0 « d } tjk = 0 (TV)1 = 0 <=> g jk 5* tjk = 0 = 5} trace t

(4.2) Proof: we have -dy

gki = Tkij = gkrT r jj

Let us derive the two sides with respect to t \ d) g' k, = | d'j tki = tkr T ry + gkr (T rij)' = i 5- ( gir t rk) = trkTnj+|gir5*trk whence (4.3)

|gir5-trk =

r g k r (T l j y

From this relation we deduce the lemma. Lemma 6. Let us suppose that the trace of the torsion is invariant by deformation and W is a function on W(M) homogeneous of degree zero at v; we then have (4.4)

_F2 trace t gjl dv ] T = Div on W(M)

Geometry of Generalized Einstein Manifolds

113

This lemma follows from the formula (7.6 chap III) and (4.2). In fact, trace t is a function on M. Under the condition of the lemma 6 in particular (1.2.11) becomes nvF.t(u, u) = *F trace t + —!— F 2 1 i j d'' T+ Div on W(M) n-2 Similarly we have Lemma 7. Let (M, gt)a deformation of a Finslerian manifold we have (4.5)

(4.6)

Ft(u,u)g 1 J 5- [F5-H'(u,u)] = 2 [trace t - n t(u, u)] H'(u, u) + Div on W(M)

This lemma is obtained by applying twice the formula (7.6 chap III). B. In the preceding paragraphs we have put (4.7)

a(m,r,i) V», R'jr, + a(m,r>i) (F2 j Km + Kvm) V0Q!jrl = 0

On supposing that the trace of the torsion is invariant by deformation the first variational of I(gt) becomes

where in virtue of (4.5) the expression of A y is defined by (3.7) becomes A\j F2 (4.9) /4ij= Hirgij(\\i+ H) dy \\> n-2 where (4.10) t,j = g',j and V|/ = T - / / / 2 . At the critical point t = 0,1'(g 0 ) = 0, (M, g 0) is a generalized Einstein manifold.

114 114

Initiation to to Global Finslerian Geometry

H y = — H g jj, A jj = Cgy where C is a constant. n H, \\i and x are independent of the direction and after (3.19) we have at t= 0 (4.11)

-n

A jj being a symmetric tensor homogeneous of degree zero at v, the derivation under the integral sign of (4.8) is : (4.12) (t i j A ijTi)'=[-2t ik tj k A ij+tij ( A ij)'+ A

ij

f y + 1 « A ij(grs - | u r u s ) t rs ]

We evaluate this expression at the point t = 0; we have (4.13)

- 2 t i k t j k i i j = -2ctijtij

(4.14)

t1Jiijgrstrs =

(4.15)

- - 1 1 J A ij t(u, u) = - - c trace t. t(u, u)

c

2

The term Aij t1 y = C g ij g" y. Now the volume W(M) = 1 after (1.10) we have (4.16) By derivation we obtain at the point t = 0 (4.17) )

Aij t'ij TI = C l(M) {tijty - trace t [trace t - ^ H(u, u)] }r)

Geometry of Generalized Einstein Manifolds

115 A

From (4.12) it remains to calculate the term tIJ {A y)' at t = 0. Now at this point \\f has vertical derivative zero; by (4.9) we then have at this point: [t1J H 'ij - (\\i+ H )' trace t - (w + H )t'J t •» _

t'J3* • v|/']t= o (n-2)

From (2.16) we obtain (4.19) tij H 'ij=H'(u, u) trace t + - tj0 d] H'(u, u) + - Ftu5- [F5*. H'(u, u)] Now the trace of torsion is invariant by deformation, in virtue of (4.2) we obtain : t 'o 5* H'(u, u)=[n t(u, u)-trace t]H'(u, u) + Div on W(M) F t i j 5 " [Fd) H'(u, u)] = (n-1) [n t(u, u) - trace t]H'(u, u) + Div on W(M) Thus (4.19) becomes (4.20) t i j (#ij)'=- n[(n+2)t(u, u) - trace t]H'(u, u) + Div on W(M) 2 For the last term of the right hand side of (4.18) we have, on taking into account (4.2) (4.21) F2 tij5; * \|/' = F2 gik d] (tjkd« v|/") = (n-2) tj0 8'j v|/' + Div on W(M) =(n-2) F ^d'j (F"1tkoi)/I)+(n-2) [t(u, u)-trace t] \\i'+Di\ on W(M) = (n-2)[nt(u, u) - trace t] \|/' + Div on W(M) At the point t = 0 we have

116 116 (4.22)

Global Finslerian Geometry Initiation to Global (y+

H)t=0=-H-C n

Now \|/ = x — H, we have after (4.21) (4.23)

— — F2 tlj d] • i|/' - (\|/ + # ) ' trace t w-2 i

~

~

= -n t(u, U)T' + - nt(u, u) H ' - trace t H' + div on W(M) On taking into account (2.17), H' becomes at the point t = 0 (4.24)

W = - - H trace t + n H'(u, u) + - F glj 5* [F 5; H'(u, u)] n 2

On multiplying the two sides by — n t(u, u) and on using the formula (4.6) of the lemma 7 we obtain (4.25) - n t(u, u) H' = - - H trace t . t(u, u) + - n trace t H'(u, u) + Div on W(M) Similarly, on multiplying the two sides of H' by - trace t from the fact that the trace of the torsion remains invariant by deformation then trace t is a function on M, as well as the last term of the right hand side of (4.24) will be a divergence; we have ~ 1 ~ (4.26) -trace t H'=- H trace t - n trace t H'(u, u)+Div on W(M) n On adding (4.25) to (4.26) and on putting them in (4.23) we obtain

Geometry of Generalized Einstein Manifolds

117

(4.27)

F2 tij d; • \]i' - (\j/+ H)' trace t = -n t(u, U)T' n-2 1 ~ 1 ~ 1 2 + - H (trace t) — H trace t t(u,u)—ntrace tH'(u,u)+Div on W(M) n 2 2 From (4.22) we have (4.28)

-(y + H )t=o tijtij = (C - - H) tljtij

Thus on adding side by side the relations (4.20) and (4.27) and (4.28) we get the expression t'J( A y)' at the point t = 0 then on adding the result thus obtained to (4.13), (4.14) (4.15) and (4.17) we obtain finally : (4.29) I"(g0)= f {-H[(2(t,t) - (trace t) 2 + Jtrace t. t(u, u) ] - - [(n+2)t(u, u) - 2 trace t] H'(u, u) +n t(u,u) x'}t=ori where x = i (D,D0Tj + 5; D0D0Tj),

(t, t) - i t1J tjj

(4.30) H'(u, u) = H'ju'u1 - F"2 Vi (Vot'o - V'U ) + F"2 (Vot'o - V't00) V0T, - F'2VO [ i (Vot'i + Tj V4oo) -TjVot'o ] Formula of the second variational. Theorem. Let (M, gt) be a deformation of a compact Finslerian manifold without boundary which leaves invariant the torsion trace. The second derivative of the integral I(g,) defined by (3.1) at the critical point (t = 0, gt = g0) is given by the formula (4.29)

118 118

Initiation to Global Finslerian Geometry 5. Case of a Conformal Infinitesimal Deformation.

Let us suppose (M, g£) be a conformal infinitesimal deformation. Then we have (5-1)

tij = g1ij = 2cp(x)gij

where cp is a differentiable function defined on M. We suppose in what follows that x is everywhere zero. Therefore H is a constant. Also after the lemma 5 it follows that torsion tensor is invariant by infinitesimal conformal deformation. We are going to evaluate the expression under the integral sign. The term containing H in I"(gt) is (5.2)

2(2-n)q> 2 //

For the term containing H'(u, u) we use the expressions G'' and G' j determined by (2.5) and (2.6) and for a conformal infinitesimal deformation (5.3) -n(n-2)F "2 (pDocpo - n(n-2) (p[gIJ D, (ft + Do (Ttyi) + 2 (p'D0 TJ In virtue of lemma 3 we have (5.4)

F' 2 D o cpoq> = - ( — f + Div = - -cp'cpi + Div F n

= - cpgij Dj (pj + Div on W(M) n Now x| t=o is zero, therefore we have (5.5)

(pcp'Do Ti = i gijDj (cp2D0 TO - -
Geometry of Generalized Einstein Manifolds

119

It remains to evaluate the term q> D0(T'(pi) at the same time as the term 2nx'|t=o • For this last on making explicit the formulas (2.5) and (2.6) for a conformal infinitesimal deformation we have the torsion trace being invariant (5.6)

(Do Tj)' = F 2 V- Tj cp1 + Vj (PiTi+ (p0Tj

On the other hand x I t=o ~ 0, the derivative of T becomes: T'| t=0 = [g1J (DiDoTj)1 + gy 5. (DoDoTj)' ] whence on multiplying by 2n cp(x) 2ncp (x) T'|t=o = 2n (pgij (DjD0Tj)' + Div on W(M) On using (5.5) and (5.6) we obtain , after simplifying 2ncp(x)x1 t=o = -2n [F2 V* Tj cpV + 2 T1 cpicp0] + Div on W(M) Thus the expression under the integral sign is (5.7) 2(n-l)(n-2) (Acp - —*— //) - 2n F2 V" Tj q>V + n(n-6) Vyty0 n-\ + Div on W(M) On direct calculation we obtain F 2 V* Tj
120 120

Initiation to Global Global Finslerian Finslerian Geometry Geometry Initiation

(5.8) I"(go)= f

[2(n-l)(n-2)(A(p--L-£q>,
We thus obtain the formula for the second variational for an infinitesimal conformal deformation^ After a formula established in ([2] page 224) we have +(n+2) F2Qy cpty =Div on W(M)+(n-2) 9 1 (Do Ti)1

(n-2) F 2

On using x = 0, and the relations (2.5, II), (2.6, II) and (7.6 and 7.9 chap III) we prove that the last term of the right hand side of the above relation is a divergence. In this case we have

(5.9)

(n-2) F 2

(n+2) F2Qy cpV = Div on W(M)

On the other hand, on taking into account the lemmas 5 and 6 of Chapter VIII, the relation (5.15 Chapter VIII) becomes (5.10) (Acp, 9) =

i

n-\

2

Q,j 9 ] 9 J

(/i-l)(«-2) + Div on W(M)

Taking into account the relation (5.9), (5.10) becomes

+ Div W(M), From this we deduce, by integrating over W(M),

(5.12)

> 0 w-1

Geometry of Generalized Einstein Manifolds

121

If the integral on taking into account the above relations and (5.12) and (5.8), we have (5.14)

I"(go) > 0.

Theorem. Let (M, g) be a compact Finslerian manifold without boundary (n^2). We suppose that r is everywhere zero, x' | t=o = 0 and the Ricci vertical curvature Qy satisfies (5.13). Then at the critical point g0 e F°(g t), (t—0) of the integral I(gt) defined by (3.1) and for a conformal infinitesimal deformation, the second derivation is positve[6J. Remark. Let A, be a function such that A(p=Axp and X\ the least value of A, at the point y eW. Let us put X-i = min yew A,j(y). Let us suppose H to be a positive constant. From the relation (5.12), it follows that n-\

122 122 CHAPTER V Properties of Compact Finslerian Manifolds of Non-negative Curvature (Abstract) The objective of this chapter is to obtain a classification of Finslerian manifolds. Let (M, g) be a Finslerian manifold of dimension n, and W(M) the fibre bundle of unit tangent vectors to M. The curvature form of the Finslerian connection (Cartan) associated to (M, g) is a two from on W(M) with values in the space of skew-symmetric endomorphisms of the tangent space to M. It is the sum of three two forms of type (2, 0), (1, 1) and (0, 2) whose coefficients R, P and Q constitute the three curvature tensors of the given connection. In the first part we study the Landsberg manifolds, manifolds with minimal fibration and Berwald manifolds. The manifold M is called a Landsberg manifold if P vanishes everywhere. This condition is equivalent to the vanishing of the covariant derivative in the direction of the canonical section v: M -> V(M) of the torsion tensor. For a Riemannian metric (0,2) on V(M) this condition means that for every x e M the fibre p~'(x) becomes a totally geodesic manifold where p: V(M) -> M (see [5 and §7]). We examine the case when V(M) -> M is of minimal fibration as well as when M is a Berwald manifold. When M is compact and without boundary we put some global conditions on the first curvature tensor R or flag curvature of the Cartan connection. In the second part we study by deformations the metric of compact Finslerian manifolds in order that their indicatrix become Einstein manifolds.

Let M be a Finslerian manifold. We suppose M to be compact and without boundary. The torsion and curvature tensors of a Finslerian connection satisfy the Bianchi identities, one of which is the following (see [8.17, chap II]) (0.1)

V ; R'p +

I%VoK

^

j

^

-PJ,, V0Tkrm +Q'jrmRrM

=0

We can put on the fibre bundle V(M) a Riemannian metric of the form

Properties of Compact Finslerian Manifolds (0.2)

123

g = g1Jdx1dxJ + gij VVV v*

For all x G M , the fibre p"!(x) is a Riemannian submanifold of V(M). We say that (M, g) is a Landsberg manifold if p~*(x) is a totally geodesic manifold_(see [4]). In this case P = 0. Equivalently V 0 T = 0 where T is the torsion tensor. Similarly, we say that V(M) is a minima fibration if VOT* = 0, (necessary condition where T * = trace T) [see §7]. 1. Landsberg Manifolds. Let us denote by R the symmetric tensor defined by

where o denotes the multiplication contracted by v. The fact that Riojo is symmetric is a consequence of the first identity of Bianchi (8.15 chap II). On the other hand let ay(z) be the symmetric tensor defined by (1.1)

a,j(z) = F 2 V
where Qij = Q.rj and the last is the third curvature tensor of the Cartan connection:

0-2) Lemma 1. For a compact Finslerian manifold without boundary we have (1.3)

(V ( J, V J > =

j (Vo T, Vo T ) 77 = - (a,R W(M)

124 124

Initiation to Global Finslerian Geometry

where ( J and Q denote respectively the global and local scalar products over W(M). Proof. Let us multiply the two sides of the Bianchi identity (0.1) by v1 and v*. We then obtain 0-4)

V O V , X +TkrmR'om+V'mR'jklvW

=0

Whence, on multiplying the two sides by Tkm, (1.5)

Tkm V o V o Tkm + Tkm Tkm R!m + Tkm ( V ; R'jkl ) v V = 0

The first term can be written, on taking into account the divergence formula (1.6)

T,km VOVoVkm = V0{TkmVJL)

-

(VoT,kmVoTL)

= iv o V o (T,T)- (V0T, V0T) = Div on W(M) - (V o T, Vo T) Similarly the last term of (1.5) can be written, on taking into account the formula of vertical divergence (7.6 Chapter III) (1.7)

Tkm ( V ; R'jkl )vV = V; [(Tkm R)kl) vV ] - ( V ; T,km R)kl W) - Tkm R'Jkl{ Siv] + 8>y) = - (V- Tk + Tm Tkn) R'oko + div on W(M),

On substituting (1.7) and (1.6) in (1.5) and taking into account the expression of the Ricci tensor Qji defined from (1.2), we obtain (1.8)

(V o T, Vo T ) = - (V' Tk - Qk) R'oko + div on W(M)

We can add the terms vk T; + ^v; Tk. The expression of the right hand side does not change since

Properties of Compact Finslerian Manifolds

(1.9)

125

(V o T, Vo T ) = -(a, R ) + Div on W(M)

where a is a 2-tensor defined by (1.1). Since we suppose M to be compact, and without boundary, by integrating over W(M), we obtain the lemma. From the above lemma we obtain the following theorem. We say that a Finslerian manifold has a non-negative curvature in the large sense if the scalar product of the flag curvature by a symmetric tensor of order two (R , a) is non-negative. Theorem 1. Let (M, g) be a compact Finslerian manifold without boundary.

If the symmetric tensor R is such that (a,R\

is

everywhere non-negative, then (M, g) is a Landsberg manifold^ 2. Finslerian Manifolds with minima fibration. We have seen that the necessary condition for the fibre of V(M) for every x e M be a minimal submanifold is that V0T* = 0. (See [5]). In fact V0T* is the trace of the fundamental form of the submanifold p"1 (x) of V(M) (See [5]). We are going to study in this paragraph when M is compact the conditions for V0T* = 0. On contracting i and m in the formula (1.4) we obtain: (2.1)

Vo Vo Tk + Trkl R'oro + vV V,. R']kl = 0

From it, we deduce, on multiplying the two sides by Tk and on using the divergence formula (2.2)

(V o T., Vo T.) + div on W(M) = TkTrki ^ r o +Tk V,. R'jkl vV

126 126

Initiation to Global Finslerian Geometry

Let us put (2-3)

Y i = F1T*tftato

(2.4) = V, Tk R[)ko - Tk Rok + T1 Tk Rl0k0 + vV Tk V; R'jkl where Rok = R' 0lk

On putting the last term of (2.4) in (2.2) and on simplifying we have (2.5) (V o T., Vo T.) = - (8] Tk + T,Tk) R'oko + Tk Rok + Div on W(M) First, let us note (see [3])

Whence, by contracting i and 1: Kok - - 1 Rokl - - - 1 o; ^ o f o + - 1
= -x- Tk a;g i l // / o t o -iT k g i l a : Hloko + ^ T k a;//:,„ = I TkT' tf^-i g" 5' (Tk Hl0k0)+±7kd'k H'mo+yxd]Tk Hlo (2.6)

Tk Rok = |

Tk T1 Hloko + I gjl 5; Tk // tofo

- Tk 8-k H'0l0 + div on W(M) Now the last term can be written

(2.7)

i Tk a; ^

= I gkl d'k (T, //;,J - j gkl 5; T,

Properties of Compact Finslerian Manifolds

127

The first term on the left hand side is a divergence. On taking into account the relations (2.6) and (2.7), the relation (2.5) becomes (2.8) ( V J . , V 0 T . ) = - i (2 5- Tk + T j Tk+ /,* grs d'r T s) R'oko +div on W(M) where h\ = 8kt -uku ,(u= v/F). We can add the terms F "'(uk Tj +UjTk) to what is in the parenthesis of the right hand side of (2.8). For as a result of the properties of the tensor R'nko, the relation does not change. Let us put (2.9)

F"26f = 2[ d^Tk+ F'(u k T* + u, Tk )] + T i Tk+ h\ grs d\ T s

Thus (2.8) becomes (2.10)

(V o T*, Vo T*) = - - (b, R ) + Div on W(M)

Now M is compact, without boundary. we obtain

By integration on W(M)

v0r,,v07;) = - | ( Theorem 2. Let (M, g) be a compact Finslerian manifold without boundary. If the symmetric tensor R is such that(b,Rj is everywhere non-negative, then (M, g) is a manifold with minimal fibration(Wo T*= 0) 3. Case of Isotropic Manifolds. Let (M, g) be a Finslerian manifold with minima fibration (V o T* = 0) and isotropic, that is to say (see chapter VI)

128 128

Initiation to Global Finslerian Geometry

where J | i s the Kronecker symbol and K is different from zero. By Bianchi identity (0.1) (see[l]) we obtain

(3.2)

dK 3 F —— = —K am where crm = - A m = -FT m dv (n +1)

where T is the trace co-vector of torsion. Let x 0 e M a fixed point and C: R—> Sx a differential map in the indicatrix SXi such that C(t) is the trajectory of the vector field a (t) = -A(t). We have dC

(3.3,

..

du

-5—W-*

Now the coordinates in the fiber Sx are the u1 ( w|| = 1) where v1 Fu1. We have dK _ du'

dK dv1

Thus, by (3.2) and (3.3), we have for the fiber at x 0 dK _ dK dt

du'"

du'"

F

dt 3

dK du'" dK F dv'" dt dv' K ( )

On supposing K to be different from zero, we deduce from it d 3 logK — dt l o g K = (n +1) dt

()

Properties of Compact Finslerian Manifolds

129

Hence the solution K(t) = K(0)exp(—^— f (
J 0

When t -> oo, K(t) -» QO . Since ( a , a) and K(t) are bounded functions on the compact SXo. This is impossible since K is constant. Since K is different from zero, we have am-

-Am = 0.

Therefore (M, g) is a Riemannian manifold. Theorem 3. A Finsleran manifold with curvature isotropic and with minimal fibration (V 0T* = 0) is Riemannian. A fortiori, it is so if(M, g) is an isotropic Lands berg manifold with K ^ 0. Corollary. A Finslerian surface ( dim M = 2) with minimal fibration (V0 T*= 0), K ^ 0 is Riemannian. 4. Calculation of ( 6 A)2 when (M, g) is a manifold with minima fibration. We suppose (4.1)

V0T* = 0

By vertical derivation we obtain (4.2)

VjTj=-VoV'Ti

Now A = FT ; so we have, using (4.1), (4.2). (4.3)

-SA

= F V i T i = - F V 0 V- TJ

Hence (SA)2

=F 2 V 0 V' T1 V0V« Tj

130 130

Initiation to Global Finslerian Geometry

= F 2 v 0 [v i T i v0VjTj]-F2v« rv o v o vvT j = Div - F2 V; [Tj V o V o V*. Tj]

(4.4)

+ F 2 T i [V 1 V 0 V-.T j +VoV i V-T J ] F 2 T j Vi V 0 V}T J On using the identity of Ricci (§ 9 Chapter I) we get V i V o ( V - T J ) = V o V i V - T J - V' m VjT J C, Putting this in (4.4), we get (4.5)

( 8 A)2 = Div - F2 Tj V ; Vv Tj C ,

Let us put (4.6)

A = FT*,B m = F 5 ; f, f = F 2 V;TJ , u = v/F

The relation (4.5) becomes Lemma 2. For a manifold with minima fibration we have (4.7)

(8A)2 =-R (A, B) + Div on W(M)

where 8 is the co-differential and A and B are defined by (4.6) and R (A,B) = (R(A, u)u, B) .

131 131

Properties of Compact Finslerian Manifolds 5. Case where (M, g) is a Landsberg Manifold.

The

Calculation of V t A } Let now (M, g) be a Landsberg manifold. Then V oT* = 0. Hence the horizontal covariant derivation of A j is symmetric. On putting R rj = R'rij and using the Ricci identity we obtain (5.1)

V.A

:

iTj = F2 V i ( T i V j T j ) - F 2 T j

-T?2

-F2Tj[ Vj V jT j+TrRrj - V; T' Rr0lJ ] + Div on W(M) =-F^V-fT*' V* T') + F 2 V • T^ V* T' - F 2 /? T r TJ JV

/

/

J

/

rj

+ F2 Tj V; Tj Rr + div on W(M) = (8 A)2 -F2 R

V«jT rTj Rr + div on W(M)

Now V r Tj is symmetric. So, the second term becomes (5.2)

F 2 V-' Tr TJ Rr. = F V-j [T j T r Rr0IJ F] - TJ T r Rr00J + F 2 R i j T i T J - F 2 T j T r v m Vli Rrmi] m

Since the tensor P = 0, by (0.12) the last term of the right hand side vanishes. Hence (5.2) becomes, on taking into account (0.11) (5.3)

F2 g lk V" TrTJ Rrokj = -(n-2) RloJO T' TJ + F 2 R y T j T j + Div on W(M)

On putting this relation in (5.1) we obtain : Lemma 3. Let (M,g) be a Landsberg manifold of dimension n. Then we have (5.4)

V^JI = (S A)2 - (n-2)( R (A, A)) + Div on W(M)

132 132

Initiation to Global Finslerian Geometry 6. Case of Compact Berwald Manifolds

Let (M, g) be a compact Landsberg manifold without boundary. The tensor T satisfies VoT = 0. By vertical derivation we conclude that the covariant horizontal derivative of T is completely symmetric. We calculate its square: (6.1)

F 2 V'T ijk V,Tijk = F2 VjTiklV,Tijk = F2Vj[TiklV,Tijk] - F2TlklVjV, T/k =Div-F2 Tllk [V1VjTjik+rik Rrl +Tjrk R'rjl +Tjir Rkrjl -

V'rVlkRrojl]

On using the symmetric properties of the tensor T we obtain (6.2)|FV,7; t || 2 = Div - F2 V, [T>k Vj Tjik ] + F2 V, T'k Vj TJlk - F 2 T[k Trik Rrl -2V2TlkVrkR'rjl

+F2Tl'k V;T j i k i?; ;

=Div+|V / 4 t || 2 -F 2 7^T rik 7^-2F 2 7;{Tjik R^ + F2 Tjk V}T;k Rrojl where we have used the fact the vertical covariant derivative of the tensor T is also completely symmetric. Now the last term becomes (6.3)

F

2

^ v>r;ki?;7/=FVJ[F7;.; 1

ik

L

r

K

ool " r

r;kRr0Jl] 1

ik lr

V

A

o/7

On taking into account the formulas of vertical divergence (0.10) let us put Yj = F72 TkRrQ]l Yj

=YJ-UJYO/F,

(uJ = v j /F)Yo = viY1

Properties of Compact Finslerian Manifolds We have

-8 Y = F V j Yj -(n-1) ^

133 133

+FT j Yj

Thus the relation (6.3) becomes (6.4)

r

F2Tjk V ' j r * Rrojl = -8 Y- F 2 T J T'lk Tr* Rrojl +(n-^ tlk Tr* R ool

Now, since P = 0, from Bianchi identity (0.1) we obtain

Thus the last term of (6.4) becomes

Therefore

F 2 Ti V J Tr* Rrojl = - S Y + ( n - 2 ) tlk T? Rr00, + F 2 T>k Tkr Rrl Let us put this expression in (6.2) T's T; - 2Y2T'S Tjrs R'rjl + div on W(M)

On taking into account the expression of the curvature tensor Q defined by (1.2) and the properties of the tensors R and T , the last term of the right hand side of (6.5) becomes

Now P= 0. So we have (see [4] p.289) R'jkl ~ H'jkl

+T

jr

Kkl

134 134

Initiation to Global Finslerian Geometry

Finally (6.5) becomes (6.6)

| K ^ | | 2 = ll V /4| 2 -(n-2) T<s 77 R;)h-F2 Q)kl H* + div on W(M) Let us note that the tensor H is (see [4]) H)kl=\

d) (d-, H-Oko-d'k H'olo)

So the last term of (6.6) becomes : ^QJk,H!kl (6.7)

= | F^ m d) [F Qjl

=h2Q!kl

d-jSjH'^

Sj H'oko ] - | F^ m d) [F Qj ] 8] H'oko = Div - V* 8] H'oko

We have put, on using the fact ihzXQim is skew-symmetric with respect to (k,l)) and (i, m)

(6.8)

¥*= 1 fT d'jQ?1 Qj)

Since viJ'k is homogeneous of degree (-1), (6.7) becomes (6 9 F O'

HJkl ta g <$;

Let us put

Properties of Compact Finslerian Manifolds

135 135

and (6.11) Thus we can the relation (6.6) (6.12)

- 4*//; i o +divonW(M)

On using the lemmas 2 and 3 and on putting

where Bj is defined by (4.3), we finally obtain FV,TIJk f = - F""22 J\J\ H'oko + div on W(M) On integrating on W(M) we have Theorem 4. Let (M, g) be a compact Landsberg manifold without boundary, dim M = n > 2. If Jkt H'oko is non-negative everywhere on W(M) then (M, g) is a Berwald manifold. 7. Finslerian manifolds geodesic or minima

whose fibres are

totally

The Finslerian connection defines at each point z e V(M) a decomposition of the tangent space to V(M) at this point. We have TZV(M) = Hz ®VZ where Hz is the horizontal and Vz the vertical space. At the point z e V(M), the Pfaffian derivatives (b\ = 8k - VrOk 8?, Sk') define a frame adapted to the decomposition of TZV(M). Let us put the Riemannian metric on V(M): (7.1)

ds2 = gjj dx1 dxJ + gy(z) Vv'Vv1, z e V(M)

Let EV(M) be the principal fibre bundle of linear frames on V(M) with the structure group GL(2n, R). Let D be the Riemannian

136 136

Initiation to to Global Finslerian Geometry

connection associated to (7.1). This connection has no torsion and D G = 0. Let nf{a, P=i, z=l,2, ...n) be the matrix of this connection relative to the adapted frame, we have (7.2)

b(dV)=n%da,

(^=r&
where ox = (dx1, Vv* ). D being Riemannian, we have (7.3) rafi= 1 GrX{da Gpx + dp Gxa- dx Gap)- ±{GrA[8p, +

dx]a

[da,dp]y-GrA[dx,da]p}

where the bracket [ da, dp ] is defined by (7.4)

[a,,5j] = -R r 0 1 J ^

(7.5)

[ d , , # ] = G r ij#

(7.6)

[Sr,S}] = 0

where the Gry are the coefficients of the Berwald connection associated to g. Calculating the right hand side of (7.3) and taking note of the bracket expression we obtain the 1-form of the Riemannian connection nf with respect to the adapted frames :

(7.7)

n) = - (TV - 1 R' n) = (Tjjk + 1 Rojik)dxk + VoTVVv15 n) = a>)

where co) represents the 1-form of the Finslerian connection associated to gij and where T and R are the torsion and curvature tensors of V. Let x = XQ be a fixed point of M and the fibre

Properties of Compact Finslerian Manifolds

137

manifold p~'(xo) = Vn a submanifold of V(M) with the induced Riemannian metric : da 2 | p-i(Xo) = gij(x0, v ) dv'dv1

(7.8)

Let Xand Y be two tangent vectors in (xo, v) eV n to p'(xo) we have (7.9)

DfX

= DfX

+ A(Y,X),

where D is the induced connection and A the second fundamental form of the submanifold p~'(xo). Let us make explicit the right hand side. If X = (Sj) and F=(4*) w e have, with respect to the adapted frame Ds.Sj=

Kf {S-k)= 7ryk Sr + ^ 5 i

Taking into account (7.7) we obtain (7.10)

Ds. Sj = T)k (x0, v) Sr + Vo T' k (x0, v)5i

where the vectors d, and t^are orthogonal with respect to the metric (7.1) G(Sr ,8\ ) = 0. From this formula it follows n immediately that for V = p"'(xo) to be a totally geodesic (respectively minima) submanifold, it is necessary and sufficient that Vo T jk= 0 (respectively necessary g1 Vo T jk = Vo T'= 0). This condition is equivalent to the vanishing of the second curvature tensor P of the Finslerian connection. Theorem. In order that the fibres of p:V(M) —> M be totally geodesic (respectively minima) it is necessary and sufficient (respectively necessary ) that the second curvature tensor of the Finslerian connection P (respectively VQ Tt = 0) is zero [5J.

138 138

Initiation to Global Finslerian Geometry II COMPACT FINSLERIAN MANIFOLDS WHOSE INDICATRIX IS AN EINSTEIN MANIFOLD

1. The first variational of / (gt) = J F2 Qt 77 w

Let F(gt) be a 1- parameter family of Finslerian metric We denote by F° (gt) the sub-family of metrics such that for every te [-s, s] the volume of the unitary tangent fibre bundle of M corresponding to gt is equal to 1. Let us denote by Qt the scalar curvature corresponding to the third curvature tensor Q (1.2.1) of the Finslerian connection. Let us look for gt e F° (gt) such that the integral I(gt) defined by

0-1) w

(1.2)

(Q = g'JQij)

J 7t=l

be an extremum with the volume constant equal to 1. We suppose that the torsion tensor is invariant by deformation. On taking into account (1.1 II chap IV) we have (1.3)

( F 2 Q t / 7 ) ' = -[F 2 Q 1J -F 2 Qg 1J )t 1J + ( ( - - l ) F 2 Q t ( u , u ) ] / 7

since T is invariant by deformation its trace is also invariant by deformation. Let us put (1.4)

n 4 ^ ( |

2

and use the formula (4.3 II chap IV) So (1.3) becomes (1.5) with

(F 2 Q t 7 7 )' = -q 1J t 1J ?7

Properties of Compact Finslerian Manifolds

139

q,j = F2 Qij + (*F - F2 Q) gij + - L - F2 S;; V n-2 ' From (1.4) we have (1.6)

(*F- F 2 Q) = -(

(1.7)

Since we have assumed the volume of W(M) to be constant we have the relation (3.9, II, chap IV). If for t = 0; qy satisfies (1.8)

where c and b are arbitrary constants. Then the first derivative of I(g) at this point go e F (gt) vanishes and go e F°(gt) for t = 0 is a critical point of I(g). On multiplying the two sides of (1.8) by u1 and UL on one side and by g'J on the other part we obtain successively on taking account of the property of the tensor Q ( QOj

(1.9)

- — F2Q = c + b In

(1.10)

- - F 2 Q + — ^ - F2 8'J l F = nc + b

From the relation (1.9) it follows that at the critical point F2 Q is constant. On putting it in (1.10) we obtain u V

/

n-2

dr

T =

2nc-(n + l)(n-2)b (n + 2)

The first part is a divergence (See 7.6 chap III), M being compact and without boundary and volume W(M) = 1 on integrating (1.11) on W(M) we get

140 140

Initiation GlobalFinslerian Finslerian Geometry Initiation totoGlobal Geometry

c = ( " - 2 ) < "+ In and from (1.11) we have for n * 2,

(1.12)

1)

b

F2 glj 5 " ¥ = 0 On multiplying the two sides of this relation by m, by integration we obtain

(i.i3) - J F2gij a; a*

V2JJ =

J F2giJa; y a;.

T/7 = O

Let us put the expression defined by (1.12) in (1.9). We then have F2Q = -(n-l)b

(1.14) And by (1.8) we finally get: (1.15)

F 2 Q ;j =

_LF2Q(glj-uij)=-LF2Qhij n-\ n-\

where F2 Q is constant. If the indicatrix Sx is represented by v1 (ta) = v1 (a= 1, ...n-1), then the metric g defined by (0.2) induces on Sx the metric dv' dvJ

_

,

a , fi= l t . . . n - l and the curvature tensor of Sx (see [4]) is represented by R apy?. = Q aPyA + gayg^p ~ gaA. gpy

where Q is the projection of the curvature tensor Q on the indicatrix Sx On multiplying by gay:

Properties of Compact Finslerian Manifolds

141

R vx = F2 Qpx + (n-2) This is the Ricci tensor of the induced connection (See [4]). But 2

1

after (1.15) we have by projection on S x: F Qp^ =

2

F n-\

Thus (1.16)

i?p

2

^

where F2 Q is constant. Thus S x is an Einstein manifold ( n ^ 2) Theorem. Let (M, g) be a deformation of a compact Finslerian manifold without boundary and ( n ^ 2). At the critical point go G F° (gt) for t = 0 of the integral I(gt), the indicatrix S x is an Einstein manifold by (1.16) where F2 Q is constant. 2. The Second Variational The derivative under the integral (1.1) is defined by (1.3). We always suppose that the torsion tensor is invariant by deformation. By (1.5) the first derivative is (2.1)

I'(g,) =

j A(g) TJ W(M)

where A(g) = [F2 Qy - F2 Qg y ) tij + ( ^ -1) F2 Q t(u, u) ] 2 In the second derivative appear the expressions g1J, t y, t'(u, u). We obtain them by deriving the relations (3.9 chap IV): (2.2)

(2.3)

J t'(u, u) TJ =. W(M)

(2.4)

W(M)

J gij t' y TJ = W(M)

J [ t ( u , u) trace t - ( | +1) t2 (u, u) ] TJ

j

W(M)

[(t, t) - trace t (trace t - ( - ) t (u, u) ] TJ

142 142

Initiation to Global Finslerian Geometry

On deriving (2.1) and using (2.3) and (2.4) as well as the values at the point t = of F2 Q y and 1F2 Q defined by (1.15) and (1.14), we obtain after long calculations

(2.5) I"(go) = -b J [(t,t)-(tracet)2-n(^-3)t2(u,u) W(M)

+ (3--l)tracett(u,u)]77 By an infinitesimal conformal deformation we have (2-6)

tij = g'ij =2q>(x)gij

(2.7)

I"(go) = - 4n(n-3)b W(M)

Theorem The second variational of the integral I(gt) where go <^F° (gt) is a critical point and defined by the formula (2.5) and for an infinitesimal conformal deformation we have (2.7) at this point point, for n> 3, according as constant scalar curvature is positive, zero or negative. In case n = 3, I"(g0) = 0, the Weil curvature tensor of the submanifold p'1 (x) czV(M) vanishes. Since p'1 (x) is an Einstein manifold for g0, it has a constant sectional curvature, and by projection we conclude that the indicatrix has a constant sectional curvature.

143 143

CHAPTER VI Finslerian Manifolds of Constant Sectional Curvature [4] (abstract) This chapter is a study of isotropic and constant sectional curvature Finslerian manifolds. We first recall briefly the basics of Finslerian manifolds, define the isotropic manifolds and single out the properties of their curvature tensors. We then give a characterization of Finslerian manifolds with constant sectional curvature, generalizing Schur's classical theorem. We next determine the necessary and sufficient conditions for an isotropic Finslerian manifold to be of constant sectional curvature. Our conditions bear on the Ricci directional curvature or on the second scalar curvature of Berwald. We show that the existence of normal geodesic coordinates of class C2 on isotropic manifolds forces them to be Riemannian or locally Minkowskian. We also deal with the case of compact isotropic Finslerian manifolds with strictly negative curvatures. In chapter HI we give a classification of complete Finslerian manifolds with constant sectional curvatures. We prove that all geodesically complete Finslerian manifolds of dimension n > 2 with negative constant sectional curvature (K < 0) and with bounded torsion vector are Riemannian. We show that all simply connected Finslerian manifolds of dimension n > 2 with strictly positive constant sectional curvature and whose indicatrix is symmetric and has a scalar curvature independent of the direction is homeomorphic to an n-sphere. In the case when the Berwald curvature H vanishes and torsion tensor as well as its covariant vertical derivative are bounded we prove that the manifold in question is Minkowskian. In the last chapter we establish the 'axioms of the plane'. By defining the totally geodesic, semi-parallel and auto-parallel Finslerian submanifolds we establish the criteria that permit to identify if a Finslerian manifold is of constant sectional curvature in the Berwald connection (axiom 1), in the Finslerian connection (axiom 2) or is Riemannian (axiom 3).

144 144

Initiation to Global Finslerian Geometry I Isotropic Finslerian Manifolds

Notation and Recalls 1. Finslerian Manifolds. Let (x1) (i = 1, 2, ...n) be a local chart of the domain U 0 and CK on V(M) 2) F(x, X\) = A, F(x, v),

XtR+

1 82F2

3) gij(x, v) = -

. is positive definite.

We say the manifold is pseudo-Finslerian if gij defines nondegenerate quadratic form. Let us recall that gy is a tensor, homogeneous of degree zero in v. So we have

g i j (x,v)vV

= F2

where g( , ) denotes the local scalar product at z e V(M). If gy(x,v) is independent of v, we then have the structure of a Riemannian manifold. It is shown in [Chap II ] that there exists a unique regular connection attached to F such that

FinslerianManifolds Manifoldsof ofConstant ConstantSectional SectionalCurvatures Curvatures Finslerian

145 145

1) V-g = O (1.1) 2) S(X,Y) = O 3) g ( T ( l , Y ) , Z ) = g ( T ( l , Z ) , Y ) where i = n ( V l ) , Y = p ( f ) , Z = p(Z ), X , 7,ZGTV(M) The connection V thus defined is called the Finslerian connection. From the preceding conditions it follows that the covariant derivation V is determined by (1.2)

2g (V- Y, Z) = X g(Y, Z) + Y g(X, Z) - Z g(X, Y)

With respect to the natural frame induced at z e p"'(U), 5

^

(1.3) From the first structure equation, it follows immediately that the torsion tensor T in these co-ordinates coincides with C. From (1.2) we obtain: (1.4)

T ijk = - Sk gjj, T ijk =g ir T r jk

Taking into account the homogeneity of the right hand side we get

(1.5)

T 0 j k = Tj ok = TjjO= 0

where o denotes the multiplication contracted by v.

146 146

Initiation to Global Finslerian Geometry Let dj = 8j - Tgj 5k the horizontal vector field over 8j: = Oj, Va,V=0).

Let us put (1.6)

Va,8i.= f *8 k

By(1.3)wehave

fj=

r ff 4 -T* r ;

Now, by (1.1)2, Tis symmetric with respect to the indices i and j . On taking three horizontal vector fields d \ ,<3j, and dk over 5 j, 6 j , and 8 k respectively and on using (1.2) we obtain : (1.7) •

*

»

2 T iy• = (di gjk + 3j gik - 4 gij), (T iy = gkr r r,j and df= 8j - r * <$)

In the same way one shows that exists a connection regular, non-metric and torsionless, attached to F, called the Berwald connection [chap II]. If D is the covariant derivation in this connection there is a relationship between D and V: (1.8)

DH-Y=

V w iY + (VvT)(X,Y)

(1.9)

Dv~ Y = V I . Y

where v is horizontal above v (pv=v). From (1.8) it follows that D and V define the same splitting on the tangent fibre bundle to V(M). We have (1.10).

V f v= Dk v

for all vector fields X on V(M). If Y is the projectable by p*, then Dv^ Y = 0. D admits two curvature tensors denoted by H and P = -G. Let us put

Finslerian Manifolds of Constant Sectional Curvatures ZM80 = Gkij8k,

(1.11).

147

£&,(8)i = 0.

Then, by (1.8) we have the relation between G and Y G i j k (x,v)=f i jk +VoT i jk

(1.12).

2. Indicatrices. For xo GM fixed, the fiber p~'(xo) = Vn is a differentiable submanifold of V(M). It will be equipped with a Riemannian metric g(X,Y) where X and Y are two vertical tangent vectors to V at z e p~'(xo) . we call the indicatrix at xo eM the hypersurface S of Vn of the equation (2.1).

S:F(x o ,v)=l

If f is a homogeneous function of degree zero in v, we denote by S,f the partial derivatives of f with respect to the unitary vector and we have

where ||v|| is the norm of v. The hypersurface S can be represented by (2.3).

v^v^f),

a=l,...n-l

Let f be a function defined on S; its differential is written df(v(t)) = S,- f dvj = d a fdf = v U ' f dta whence

148 148

(2.4).

Initiation to Global Finslerian Geometry

3a=<Sr,

The da define (n-1) tangent vectors to S, the matrix (v'a) is of rank (n-1), the induced tensor metric on S becomes (2.5).

g a p = gij v' a V*jJ

Let v = v1 3j the tangent vector to the fibre passing through z, on differentiating F (xo, v) = 1, we have

Thus the vector v is normal to (n-1), vl« tangent vectors to S. In this way, the set (v' a , v) define at z(x0, v) n-vectors linearly independent. Therefore (2,5) can also be written (2.6).

gap = hjj V'a Vp

(h,j = gy - V,Vj)

where v, = g,jV\ |v| = 1. It then follows (2.7).

ge = gafJ vja vip + vV

Similarly, the matrix (v!a, v1 = v1,,) is of rank n. On denoting by the same letter its inverse, we have (2.8).

v'j v \ = vja v \ + v\ v \ = = 8V (vn| = vO

Thus from (2.6) we get (2.9).

hij - g a p V^j VPj

At a point z = (xo, v) e Vn = p"'(xo), the coefficients of the Riemannian connection associated to the vertical metric g(X, Y) are Tjk(xo» v) where T is the torsion tensor of the Finslerian connection. The corresponding Riemannian covariant derivation is determined by

Finslerian Manifolds of Constant Sectional Curvatures (2.10)

DdkdJ-T)k(x0,y)d)

149 149

(^ = ^ r )

Let us denote by V the connection induced on S and by a, the second fundamental form of the hypersurface S. If X and Y are two tangent vectors to S, the Gauss equation becomes (2.11).

DYX= V Y x + a(X,Y)

Now v is normal to S. So we have (2.12).

g(D Y X, v) = g(a(X, Y), v) = -g(X, DY v)

Now, by (2.10) we have (2.13).

DYv=Y

Thus a(X,Y) = -g(X,Y)v The equation (2.11) becomes (2.14).

DYX= V Y X-g(X,Y) v

If we put in (2.14), X = da,Y = dp and V5 ^ da = f

x aP

dx where f %

are the coefficients of the Riemannian connection induced on S, we obtain (2.15).

V pv'a-A'jkvUygapV, ( A'jk = FT'jk)

Let us denote by R the curvature tensor of the connection induced on S and by Q the projection on S of the third curvature tensor of the Finslerian connection. From the relation (2.15) we have

150 150 (2.16).

toGlobal Global Finslerian Geometry Initiation to Ra$yx = F2 Qaf,7X + gaygX(5 -gaX-gpy

By contraction we obtain the relation between the scalar curvatures (2.17).

i? = F 2 g+(n-l)(n-2)

where Q = gay gp" 0 aPyX and R = gay g p ' R

am

3. Isotropic manifolds Let P(v, X) c T(M) be a two plane generated by two vectors v and Xe Tpz where v is the canonical section. After Berwald the sectional curvature following P in the Berwald connection is defined by (3.1)

K(z,v,X)=

g(X,X)g(v,v)-g(X,v)

K is a homogeneous function of degree zero in v. If we substitute X by the vector Y = aX + bv (a, b eR), K remains unchanged. Thus K does not depend on the choice of X in the 2-plane P. (M, g) is to be isotropic at x = pz eM (scalar curvature in Berwald's terminology) if K is independent of X. We say that (M, g) is isotropic if it is isotropic at all the points. In this case the curvature tensor H of the Berwald connection is defined by (3.2).

H(Y, v) v = K(z) [g(v, v) Y - g(Y, v)v], z e V(M)

With respect to a local chart (x1) of M, (3.2) becomes (3.3)

H'ojo = F 2 K(z)h'j, hjj = (S'j-u'uj), ||u|| = 1

where 8 denotes the Kronecker symbol. Hjki is obtained from H'oko by

But the curvature tensor

Finslerian Manifolds of Constant Sectional Curvatures (3.4)

H'ju^aKdjH'oko-^H'oio)

151 151

(dj =

On taking into account (3.3) we obtain (3.5) jrl = K(5 rgjl - 5 lgjr) +(8 rVi - 5 lVr)Kj + -*• (2Vj§ r - 0 jVr - V gjr)Ki

- 1 (2VJ51, - 5jjV, - v' gjl )K r + 1 F 2 (hVKj, -h1, Kjr) where Kj = 31/K, Kjj=d2" K from (3.5) we have the Ricci tensor H y = Hr,rj,

(3.6)

Hij = (n-1) (K(z)gy + VjKi) + 1 (n-2)(2v,Kj + F 2 ^ )

If (M, g) is Riemannian Hy is symmetric, by (3.6) it follows immediately that Kj is zero. On using the Bianchi identity we obtain for n > 2 that K = constant. Thus an isotropic Riemannian manifold (n > 2) in the above sense is of constant sectional curvature. 4. Properties of curvature tensors in the isotropic case By (1.8) the curvature tensor R of the Finslerian manifold is related to the curvature tensor H of the Berwald connection by (4.1).

Rjki = Hjki + TjrRoki + VjVoTjk - VkVoT'j, + VoTrjkVorrl - VoTVkVorj,

where Vo = v'Vj

On multiplying the two sides of this relation by v* and on taking into account (3.5), we obtain

152 152

Global Finslerian Geometry Initiation to Global

(4.2). Rjorl ^H'ori =K(8V vi-81, v r ) + 1 F2(hjr K, -h\ Kr),

(K,= df K)

From the relation of commutation ([8.27, Chap II],) R-lijk " Rjkli

=

Tij r R oik + Tkir R Ojl + Tj| r R oki + T]k r R oij

So, in virtue of (4.2) (4.3)

R lijk = Rjkn

From the Bianchi identity (see Chap II) we obtain (4.4)

CT(m5k,l) Vm Rjkl + a(m,k,l) Q jrm R Okl + <^(m,kj) V k V 0 Q j m i = 0

where a denotes the sum of terms obtained by permuting cyclically the indices m, k and 1. On multiplying the two sides of (4.4) by vk and on taking into account (4.2) and (4.3) we then have in the isotropic case (4.5)

RIjr, = K(girgj, - gqgn) + kF2 Q,jrl + 1V 0 V 0 Qijrl + \ (gjlVr-gjrVl)Ki+ 1 (grfVi -gilVr)Kj + \ (girVj -gjrVi)K, + 1 (gjiVj -gliVj) Kr + ^ F [hji V;. Kj + hir V} Kj - hjr V* Kj - hn V? Kj]

where Vr denotes the vertical covariant derivation. It follows that in the isotropic case the Ricci tensor Rji = Rrjri is symmetric. By (8.15 Chap II), the first two Bianchi identities containing the curvature tensor R reduce in the isotropic case to (4.6)

(4.7)

O(j)k>i) Rjki - 0

a(m,r,D Vi» Rrjik + a(m,r,i) (F 2 j Km + Kv m ) VoQ^i = 0

Finslerian Manifolds of Constant Sectional Curvatures

153

where Km = d% K. Finally in the isotropic case, from the Bianchi identities (see chap II) we obtain for the curvature tensors of the Berwald connection the identities (4.8)

<j(j>k,i) tfju = 0

(4.9)

a(m,k>i) D m HV, - 0

(4.10)

D, G'jkm - D K G\lm+d% HV, = 0

II Finslerian manifolds with constant sectional curvature 1.Generalization of Schur's theorem. A. CASE OF BERWALD CONNECTION 1) Berwald studies the Finslerian manifolds that satisfy a) (M, g) is isotropic b) K is independent of v. By (3.5), we then have (1.1)

Hijki = K(8 i k g J ,-5 i lgjk )

On contracting i and k in the identity (4.9.1) and on multiplying the two sides of the relation so obtained by v1 and v successively we obtain (1.2)

D i H U + DoHc-DiHoo-O

On taking into account (1.1) the above relation becomes for n * 2

154 154 (1.3)

Initiation to Global Finslerian Geometry v,D0K = F2D,K

whence, by vertical derivation gj, D0K + v,DjK = 2 Vj D,K On multiplying the two sides by g*1 we get D0K = 0 From the relation (1.3) it follows that K = constant (n> 2). Thus an isotropic Finslerian manifold (n > 2), with K independent of the direction, is of constant sectional curvature in the Berwald connection. Now we modify Berwald's definition and introduce a function defined on a suitable fibre bundle. 2. Let G2 (M) be the Grassmannian fibre bundle of 2-planes on M. We note by TI' 1 G 2 (M) - • W(M) the fibre induced on W(M) by 71: W(M) —» M. Let P sn'1G2 (M) a 2-plane generated by the vectors X and Y linearly independent at x = Try eM. Definition. The sectional curvature at a point y e W(M) following the 2-plane P defined by the vectors X and Y at x = rty e M in the Berwald connection is defined to be the function K :n'G2 (M) —> R (1.4)

Ki(y,X,Y) =

where A is the anti-symmetric tensor of H, defined by (1.5)

g(AX, Y)Y, X) = 1 [g(H(X,Y)Y,X) -g(H(X,Y)X,Y)]

Ki(y, X, Y) is a homogeneous function of degree zero in v and remains unchanged if one replaces X and Y by the vectors

Finslerian Manifolds of Constant Sectional Curvatures

155

Xi = aX + bY, Yi = cX + dY, with ad-bc * 0, a, b, c, d, eR. Thus Ki does not depend on the choice of the vectors X and Y in the plane P. let suppose that Ki is independent of the plane P, then Ki is a fortiori independent of the plane passing the canonical vector v. Now, Ki(y, v, X) = K(y,v,X). Therefore (M, g) is isotropic (scalar curvature) in the Berwald sense. On the other hand, from (4.1.1) we have (1.6).

Aijkl = RUki + VoTirkVoTrj, - VoTjrkVoTri,

One easily verifies that in the isotropic case, in virtue of (4.3.1) and (4.6.1) the tensor A satisfies (1.7)

1°

Aijkl = -A jikl

2°. Ayiti = - Ajjik 3°. CT(J;k,i) Ayid = 0 4°. Aijki

Now K] is independent of the plane and A satisfies the relations 1°, 2°, 3°, 4°. So by a classical reasoning ([23], tome I, p.200, proposition 1.4) that A must be of the form Aijrl= K (gi r gj| - gilgjr)

Whence on multiplying the two sides by V A i 0 rl = H ior i= K (Vigir - Vrgii)

(M, g) being isotropic in virtue of (4.2.1) we obtain hVK^h'.Kr,

hV= SV-u'ur

(O=l)

156 156

Initiation to Global Finslerian Geometry

On contracting i and r, we have, for n > 2, Ki = 0. Thus K is independent of the direction. By a reasoning identical to the above case one finds that K is a constant. Theorem. - For K; (y, P) to be independent of the 2-plane P (dim M > 2) it is necessary and sufficient that curvature tensor H of the Berwald connection be defined by (1.1) where K = K; is an absolute constant[4]. In this case, after (4.5.1), the curvature tensor R of the Finslerian connection is defined by Ryu = K (gikgji - gjkgii) + K F 2 Qpi + | VoVoQyki

(1.8).

B. CASE OF THE FINSLERIAN CONNECTION By means of the curvature tensor R of the Finslerian connection one defines the function K2 : 7i~'G(M) -» R (1.9)

K2(y, X, Y) =

g(R(X&Y,X)

K.2 will be called the sectional curvature at y e W(M) following the 2-plane P(X, Y) generated by X and Y in the Finslerian connection. We have (1.10)

K 2 (y,v,X) = K(y,v,X)

As in the preceding case one proves Theorem. [1] For K2& P) to be independent of the two plane P(X, Y) (dim M > 2) it is necessary and sufficient that the curvature tensor R of the Finslerian connection be defined by (1. 11).

R(X,Y)Z = K[g(Y,Z) X-g(X, Z)YJ

where K is a constant andX, Y, Z e Tx (M)[4J.

Finslerian Manifolds of Constant Sectional Curvatures

157 157

One shows that in this case for K # 0 the second curvature tensor Pjjki is symmetric with respect to the last two indices k and 1 and the third curvature tensor of the Finslerian connection vanishes everywhere. (1.12).

Q = 0.

Such a manifold is a fortiori of constant sectional curvature in the Berwald connection. The converse is true if the condition (1.12) is satisfied. 2. Necessary and sufficient conditions for an isotropic Finslerian manifold to be of constant sectional curvature. A. With the help of the Ricci curvature Hy or Ry one defines a scalar-valued function on W(M) by p= H(u, u) = Hy (x, u) u it^Rjj (x, u ) ujuj. ( u - 1). We call p the Ricci directional curvature. Theorem [4] For an isotropic Finslerian manifold to be of constant sectional curvature in the Berwald connection (n > 2), it is necessary and sufficient that the Ricci directional curvature H(u, u) satisfy the condition (2.1).

V0H(u,u) = 0

Proof. - By the identity (4.10.1) one obtains, on multiplying the two sides by v1 (2.2).

D0Gij,m + vr5»,Hijlr = 0

(M, g) being isotropic, in virtue of (3.5.1), the above relation becomes

158 158 (2.3).

Initiation to Global Finslerian Geometry Do G'jbn = 2 KV Tjlm - a

a>,, m)

j (5j,Vj + S^v, -2v i g jl ) Km

- cr (j, i,m) 1 (F28'i - v1 vi) Kjm where a ^ i,m) denotes the sum of the terms obtained on permuting cyclically the indices j , i, and m, Km = dh K, Kjm = d2]m K. Let us contract i and 1 in (2.3)

(2.4).

Do Gjm = - ^

(VjKm + vmKj + F2 Kjm),

(Gjm = G'Jim)

Let us multiply the two sides of (2.4) by g*"1. We obtain (2-5)

F2 g1J -^-+ -*— D0G = 0, dvdvJ (n + l)

(G = g^mGjm)

(M, g) being isotropic, we have (n-l)K = p. Now by hypothesis p satisfies the condition (2.1). From (2.5) we obtain (2.6). The right hand side of (2.6) is a divergence on W(M). If M is compact, without boundary, by (2.6) it follows by integration over W(M) J F2 gij KjKjTiCg) = 0 W(M)

where r|(g) is the volume element of W(M). Hence 3, K = 0. Thus K is independent of the direction. So (M, g) is of constant sectional curvature in the Berwald connection. We will extend this result to the non-compact case. From the Bianchi identity we have obtained the relation (1.2). Now (M, g) is isotropic from (4.2.1). So we obtain, on contracting the indices i and r

Finslerian Manifolds of Constant Sectional Curvatures (2.7)

H0, = (n-l)K(z)v,+ ^ = ^ F 2 K , ,

159 (K, = d,K)

Thus (1.2) becomes (n * 2) F 2 DiK=V|D 0 K+l F2D0K, In virtue of (2.1), the above relation reduces to (2.8)

D,K = 1 D0K,

On the other hand, let us derive vertically the relation (2.1) D,K + D0K,= 0 Now, on taking into account (2.8) (2.9)

D,K = 0= DoK,

By vertical derivation, we obtain successively (2.10)

D,(Hoj + HJO) = 0 2Di Hjm =(H or + Hro)Grjlm

where we have put j m

- l ~2

Now the curvature tensor Grjim is symmetric with respect to the indices j , 1, and m. So from (2.10) we get (2.12) Let us write down the Ricci identity for the tensor H im

160 160

Initiation to Global Finslerian Geometry DkDi H j m - DiDk H j m = - H n,,H jki - H j r H mki - & / / j m H oki

Let us multiply the two sides this relation by v1. Then on taking into account (2.10), we have H rm H okl + H or H mkl = 0

On multiplying this relation by v and then by gm , we have (2.13)

H o o^rm-H m o kg

H or = 0

Now from the expression p = F~2HJJVV , we have by vertical derivation and on taking into account (2.2)

2 / / m r = 2 p g m r + 2 v m 5^p+2vr dmp+¥2

Thus from (2.13) we have (2.14) (M, g) being isotropic, on calculating Hrom0 and Hrmokby (3.5.1), D(p) becomes (2.15)

2(n-l) D(p)= F2 g1J ^ -

= 0,

(p =(n-l)K)

Let us denote once again by p the restriction of p to the indicatrix Sx. we get

Finslerian Manifolds of Constant Sectional Curvatures

161 161

d p d a p 2 = F d pVa d, p2 + F2 vja v>p dj 5, p2 + FV p dj Fvja 5; p 2 Nowv i p a,-F = 0. So 5p5a

2 P

= F 5 P v'a 5, p2 + F2 vja VpSy d, p2

On taking into account (2.7.1) and on multiplying the two sides by g ® we get a

g aP d p d B p2= F2 gij 5, &, p2 + F g ap 5 pvja 5, p2 In virtue of (2.15.1), (2.15) becomes on Sx (2.16)

P

2

2

where A a = gx® A\p. Now the indicatrix Sx is compact, it follows from the preceding equation using the maximum principle of Hopf that p 2 is constant on Sx. From (2.9) it then follows that K is an absolute constant. B. Let us suppose that (M, g) is of constant sectional curvature in the Berwald connection. From (2.5) it follows that the second scalar curvature G must satisfy (2.17)

(G = g'1Gjii)

D o G = 0,

where Do - v1 Dj. Conversely let suppose that (M, g) is isotropic and besides the scalar curvature G satisfies (2.17). From (2.5) we obtain ^

ii

d2K

On restricting to the indicatrix Sx, we prove as in the preceding case that the equation (2.18) on S becomes

162 162

Initiation to Global Finslerian Geometry g aP V p 5 a K + A a a a K = O

(a,(3= 1, ...n-1)

where we have denoted by K the restriction of K to S. reasoning analogous to the preceding case, we obtain

By a

Theorem. [4] - For an isotropic Finslerian manifold M (dim M > 2) to be of constant sectional curvature in the Berwald connection it is necessary and sufficient that second scalar curvature G satisfies (2.17). If dim M = 2, then (M, g) is isotropic and the tensor Q vanishes. If the scalar curvature G satisfies (2.17), then K is independent of the direction, and the tensors H and R become R jki = Hjki

=

K (8'kgji - 5'igjk)

where K is not necessarily a constant. 3. Locally Minkowskian manifolds Let y : [0, b] -» M be a Finslerian geodesic, parametrized along its arc length on [0, b], y is defined by a differential equation of the second order (3.1) where Gjk are the coefficients of the Berwald connection. We know that the data for dx' 1

j 1

s = 0, x'o = x 1 s=o and -j-\ s=o = a

Constant Sectional Curvatures Curvatures Finslerian Manifolds of Constant

163 163

determine a unique geodesic starting from x0 e M tangent to the vector A(a') at x0 and situated in a sufficiently small neighborhood of x0. This geodesic is represented in the chart by [4]

(3.2)

x^fW.yO,

yi = ajs

From (3.2) we obtain ds

dyJ ds

dyJ

whence, at the origin x0

d i dy<' ** Thus the Jacobian of (3.2) is different from zero at the origin x0. The equation is invertible in the neighborhood of x0. This can be considered as change of local coordinates. In the new chart U(y) the geodesic y is represented by the linear equation y =as. A second derivation gives us

d2f' l i m n ^ = - limG'jk(xo>y) Now the right hand side is indeterminate as y —» 0, since it is homogeneous of degree zero in y. Thus the map y —> x, defined by (3.2) is a Cx-diffeomorphism of an open set of the zero section of TM over an open set ofM. We call the coordinates (y) the normal geodesic coordinates. Theorem [4]. - For the map f to be class C2 it is necessary and sufficient that the second curvature tensor of the Berwald connection vanishes (3.3).

G'ju = 0& VkTX]l = 0

Definition. - A Finslerian manifold is called locally Minkowskian if there exists a local chart U(x') such that the fundamental function F defined onp~'U(x', v') does not depend on x [13].

164 164

Initiation to Global Finslerian Geometry

For a Finslerian manifold to be locally Minkowskian it is necessary and sufficient that two curvature tensors H and G of the Berwald connection vanish. This condition is equivalent to R = 0, and G = 0. Let us suppose that (M, g) is isotropic and in addition the tensor G vanishes. From the preceding considerations it follows that (M, g) is of constant sectional curvature. From (2.3) we have either (M, g) is Riemannian or K = 0. Thus, we have Theorem. An isotropic Finslerian manifold admitting normal geodesic coordinates of class C is either Riemannian or locally Minkowskian[4]. In dimension 2, this result is due to Busemann [11]. 4. Compact isotropic manifolds with strictly negative curvature. Let us suppose that (M, g) is isotropic. Let us multiply the two sides of (2.2) by Vj. We then have (4.1).

D0D0Tjlm + KFTjlm + ±F 2 (hj,Km + h,mKj+ hjmK,)= 0

with hjj = gij -u,Uj, (UJ = F"1 Vj, Kj = djK)

If K is independent of the direction, we obtain (4.2).

D0D0 Tjlm + KF2 Tjlm = 0

whence (4.3).

1D 0 D 0 g(T, T) + KF2 g(T, T) - g(D0T, D0T) = 0

Let us suppose that M is compact without boundary and K < 0. Then the first term of (4.3) is a divergence on W(M). On integrating over W(M) we obtain D0T = 0. From (4.2) we have T = 0. Thus (M, g) is Riemannian.

Finslerian Manifolds of of Constant Sectional Curvatures

165 165

More generally Theorem. - A compact without boundary isotropic Finslerian manifold (M, g) (dim M > 3), with symmetric indicatrix and with strictly negative sectional curvature is a Riemannian manifold with constant sectional curvature. Proof. For x = x0 fixed, let gap(v(t)) = gap(x0, v(t)) the metric of the indicatrix So defined by (2.51). Let g ap be a metric conformal to gap (4.4)

iap=e2CT(v(t))gaP a , p = l,...(n-l)

We denote by D the Riemannian covariant derivative associated to g ap, we have (4.5) D p v ^ V p v W S V a + S^Gp-gapo^V,

(aa=|g-)

where Vpv'a is defined by (2.15.1). We call pseudo-normal the vector of components (4.6).

Nj = ^

gaP V pv'a = - - L A V x - vj

(Ax = g a P A\ p )

It is clear that the (M, g) is Riemannian if and only if N is normal toS(A = 0)[21]. Let us put (4.7).

Z

Let us choose (4.8). (4.9) (4.10) (4.11) With

oo= e 2a Z1 = 2ax v\ V e^VaZ^a^v'^ Dpvja =C^ aP v\+

166 166

Initiation to Global Finslerian Geometry

(4.12). \xA p +gap Ax

(4.13)

Now, M is isotropic; by (4.1) it follows, on eliminating the vertical derivative of K (4.14) with (4.15)

D 0 D 0 Tjlm+ KF 2 xj,m= 0 Xjim = Tji m -(l/(n+l))(hj|T m + hi m Tj+ hmjT|) (T m = T'im)

On multiplying the two sides of (4.14) by x*1"1 we obtain

(4.16)

lD 0 D 0 |H| 2 =||Ar| 2 -KL 2 H 2

where |r| 2 is the square of the norm of x. The left hand side of (4.16) is a divergence on W(M) [7.9 Chapter III]. M supposed to be compact without boundary, we have by integrating over W(M)

W(M)

Now K < 0, Then Do x= 0. From (4.14) it follows that (4.17)

x jlm =0

On projecting this relation on the tangent space to the indicatrix Sxo we obtain Cxap = 0. Thus (4.10) and (4.11) become (4.18)

e2oVa

(4.19)

Dpv'a

Finslerian Manifolds of Constant Sectional Curvatures

167 167

From (4.19) it follows immediately by derivation, on taking into account (4.18) and on using the properties Riemann curvature tensor R associated to b (4.20)

i?\xyP=H>(gapoV £ aY 5"p)

(4.21)

V p a a + gap a ^ = x gap

*|/ = e"2o(l - x)

After (4.20) the indicatrix Sx0 (n > 3), equipped with the metric g ap is of constant sectional curvature and we have (4.22) V a Z' = - \\i V a v', \\i - constant By integrating we obtain (4.23)

Z ^ - y v ' + c1 , V a C j = 0

where c ' is a constant vector. Now the indicatrix Sx0 being supposed to be symmetric, that is to say F(x0, -v) = F(x0, v), by (4.23) it follows that on changing v into -v and on taking into account of the expression Z defined by (4.9) that Z is transformed into -Z. Consequently, Z1 = - \y v '• Thus Sx0 is a sphere and (M, g) is a Riemannian manifold with constant sectional curvature III. Complete manifolds with constant sectional curvatures 1. Operator D1. The Isotropic Case We suppose that (M, g) is isotropic. For every function f: W(M) —»R, we put

(1.1).

D'f= A A A f +4K A f + 2 AK.f

where u = F"1 v is horizontal over u (p u =u). If f and i|/ are two functions on W(M) we have g(D'f, \if) + g(f, DV) = div on W(M)

168 168

Initiation to Global Finslerian Geometry

From this we deduce that if M is compact, D1 is anti-auto-adjoint. Thanks to the tensor T, defined by (4.15.11) we construct the tensor (1.2).

S jkl = X h-T jk - x'krX jl

(M, g) being isotropic, on taking into account the relation (4.14.11) we obtain (1 .3).

D1 (F

Sjki) - 0

It follows, by contraction D1 (L:•S) = 0 With (1 .4).

S

=411' M

where T* denotes the torsion trace vector. We note that the tensor T majors the tensors x and T*. The relation (1.3) shows that the kernel of D1 is non-empty. Let Q be the third curvature tensor of the Finslerian connection [See Chapter II] (1.5).

Qijki = T i Ir T r jk -T i kr T r j,

Let us denote by M the tensor (1.6).

Mrijki = F(hlkTrj, + hj,Trlk -hi,Trjk- hjkTr,,)

w i t h hy = gy - UjUj, u = F"1 v

In the isotropic case, we obtain (1.7).

D'(F2Qljkl) = D0Mrijk, Kr + 1 Mrljld Do Kr

where Kr = dr K. The tensor S defined by (1.2) is related to Q by

Finslerian Manifolds of Constant Sectional Curvatures (1-8).

with (1.9).

169

Sjjki = Qijki + hn Tjk + hjkT n -hik Tji -hji Tjk

Tjk = ^ _

[TjTk

+ 1 h j k T r r - (n+l)Tjkr Tr]

From (1.4) it follows (••>o).

2. Complete manifolds with strictly negative constant sectional curvature Let us suppose that (M, g) is isotropic and n > 2. We have shown in paragraph (2.II) that if D0K is zero everywhere, then the manifold is of constant sectional curvature. The operator D1, defined by (1.1) becomes (2.1).

D2f = A A A f + 4K A f,

(K = constant)

It is clear that the tensor F2 S'jki is in the kernel of this operator. In the same way are the tensors, by (1.7), F2Q jki and the contracted tensors F2Qj,, F2 Q (2.2).

D2(F2 S) = 0,

D2(F2Q) = 0

In virtue of (1.10) it follows that

(2.3)

D 2 \\FT*f = 0, D 2 |F7f = 0

Let us rewrite (2.3)i (2.4)

A A A F2 \\T*\\2 + 4K A

2

=

170 170

Global Finslerian Geometry Initiation to Global

Let us put

(2.5)

A L 2 |fT*||2= f, 4K = - C2

Now y is a geodesic on M, parametrized according to its arc length s, the equation (2.4) along y becomes (2.6)

f'-C2f=0,

( f ' = ^

The roots of the characteristic polynomial are ± C, whence the general integral (2.7)

f = C i e C s + C2e-Cs

where Ci and d are arbitrary constants on y . In the following we suppose that (M, g) is geodesically complete. On choosing A and B such that (2.8)

A - + 2-y/C,C2

B-

f is put under the form f=Ac£(Cs + B) Let us denote 9= \FT^ • Along the geodesic y

where D is a constant on y. Let us suppose that the norm of the torsion vector is bounded on W(M). If s —» ± oc, sh (Cs + B) —> ±<x, and cp is bounded only if A =0. That is to say f= 0. By (2.5) one obtains along the geodesic y

Finslerian Manifolds of Constant Sectional Curvatures

171

g(T*, A T * ) = O A second derivation gives us g(T*, A A T * ) + g(AT*, A T * ) = 0 In virtue of (4.2.II), the above equation is reduced to -Kg(T*, T*) + g(D- T*, A T*) = 0 Now K is strictly negative, from the above relation it follows that T* vanishes along y. But the geodesic y is arbitrary. So T* vanishes everywhere. Hence (M, g) is Riemannian. Theorem. - A geodesically complete Finslerian manifold (dim M > 2) with strictly negative constant sectional curvature and with bounded torsion vector is Riemannian[4]. 3. Complete manifolds with strictly positive constant sectional curvature In this paragraph, we suppose that the indicatrix Sx is symmetric

and (M, g) is of strictly positive constant sectional curvature. In addition we suppose that (M, g) is metrically complete. By the generalization of Myers' theorem, M is compact. The tensors F2Qjki and Rjki belong to the kernel of the operator D2. We therefore have

(3.1)

D2Q = A A A Q +4KA 0 = 0

where Q = F2 Q. Let y be a geodesic of M parametrized by its arc length s, from (3.1) one obtains along y

172 172 (3.2)

Initiation to Global Finslerian Geometry ^ + k

where
2

^ = 0 ,

(k2 = 4K)

In the following we assume that Q = F2.Q is

independent of the direction. By (2.17.1), the indicatrix &o for x — xo fixed has a constant scalar curvature. Thus Q is a function defined on M. The preceding equation determines a solution for Q along y (3.3)

Q = A. cos ks + B. sin ks + C

where A, B, and C are constants on y. Now M is compact. So the function Q admits an absolute maximum and minimum on M. Let us suppose that this maximum is attained for s = 0 and that the value of Q at this point is + 1. Then we have (3.4)

Let Po eM, a point corresponding to s = 0. Q admits an absolute minimum for nfk. Let us denote by Pi € M this point and let us suppose that the value of Q at this point is equal to - 1 . We have (3.5) From (3.4) and (3.5) it follows that (3.6)

Q = cos ks

One proves (see [3] pp 67-71) that Q admits only Po and Pi as critical points. If in addition one assumes that M is simply connected, then by a result of Milnor [30], M is homeomorphic to an n-sphere (n > 2).

Finslerian Manifolds of Constant Sectional Curvatures

173

Theorem. - Let (M, g) (dim M > 2) be a simply connected Finslerian manifold, complete, with a symmetric indicatrix of strictly positive constant sectional curvature in the Berwald connection. If moreover the scalar curvature of the indicatrix at each point of the manifold is independent of the direction, then (M, g) is homeomorphic to an n-sphere[4J.

4. Complete curvature

manifolds

with

zero

sectional

If the sectional curvature in the Berwald connection vanishes, by (3.1.1) (3.4.1) and (1.4.11) the curvature tensor H vanishes everywhere. The equation (2.3)2 becomes

(4.1)

A A D,\\FT\f=0

If we denote by (p- \FT\ we have along a geodesic y (4.2).

q> = as2 + bs + c

where a, b, and c are constants on y. Suppose that

(4.4).

A A F T =0

174 174

Initiation to Global Finslerian Geometry

Let us derive the relation (4.3) along y, taking into account (4.4). We obtain

g(AFT, A The geodesic y being arbitrary in the above relation, it follows (4.5).

V0Tijk = 0 o Pijk, = 0

Thus (M, g) is a Landsberg manifold. Now H is zero by (4.1.1). So it follows immediately that curvature tensor R of the Finslerian connection vanishes everywhere (R= P= 0) Let us derive vertically the relation (4.5) (4.6).

V,T ljk =-V 0 V, Tijk

On multiplying the two sides by V T'^k we obtain V' TijkV,T,jk = - I Vo( VA Tljk V, Tijk)

(4.7).

V' TijkV,Tljk= | Vo Vo ( V' # T ijk v . i Tijk)

But the covariant derivative of the type Vo = Do the left hand side is zero. So we have

(4.8)

A A A |*V7f =0

where V T is the vertical covariant derivation of T. Let us now suppose that vertical covariant derivative of the torsion tensor is bounded on W(M). On integrating the equation (4.8) along a geodesic ;rand on reasoning as above, we conclude that the

Finslerian Manifolds of Constant Sectional Curvatures

175 175

geodesic ^and on reasoning as above, we conclude that the |2

derivative of FV71 along the geodesic y is zero. So the left hand side of (4.7) vanishes on y. But y is an arbitrary geodesic so that Vj Tjki vanishes everywhere. Therefore (M, g) is locally Minkowskian. Theorem. - Let (M, g) be a Finslerian manifold (dim M > 2), geodesically complete with zero Berwald curvature. If the torsion tensor and its vertical covariant derivative are bounded over W(M) then (M, g) is locally Minkowskian[4]. Corollary. -All compact Finslerian manifolds (dim M> 2) with zero Berwald curvature are locally Minkoswkian[4]. IV. The Plane Axioms in Finslerian Geometry 1.Finslerian submanifolds [7] Let i : S -> M be a k-dimensional submanifold of M. We identify a point x in S with its image i(x) and a tangent vector X e TX(S) to its image i*(X) where i* is the linear tangent map. Thus TX(S) becomes a subspace of TX(M). The canonical imbedding i induces a map 7 : V(S) -> V(M) where z e v(S) is identified to 7 (z). Thus V(S) is fibre sub-bundle of V(M) and the restriction of p to V(S) will be denoted by q : V(S) - • S ; we also denote by T (S) = i"1 T(M) the fibre bundle induced from T(M) by i. The Finslerian metric of M induces over S a Finslerian metric which we again denote by g. At a point x = qz e S (z e V(S)), we denote by Nqz the orthogonal complement of Tqz(S) in T qz(S). (T x(S) = Tx(S) + Nx(S)), Tx (S)n Nx (S) = 0. We denote by Pi: f (S) ->• T(S), and P2 : T (S) -» Nx the projections and we put (1.1)

q'177(S) = q

where N is called the normal fibre and is identified canonically to

176 176

Initiation to Global Finslerian Geometry

q"1 f (Syq-'TXS). If TV(S) is the tangent bundle to V(S) we denote again by p the canonical linear map of TV(S)->q"1T(S). Let X and Y be two vector field on V(S). For z e V(S), (V k Y)z belongs to T qz(S). We have by (1.1) (1.2) where V is the covariant derivation in the Finslerian connection. From (1.2), it follows that V is a covariant derivation in the fibre bundle q'!T(S) -+ V(S) and is Euclidean ( V g = 0). Now a(X, p( Y)), having values in N, is bilinear in X and Y. We have by (11.1 Chap I) (1.3)

(1.4)

f(X,Y)

if

=? n ( X , Y ) =

V-Y-V

i i \

f

X-p[X,Y]

l

i

where f is the torsion of the connection V . Let v : S -> V(S) the canonical section and v the vector field on V(S) over v . (p v = v). From (1.2) we get (1.5)

V . v = V.v

+

a(X,w)

For x e S the fibre q"'(x) is a submanifold of p"'(x). Every vertical tangent vector V I to q"'(x) at a point z G q"'(x) can be identified to a vertical vector of V(M). From (1.5) we get v= After (1.4), on taking into account the homogeneity of the torsion tensor T we get

(1.6)

i

i

Constant Sectional Sectional Curvatures Curvatures Finslerian Manifolds of Constant We therefore have (1-7)

V-v=V

177

-v

Thus if FzV(S) is the vertical tangent space to q"'(x) at z e V(S) the restriction of the map uz to VzV(S) (z e V(S)) defines an isomorphism of this space with T qz(S). On the other hand, the relation (1.5) proves that if X is a V-horizontal vector field on V(S), it does not belong to the horizontal distribution defined by V. Thus x(hX, hY) * 0, where hX and hF are V- horizontal We put (1.8) T(X,

Y)= f (\X,hY),

S(X, Y)= f (hX

where X= V ^ v , X = p(X), Y= p(F).

,hf)

Finally if X, Y and

Zare three vector fields on V(S) projectable by q, by (1.1.1) we obtain (1.9)

g(f(X,Y),Z) = g(f(X,Z),Y).

The torsion tensor corresponding to the Finslerian case admits the same property of symmetry. We call V the induced connection and a(X, Y) the second fundamental form of the submanifold S. To the induced metric we can associate also a Finslerian connection V which will be called the intrinsic connection of the submanifold. In order that the two connections V and V coincide it is necessary and sufficient that the tensor S is zero.

178 178

Initiation to Global Finslerian Geometry

Let x e S and a U (x) a neighbourhood of M containing x, endowed with local coordinates (x a , xp ) (a, p = l...k, a, b=k+1... n). Let U be a neighbourhood of x such that U = {Pe U (x) | xk+1(P) =...=xn(P) = 0} The (xa)|u define a system of local coordinates of S. With respect to this system the induced connection V and the 2-form a( X, Y) become for X, and Y e TV(S),

(1.10) (1.11)

? i

A i a p c

ox

where kac is the inverse of gab and co is the Finslerian connection of M. If we denote by co^on q"1 (U) the 1-form of the intrinsic Finslerian connection on S, we obtain after (1.7.1),

(1.12)

2£ Y =fyapv a v P

= - V a VP (8 a gpy + 5pg a y - 8 y g a p),

The expression of the right hand side of (1.12) is identical to that coming from the ambient connection V that is (1.13)

2 Gr = 2gyiGi = 2gyi (G1 - gickeaGa)

since gTj glc = 0. After (1.10) we put (1.14)

G j = G'-g ic kcaGa

Finslerian Constant Sectional Sectional Curvatures Curvatures Finslerian Manifolds Manifolds of Constant Now G b = 0, the relation (1.13) becomes (1.15) From this we conclude that GA = G \ Thus the induced connection and the intrinsic connection define the same geodesic. From the relation (1.12) we obtain, by vertical derivation, on taking into account (1.15) and the expression of the coefficients of the intrinsic connection

At the same time the analogous expression which is used in the calculation of the coefficients of the induced connection is, after (1.10),

We thus have (1.18)

V Vx = VVX - (gx\ac)

S; .Gc

2. Induced and Intrinsic Connections of Berwald Let S be a k-dimensional submanifold of M, X and Y be two vector fields on V(S) and D the Berwald connection. By the decomposition of the tangent space at the point x = qz e S, (1.1), (D % Y)x becomes (2.1)

D i Y = D . Y + p(X,Y), X = p ( i ) , Y = p ( 7 )

We deduce from it that D is a covariant derivation in the vector bundle q"'T(S)—»V(S). It is without torsion and satisfies

179 179

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Initiation to Global Finslerian Geometry

D - g = 0 where v is horizontal (D ; v = 0) over v, (p( v) = v). Now p has values in N being bilinear symmetric in X and Y. From (1.5) and (2.1) it follows (2.2)

V kv=D

j^v, a(X, v) = p(X, v),

X= p ( l )

Thus the two induced connections of Finsler and Berwald define the same splitting in the tangent space to V(S). It thus follows the geodesies of two induced connections coincide with those of the intrinsic connection. If X is a V - horizontal vector field over V(S), on taking into account (1.8.1) we obtain (2.3) (2.4)

D x Y = V ^ Y + P,(V P(X,Y) = a(X,Y) + P2(V,T)(X,Y), X = p( X)

With respect to a local chart (x a )|u ( X, a- 1, ...k, a, b,= k+l,...n) adapted to the submanifold S we have (2.5)

D{^-)B" " 1 8xa B* = ( G V g a X b G \ y ) d x v and

On the other hand the intrinsic connection of Berwald is obtained from GX = GX- gJ"akabGb by vertical derivation say —

of Constant Constant Sectional Sectional Curvatures Curvatures Finslerian Manifolds of

181 181

3.TotallyGeodesic Submanifolds Definition. - A submanifold S ofM is said to be geodesic at me S if every geodesic starting from m ofS is a geodesic ofM. It is said to be totally geodesic if it is geodesic at all its points[4]. Let us suppose that S is totally geodesic. Let y: [0,a] —»S be a geodesic of S, starting from y(0) = m, parametrized by its curvilinear abscissa over [0,a], u the vector field of unit tangents to y, y i t s canonical horizontal lift in the unitary fibre bundle w(S) and u the vector field tangent along f o v e r u- We have after (1.5) (3.1)

V u U = 0 ^ V M U = 0^CC(M,U) = 0

Let us consider the class of curves defined by / i n V(S). An element of this class is a curve above f ? let (X(SX v ( s ) = ^u(s)), X > 0 and s e [0, a]. We then have since y is a geodesic V u v =i( u )d?uu(s) where i(,) denotes the interior product. Since u is unitary we have

g(u, Vuv) = i(u)dX whence Vwv = g(u,VMv).u(s). Thus u considered as a vector in the space TV(S) becomes (3.2) u = ua -£- + g(u, V- v) u a ^ v u ' dxa dv After (1.11), the condition a(u, u) = 0 becomes

This relation is valid whatever be the geodesic y, whence

182 182

Global Finslerian Geometry Initiation to Global = r*apx (x, v) vp = 0.

(3.3)

We thus have after (1.11) and (1.6) (3.4)

a(l,v) =0

whatever be the vector field x on V(S). Conversely if (3.4) is satisfied for every X e TV(S) then S is totally geodesic. Let us note that after (2.6) this condition is equivalent to P(X, Y) = 0, whatever be X and Y over S. Theorem. In order that a submanifold S of a Finslerian manifold M be totally geodesic it is necessary and sufficient that the second fundamental form satisfies (3.4) whatever be the vector field X over V(S)[4]. From the above theorem it follows that for a totally geodesic submanifold the induced connections from Finsler or from Berwald coincide with the corresponding intrinsic connection. 4. The Plane Axioms Axiom 1.—Let (M,g) a Finslerian manifold of dimension >3. For every point x e M and every subspace E2 of dimension two of TX(M) there exists a surface S passing through x, totally geodesic such that TX(S) = E2 [4] Let (M, g) be Finslerian manifold, dim M > 3, satisfying the axiom 1. Let y = (x, u) be any element of the unitary fibre bundle W(M) where u is the unitary tangent vector to M at x= rcy where 71: W(M) —» M. Let Y and Z be two orthonormal vectors at x and orthogonal to u g(Z,Y) = g(Z, u) = g(Y, u) = 0, ||Z|| = ||Y|| = ||u|| = 1.

Constant Sectional Curvatures Curvatures Finslerian Manifolds of Constant

183 183

By hypothesis there exists a surface S, passing through x, totally geodesic and tangent to the plane defined by u and Y. Let Y and u be two vector fields horizontal on V(S) over Y and u, then we have , after (11.2.1 Chapter I) (4.1)

R(Y,u)u = V - V , u - V , V - u - V [ - a ] u = - V ^ u

S being totally geodesic, we have (4.2)

V [ -. ] u= V ^ u

where V is the connection induced over S. By (4.1) we have g(R(u,Y)Z,u) = g(R(Y,u)u, Z) = - g( V, -., u, Z) = 0 This relation is valid whatever be the vector fields Y and Z orthonormal and orthogonal to u. If one chooses Y' and Z' in the plane defined by Y and Z such that

(4.3)

Y

= V7< Y + Z )

27£(Y-Z>

It is clear that Y' and Z' are orthonormal and orthogonal to u we have 0 = g(R(u,Y')Z',u) = g(R(u,Y)Y,u) - g(R(u,Z)Z,u) The expression Ku = g(R(u, Y)Y, u) is therefore a function of (x, u) and does not depend on the plane passing through u. It thus follows that (M, g) is isotropic. Thus every Finslerian manifold satisfying the axiom 1 is isotropic. Let now x e M and X, Y, Z three orthonormal vectors at x. Let us denote once again by S the surface totally geodesic passing through x tangent to the plane defined by X and Y. Let v the canonical

184 184

Initiation to Global Finslerian Geometry

section S -> V(S) Let us denote by X and Y two vector fields horizontal above X and Y. Since g(Z,X) = 0 , we have by covariant derivation (4.5)

g(V-Z,-(V o T)(Y,Z),X) = O

Similarly, since g (Z,Y)= 0, we have (4.5)

g(V.Z-Vo(T)(Y,Z),Y) = O

Thus V . Z - (Vo T)(Y,Z) is orthogonal to the plane defined by X and Y. From (4.5) we have by co variant derivation (4.6)

g (V • V - Z - V . ((Vo T)(Y,Z)), X) = g(V f Z, (Vo T)(X, X)) - g((Vo T)(Y, Z),( Vo T)(X,X))

By the same reasoning we obtain (4.7)

g(V,V J ? Z-Vf((VoT)(X,Z)),X) = g(V . Z, (Vo T)(X, Y))-g((V0 T)(X,Z),(V0 T)(X, Y)).

Let us derive the relation (4.4) (4.8)

(V [ ^- ] Z,X) + g ( Z , V [ ^ ] X) = 0

Since S is totally geodesic p = 0, we have after (1.8.1)

On the other hand X and Y are horizontal; so we have

Finslerian Manifolds of Constant Sectional Curvatures

185

In virtue of (1.4) and the above relation we obtain

= V

^ X - P 2 T ( R ( X , Y)v,X)

On taking into account (4.10) and (4.12) the relation (4.9) becomes (4.13) g{V[kf] Z, X)=g(R(X, Y)v,T(Z, X))+ g(V . Y-V - X, (Vo T) (Z, X)) After (1.6.II) we put

(4.14) g(A(X, Y)Z,X) = g(R(X, Y)Z,X) + g((V0T)(X, X) (V0T)(Y, Z)) -g((V 0 T)(X,Z),(V 0 T(X,Y)) Subtracting the sum of the relations (4.13), (4.8) from the relation (4.7) and taking into account (4.14) we obtain (4.15) g((A(X, Y)Z, X) = g ((V ^ V0T)(X, Z), Y) -g((V - V0T)(X,Z), X)-g(R(X, Y)v, T(Z, X)) Now (M, g) is isotropic, so in virtue of the properties of the tensor A, (1.7.II), g(A(X, Y) Z, X) is symmetric with respect to Y and Z. Hence also the right hand side. Thus (4.15) becomes (4.15)' g(A(X, Y)Z, X) = g((V ^ V0T)(X, Z), Y) i V0T)(X, Y), X) -g(R(X, Y)v, T(Y, X)) Let us put Y = u with g(Z, u) = g(X, u) = g(Z, X) = 0. In virtue of the homogeneity of the tensor T the right hand side of (4.15)' is

186 186

Initiation to Global Finslerian Geometry

zero. Hence g(A(X, u)Z, X) = 0. On the other hand we have g(A(X, X)Z, X) = 0. NowY = au+bX. Thus we obtain g(A(X,Y)Z,X) = 0 Let us make the transformation defined by (4.3), we get 0 = g(A(X, Y) Y, X) - g(A(X, Z)Z, X) It thus follows that K, = g(A(X,Y)Y,X) depends only on (x, u). By the generalization of the theorem of Schur (1 -II) Ki is a constant. Theorem. Every Finslerian manifold satisfying axiom 1 is of constant sectional curvature in the connection ofBerwald. Definition. A submanifold S of a Finslerian manifold M is semiparallel if for every vector field X V -horizontal over V(S) we have for ally (4.16)

a(Z,Y) = 0

From the relation (2.2)2 it follows that S is totally geodesic and we have (4.16)'

P(X,Y) = 0, P2(V0T)(X,Y) = 0

where X and Y are tangent to S. Axiom 2. Let (M, g) be a Finslerian manifold (dim M > 3). For every point xe M, and for every subspace E2 of dimension two if there exists a surface passing through x, semi-parallel such that TX(S) = E2. Suppose (M, g) satisfies axiom 2. orthonormal vectors at x = pz, ze V(M).

Let X,Y, Z be three

Constant Sectional Curvatures Curvatures Finslerian Manifolds of Constant g(X, Y) = g(Y, Z) = g(Z, X) = 0,

187 187

||X|| = ||Y|| = ||Z|| = 1.

We denote by S the surface passing through x and tangent to the plane defined by X and Y. Let X and Y be two vector fields horizontal on V(S) over X and Y respectively. Since S is semiparallel we have g(V-Z,X) = 0 g(V-Z,Y) = 0 Thus V^Z is orthogonal to the plane (X,Y). So also is V^Z. From the above relations, we obtain, by covariant derivation, (4.17)

g(W- V-Z,X) = 0 , g ( V - V i Z , X ) = 0

On the other hand by (4.12) we have (4.18)

g(V [ki] Z, X) = g(R(X,Y)v,T(Z,X))

From (4.17) and (4.18) we get (4.19)

g(R(X,Y)Z,X) + g(R(X, Y) v, T(Z,X)) = 0

where v is the canonical section of S -» V(S). But (M, g) has constant sectional curvature in the Berwald connection. So we have R(X,Y)v = K(g(Y, v)X - g(X,v)Y) where K is a constant. Thus (4.19) can be written (4.20) g(R(X,Y)Z,X) + Kg(Y, v).g(T(Z,X),X) - Kg(X,v).g(T(Z,X),Y) = 0 The first term g(R(X,Y)Z,X) is symmetric with respect to Y and Z. So we have

188 188

Initiation to Global Finslerian Geometry Kg(Y, v).g(T(Z,X),X) = Kg(Z, v).g(T(Y,X), X) = 0

On the other hand in the tangent plane TX(S), Y has the form Y = av + bX. Thus the last term of (4.20) is equally zero. So we have (4.21)

g(R(X,Y)Z,X) = 0

By an identical reasoning we also obtain (4.22)

g(R(X,Y)Y,X) = g(R(X,Z)Z,X)

Thus the common value of (4.22) does not depend on the plane (X,Y). It is a function of (x, v). By the generalization of Schur's theorem (paragraph 1 .II) it is a constant. Theorem. Every Finslerian manifold satisfying axiom 2 has constant sectional curvature in the Finslerian connection^}. Definition. A submanifold S of a Finslerian manifold (M, g) is called auto-parallel if the second fundamental form of S in the Finslerian connection is everywhere zero. (4.23) where X and Y are two vector fields over V(S). Axiome 3. Let (M, g) be a Finslerian manifold (dim M > 3) For every point x e M et every subspace E2 of dimension two ofTx(M) there exists a surface S, autoparallel passing through x such that TX(S)=E2. Let us now suppose that (M, g) satisfies axiom 3. Let X, Y, Z be three orthonormal vectors at x e M; there exists a surface S passing through x, tangent to the plane (X,Y) and auto-parallel. If X and Y are two vector fields on V(S) over X and Y, we find by an identical reasoning to the preceding case, and taking into account of (4.23), the relation

Finslerian Manifolds of Constant Sectional Curvatures (4.24)

189 189

g(Q(X,Y)Z,X) = O

where Q is the curvature two form. We choose X and Y such that (4.25)

V K i v = X, V K f v = Y

where v is the canonical section of S —> V(S). obtain (4.26)

From (4.24) we

g(Q(X,Y)Z,X) = 0

But the curvature tensor Q has the same symmetry property as the curvature tensor R in the isotropic case. On making the transformation (4.3) we obtain (4.27)

g(Q(X,Y)Y,X) = g(Q(X,Z)Z,X)

Thus the common value of (4.27) depends only on (x, u). deduce from it easily that Q = 0 ( see [1], pp.46-47)

We

If one chooses in (4.24), X and Y horizontal above X and Y, we obtain the relation (4.21). This proves that (M, g) has the constant sectional curvature in the Finslerian connection. We have to vs.

/s

^s.

consider the case where X is horizontal and Y vertical so that p(X) = X, Vy v = Y We obtain (4.28)

g(P(X,Y)Z, X) = 0

where the left hand side is defined by (see [1] p.36) (4.29)

g(P(X, Y)Z, X) = g(( V . T)(Y,Z), X) -g(( V. T)(X, Y), X) +g(T(X, (V0T)(Y,Z), X) -g(T(X,Z),(V 0 T)(X,Y))

190 190

Initiation to Global Finslerian Geometry

where Z is horizontal over Z. written (4.30)

The relation (4.28) then can be

g(( V^ T)(Y, Z), X)) + g(T(X, (V, T)(Y, Z)), X)) T)(X,Y), X) + g(T(X, Z),( VOT)(X, Y))

where the left hand side is symmetric with respect to Y and Z. Hence it must be so for the right hand side also (4.31) g(( Vi T)(X, Y), X) + g(T(X, Z), (V0T)(X, Y)) = g((VYT)(X, Z), X) + g(T(X, Y), (V0T)(X, Z)) where Y is horizontal over Y . Now let us make the transformation (4.3). Then on taking into account (4.31) we obtain (4.32)

g(P(X, Z)Y, X) = g(P(X, Y)Z, X)

Thus the common value of this expression is independent of the plane. On choosing Y = u, we conclude that this common value is zero. From this we deduce easily (see [1], p.46) that the tensor P is identically zero. If the sectional curvature is non zero then by (4.2.II) (M, g) is a Riemannian manifold of constant sectional curvature. If the sectional curvature is zero, it follows from the above argument that the curvature two form Q is everywhere zero. Theorem. Let (M,g) be a Finslerian manifold satisfying axiom 3. Then either (M, g) is a Riemannian manifold with constant sectional curvature or the two form Q of Finslerian curvature of (M, g) is everywhere zero[4].

191 191 CHAPTER VII PROJECTIVE VECTOR FIELDS ON THE UNITARY TANGENT FIBRE BUNDLE [3] (abstract) We give a characterization of the projective vector fields on Finslerian unitary tangent fibre bundle and we prove that in the compact case the existence of projective vectors is related to the sign of the flag curvature. We define the notion of a restricted infinitesimal projective transformation, and introduce a new projective invariant tensor. We determine the necessary and sufficient conditions for the vanishing of projective invariants. We study the case when the Ricci directional curvature (R.D.C) is satisfied under certain conditions especially when the curvature is constant. (This is a generalization of Einstein manifold). We show that any simply connected, metrically complete, Finslerian manifold with a R.D.C. positive constant and admitting a proper vector field leaving the covector of torsion trace invariant is homeomorphic to an n-sphere.

1. Infinitesimal Projective Transformations Let X be a vector field over U a M; for u sufficiently small it defines a local 1-parameter group denoted by exp(uX) of local transformations of U. We denote by exp(uX) its extension to p'CU) and we have p o exp(uX) = exp(uX) o p (see chapter III). With respect to a frame adapted to the decomposition defined by the Finslerian connection X becomes at z e p-](U):

(i.D wheree —-and -§- are Pfaffian derivations with respect to 3c (dx^e^Vv1), and Vo = V,. From (11.1 Chapter I), and for a Finslerian connection it follows that we have V - v = V; X. Let

192 192

Initiation to Global Finslerian Geometry

Vv be the 1-form of splitting associated to the canonical vector field v. Its Lie derivative is (1.2)

)

[

^

for any vector field Y on V(M). If Y is vertical we see easily that the right hand side is zero (see 5.8 chapter III); then from the structure equation we obtain: (1.3)

L(l)Vv(H7) = VHf V,X + R(X, Y)v + (V*T) (Y, V,X).

where H Y is the horizontal part of Y. Definition . A vector field X over M is a projective infinitesimal transformation for the Finslerain connection if (1.4) where Y is a vector field over V(M) such that: (1.5)

V- V = P ( 7 ) = Y,

Vyf 4>=

and *F is a homogeneous function of degree one at v over V(M), T* denotes the vertical differential of *F QV*(Y)= Y1 dj^) Equating (1.3) to (1.4), we get (1.6) WHY VO X+R(X, Y)v + ( V, T)(Y, V, X) = ^*(v)Y + Let us put in (1.6), H F = v, (V^v = 0); on taking into account the homogeneity of T : (1.7)

V, V, X + R(X, V)v = 2 T*(v)v.

Projective Vector Fields

193

Conversely if X satisfies (1.7), by covariant vertical derivation we prove that X is a infinitesimal projective transformation. Let us suppose that X is an infinitesimal projective transformation , the function *F is defined, from (1.7), by (1.8)

2*F = F.g(V fi V fi X > u),

where u = F"!.v and u = F"1 v To every vector field X on M, we associate the vector field Z on V(M) defined by : (1.9)

Z = X-g(X,u)£ -g(V fi X,u)tf ,

where v = V-2-, whence ov (1.10)

Z = p(Z) = X-g(X,u)u.

If X is an infinitesimal projective transformation then Z satisfies (1.11)

V;V;Z

and conversely. Let us express the relation (1.11) in terms of the bracket [ Z , v ]. For this we have at first: V,Z=VjiX-g(VfiX,u)v=VJ?v-g(ViiX,u)

VjV=ViV

Now from the first structure equation (11.1 Chapter I), for a Finslerian connection it follows that [ Z , v ] is vertical and taking into account the second structure equation for the Finslerian connection, (1.11) becomes:

Thus the bracket [ Z , v ] is horizontal; it is therefore identically zero.

194 194

Initiation to Global Finslerian Geometry

The equation (1.11) with Z defined by (1.9) is therefore equivalent to the invariance of horizontal vector fields v by Z . Proposition 1. In order that a vector field X over M defines a projective infinitesimal transformation for the Finslerian connection it is necessary and sufficient that the vector field Z associated to X by (1.9) leaves invariant the horizontal vector field v [3]. 2. Other characterizations of infinitesimal projective transformations. To every vector field X on M, we associate on the one hand the symmetric covariant tensor of order two defined by: (2.1) t(Y, Z) = g( V -X, Z) + g( V - X, Y) + 2g (T( V,X, Y), Z) where Y and Z are vector fields on V(M); on the other hand, the vector field I(X), section of p~!T(M) defined by : (2.2)

I(X)=V 0 V 0 X + R(X,v)v.

We note that t is none other than the Lie derivative of the metric tensor by X. Lemma 1. We have the formula : (2.3)

(V,t)(Y, v) - l(V w -t)(v, v) = g(I(X), Y).

Proof. By definition we have (2.4)

( V, t) (Y, v) = V, (t(Y, v)) -1( V, Y, v) .

when v is horizontal over v (p v = v)

Projective Vector Fields

195

From (2.1) we obtain: (2.5)

t(Y, v) = g( VHf X, v) + g( V,X, Y),

(2.6)

t(v,v) = 2g(V,X,v).

On calculating the left hand side of (2.3) by (2.4) (2.5) and (2.6) as well as the structure equations we obtain the formula. Lemma 2. Let X be a vector field on M and a function i//defined on V(M) by (2.7) we then have for every Y satisfying (2.5): (2.8)

2( V, t) (Y, v) - g(I(X), Y) = 2 T*(Y) F2 + 4 \j/ g(Y, v ) .

Proof. Let Y be a vector field satisfying (1.5); from (2.7) we obtain by vertical covariant derivation (Y = V F ) (2.9) g( V . V, V,X, v)+g( V, V, X, Y)=2*F*(Y) F2 + 4 4>g(Y, v). On the other hand, by the structure equations we have f

V

V

V

V

y

•

[yy]

V

Now DrFvl=Y

V-v+V-Y = 0

V, V,X=V, V ; X+V^X=(V f T)(Y, X) + T(Y, V,X) + VHand V, V,X=V, V . X + V , [Yi]

v

v

[Yi]

[[Yi]i]

X + R(Y, v)X V

'

196 196

Initiation to Global Finslerian Geometry p [ [ t v ] , v ] = - 2 V-Y

Using the definition of t and the above formulas we obtain the lemma. Proposition 2. In order that a vector field X defines an infinitesimal protective transformations for the Finslerian connection it is necessary and sufficient that it satisfies one of the equivalent conditions [3] : (1) (2) (3)

(V,t)(Y, v) - 1 (V w -t)(v, v) = 2 4>g(Y, v). (V, t) (Y, V) = 3 ^ g (Y, v) + 4>.(Y) g(v, v). (V, t)(Y,Z)= 2 T g(Y, Z)+T*(Z)g(Y, v) +Y. (Y) g(Z, v).

Proof. The condition (1) follows from the proposition 1 and lemma 1. The condition (1) implies condition (2), since on putting Y = v in (1) we obtain the relation (2.7), in virtue of the lemma 2, we then have the condition (2). Conversely (2) implies the condition (1) as it suffices to put Y = v in (2). We then have (2.7) from lemma (2) and in virtue of the condition (2) we obtain (1). So (1) and (2) are equivalent and characterize the infinitesimal projective transformations. On the other hand (3) implies (2), it suffices to put Z = v in (3). To show (1) implies (3) we choose a local chart of M; the condition (1) then becomes in this chart: (2.10)

Votio=-£Vitoo + 2¥vi

where ty = L(X) gjj is defined by (2.1). From (2.10) one obtains, by covariant vertical derivation and on taking into account the second identity of Ricci, ([Chapter I]) the relation Vjtjo + Votij = Vjtjo + 2 ¥j. V i + 2 T g l j

where 4^ = S'j^F. On exchanging the indices i and j and on adding it to this relation we obtain

Vector Fields Projective Vector (2.11)

197 197

Votij = 2^ g l j + v ^ j + VJ 4^ Q¥, = 5-i 40

This relation is none other than the condition (3). 3. Curvature and infinitesimal projective transformations [3] Let Sx : gu(u,u) = 1 be the indicatrix at x ;we have denoted by W(M) = U XEM SX the unitary tangent fibre bundle on M, by r\(g) the volume element of W(M). The divergences of the differential forms on W(M) are calculated relative to r)(g) (see Chapter III) We have shown that if X is a projective vector field, we have (1.11) where Z is defined by (1.10) Lemma 1. Let (M, g) a compact Finslerian manifold without boundary; for every projective Xwe have : (3.1)

=f *VK

g(R(X,u)u,X)ti(g), (M )

where (,) denotes the global scalar product and where p( u )=u, Vyu = 0. We note that V £ Z is vertical part of the HftofXonW(M). Proof. If X is projective, from (1.11) we have

Z,Z) =(V«Z,V£Z)-g(R(X,u)u,X). The left hand side is a divergence on W(M), by integration we obtain the lemma. Lemma 2. Let (M, g) a compact Finslerian manifold without boundary; for every vector field X on Mwe have

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Initiation to Global Finslerian Geometry

(3.2) n[(M) with (3.3)

g(R(X, u)u, X) T!(g) = [ ( M ) O(X,

O(X, X) = [Hy + 2(VjV0Ti - VrV0Tr,j) XjXj]

where (Hy = H'irj) is the Ricci tensor of the Berwald connection Proof. By the Bianchi identity written in local coordinates for a Berwald connection, we obtain the identity ( see 4.10 I Chapter VI)) (3.4)

5-mH'jk, + D, GVm - DkG1jlm= 0,

(8-m =

whence, on multiplying the two sides by v", (3.5)

V 8-m H'jk| = 0

On the other hand by (§4.4.1. I Chap VI) the curvature tensor R of the Finslerian connection is linked to H by (3.6) Ryu = Hijkl+TijrRroki+V,VoTijk-VkVoTiji+VoTi,rVoTrjk- VoTikrVoTrj,

R being skew-symmetric with respect to the indices i and j we deduce from i t : (3.7)

Hijk, + Hjikl +2 Tijr H r okl + 2(V,V0 Tijk - V k V 0 Ty,) = 0

On multiplying the two sides by g*1 and on changing the indices : (3.8)

gkl Hlljk+ 2 Tkir Hrojk = Hy + 2 (VjV0Ti - VrV0Tr,j).

We now consider the vertical 1- form on W(M) defined by its components :

Projective Vector Fields (3.9)

r k = rk - uk ro/F

(3.10)

rk = F 1 Hiojk X1 XJ,

199 (r0 = rk vk)

where X is a vector field on M. On taking into account (3.5) and (3.8) we obtain : (3.11)

8-r = ng (R(X, u), X) -
where 5'denotes the co-differential of the vertical 1 -form on W(M) and O(X, X) is defined by (3.3), M being assumed compact, by integration on W(M) we get lemma 2. From lemmas 1 and 2 we have : Theorem. Let (M, g) a compact Finsleriam manifold without boundary and X an infinitesimal projective transformation on M. If the quadratic form &(X, X) is negative definite on W(M) then the projective transformation corresponding to X is reduced to identity; if 0(X, X) or if the flag curvature g(R(X, u)u, X) is non positive then X is an isometry and has the covariant derivation of the horizontal type zero[3]. Proof. If O(X, X) is negative definite, from the lemmas 1 and 2, it follows that X is zero if O(X, X) or the flag curvature g(R(X, u)u, X) is non positive then (3.12)

V£Z = 0 V«X = F-2

whence by vertical derivation (3.13)

VjX, + V 0 T i j h X h = F-2ViVjX0 + F 2 (gij -UjUj)V0X0.

On multiplying the two sides of this relation by glj and on contracting (3.14)

Vj Xj + V0(X jT1) = nF"2V0X 0

200

Initiation to Global Finslerian Geometry

where T1 = gjk T'jk. Now X is projective. So, by (2.11) we obtain (3.15)

¥ = (1/n + 1) V0(ViXi + T.VoX1),

(3.16)

T=F-2V0V0X0.

On taking into account (3.12), (3.16) becomes (3.17)

v

F = (l/n+l)V 0 V i X i

Similarly the equation (1.11) becomes (3.18)

XkRioko = X k H i oko = 0

Now the curvature tensor H is written (3.19)

H'jk, = (1/3) Oj[o-,H'oko -

O-RHVI

In virtue of the relations (3.18), (3.4) as well as the homogeneity of the tensor G we get (3.20)

HjiXjX' = Vr5jHirXJX' = - V0(Grjr|XJX')

Thus the first term in O(X, X) is a divergence ; we next analyze the other terms. In virtue of the proposition 2 and (3.12) we have (3.21)

(V0Tr,j V0Tr - Vr Vo rij)XjXj = 8((V,T)(X,X))- 5(A,u) - TV g(X, T*)

where T* is the torsion trace vector, X = Tyr t'r Xj and 5 denotes the co-differential relative to p(g). On the other hand by (3.14), (3.15) and (3.16) it follows (3.22)

VoVog(X, T) =

n+

Projective Vector Fields

201

On using (3.12) and (3.13) we obtain (3.23) |V;g(X,7)|2+24'.g(X,T*) + Div nwhere Div denotes the terms which are divergences ; on adding it to (3.21) we get on taking into account of (3.2) and (3.12)

whence V;g(X,T*) = 0 The relations (3.17) and (3.22) then imply that ^F is identically zero. Thus, M being compact, X is an infinitesimal isometry. Hence g(V,X,v) = 0. Then (3.12) gives V^X= 0, and we deduce immediately that X has covariant derivation of the horizontal type zero. From the above theorem it follows in particular Corollary. On a Minkowskian compact manifold without boundary there do not exist projective transformations other than the isometries P0(M) = 10(M) [3] 4.Restricted projective vector fields [3] Let G be the second curvature tensor of the Berwald connection D and X a projective vector field (Xe P0(M)). We say X is restricted if it leaves invariant the trace tensor of G : L( X) trace G = 0,

202

Initiation to Global Finslerian Geometry

where trace G = (Gjjk). After the definition of projective infinitesimal transformation it follows immediately that the function *F homogeneous of degree one at v is in fact a linear form at v, ¥ = aj(x)v', and conversely. We denote by P0(M, r) the largest connected group of restricted projective transformations. We note that if X e P0(M) and leaves the torsion trace co-vector invariant then X is restricted. Let suppose (M, g) a Landsberg manifold (P = 0), geometrically it means that if we endow the manifold V(M) with a Riemannian metric gij(z)dx'dxJ+gjjVvlVvJ where gy(z) is the Finslerian metric, then p : V(M) -» M is a totally geodesic fibration, that is to say, for every x e M, p"'(x) is a submanifold totally geodesic in V(M)[see [§7 chapter V]]. Let us suppose that it is so, then for X e P0(M, r) we obtain (4.1)

Vk(VjX' + f jh V 0 X h ) + XhR)hk = 8VFk+ 8\Vj + T j jk T ,

where 8 j is the Kronecker symbol. with scalar values on W(M): (4.2)

To X we associate a 2-form

a(X) = (l/2)(ViXj - VjX,)dx'Adxj.

On taking into account (4.1) we obtain : (4.3)

5(i(X)a(X)) = g(a(X), a(X)) - pRjjX'tf + (n-l)X'T,]

where i(X) denotes the operator of interior product by X. On the other hand we have for X projective (4.4)

L( X )V0T'jk = V 0 L(X)TV ^T' j k + vT jk 4> s .

Now (see 5.12 Chapter III) (4.5)

L( X )fjk = Vk(VJXi + fjhVoX*1) + XhP'jhk + V0Xh Q'jhk

From (4.5) we obtain on contracting i and j

Projective Vector Fields

203

L( X )Tk = Vk(V,Xj + TjV0Xj) = V k f

(4.6) with (4.7)

f=V i X' + T1V0Xi

From (4.4) we then have (4.8)

L(X)V o T k = VoVkf

From (4.15), we obtain, by vertical derivation

But by hypothesis V0T j k = 0, whence V0T k = 0, in virtue of (4.8) the above relation becomes (4.9) whence (n+l)XivFj = X'Vjf - ^FX'Tj = Vi(XT ) - fVjX1 with p= g(X, T*) The above relation can be rewritten: (4.10) (n+l)XivPi =Vi(X'f) - f(ViXj + TjVoX1) + fVop = - 8((X+ pV)f) -f2 where 5 is the co-differential [see chapter 3]. In (4.3) appears the expression XlvFj which we have just calculated; we have still to show that p*¥ is a divergence. To this effect we put

The Yj, considered as the components of a vertical 1 -form, we have (4.11)

-8-Y = Xlx¥k- nF"'g(X,

204

Initiation to Global Finslerian Geometry

Similarly if we put Zi = F we obtain (4.12)

Z = n F 1 g(X, u) T - X 1 ^ - 2g(X, T*)T,

From (4.11) and (4.12) we have 8'(Y-Z) = 2g (X, T*)^ Thus (4.10) becomes (n+l)X'Ti = - 8[(X + pv)fj- ^—

8-(Y-Z) - f2 = Div - f2.

Substituting this expression in (4.3) we obtain 8(i(X)a(X)) = g(a(X), a(X)) + "-— 5(X+ pv)2 - 2R1JXiXj + Div Let us suppose that M is compact, without boundary then on integrating on W(M):

6(X+PY)\(g)=2 It thus follows that if the Ricci curvature R(X, X) is negative definite, then the infinitesimal projective transformation corresponding to X is the identity. If R(X, X) < 0, then 8(X + pv) = 0 and a(X) = 0. Thus X is an infinitesimal isometry, and a(X) = 0 implies that the covariant derivative of horizontal type of X is zero.

Projective Vector Fields

205

Theorem. Let (M, g) be a compact Landsberg manifold without boundary (P= 0) andXe P0(M, r). If the Ricci curvature R(X, X) is negative definite the corresponding projective transformation to X is identity. If R(X, X) is non positive then X is an infinitesimal isometry and its covariant derivative of horizontal type is zero[3]. 5. Projective invariants [3] Let (x1) be the local coordinates of a point x e U c M and (x1, v1) the induced coordinates of the point z e p"'(U). Let us denote by G jk(z) the coefficients of the Berwald connection; in this chart, the curvature tensors H and G of this connection are defined by ji - G ji s G s k ) -(5 t G j k - G j ks G s i) + GVkGrji - G'i r G r j k

G jkl -

;

\1

O ](j jk,

jkl - O ](

with

U jklm - 0 m Lr jkl

2G1 = G 1 jk v i v k ,

GWkG1

If X an infinitesimal projective transformation we have L( X )Gj = W ,

(5.1)

iV

^

L( X )G\ = 5 ^ + vivPk

'

V

(5.2)

(5.3) and

L( X)G)u= 8jj8 ,^ k + 5'k5 ^ + S^Sj^k + vj5 ) 5 jD^k = D,8 j ^ k - T r Grjki

On contacting i and k in (5.2) we obtain

206

Initiation to Global Finslerian Geometry L( X )Hj, = DjT, - nD^j + D05 j¥,

(Hj, = Hrjri)

and

On eliminating ¥j in (5.2) and (5.3) we get (5.4) (H kl -tf lk )+5 j k i/jrS 1 ,H jk +v j 5j

W'jkrH'k,--^-j-[8'j With (5.5)

^ j k - n H j k + Hkj + Vr5jHkr Jf j j kl = Gjjkl - ~

(5.6) 1

[S'jG^, + 5'kGrjri + W j * + v'GVki]

2

The PF and PF are homogeneous tensors of degree zero and -1 respectively, invariant by the infinitesimal projective transformation X and are called generalized projective tensors ([3]). If X is a restricted infinitesimal projective transformation 2

1

the tensor W reduces to G and W to (5.7) ^iJki=Hijkl--^-Y[8ik(nHjl+Hlj)-8i,(nHjk+Hkj) + (n-l^/Hk, - Hlk)] If we multiply the two sides of (5.4) by v and v1 respectively we get the tensors invariant by the infinitesimal projective transformation X. Let us put (5.8)

W\=W}oko

where the right hand side becomes by (5.4)

Vector Fields Projective Vector

207

(5.8)' W\ = Hjoko

-tfkHoo + - 1 —-v i [(2n-l)H ok - (n-2)Hko] n-\ n -\

From (5.4), (5.8)' and (4.19) we obtain the following formulas: i \->-7)

yy

.

i .

i .

oki

(5.10) i

In other words the projective curvature tensor W is expressed with i

the help of the tensor W\ in the same way as the curvature tensor Hjki of the Berwald connection with the tensor ffOko- It thus follows the equivalence of the following three conditions : W\ = Q^WloM = 0<^Wlm = 0.

(5.11)

One can ask for the necessary and sufficient conditions for i

•

the tensors W and W to vanish.

We consider two distinct cases

and dim M>2. 1. Suppose (M, g) is isotropic : (5.12) Hioko = K(F25ik-v1vk) where K is a function on W(M). Whence on taking into account (3.19) (5.13)Hiok,=K(5ikv,-8ilvk)+(l/3)5-,K(F281k-vivk)-(l/3)5-kK(F25i,-viv,) From it we deduce by contraction of i and k (5.14)

Hoi = (n-l)K v, + |(n-2)F 2 5-, K

208

Initiation to Global Finslerian Geometry

Similarly by contracting i and k in (5.12) and on deriving vertically and taking into account (5.14): Hl0 = (n-1 )K.v, + 1 (2n-1 )F25 ,K

(5.15)

i

On substituting H'oto, Hoi and Hi0 in (5.8)', we note that vanishes.

By (5.11) it follows that the projective tensor W

vanishes. Conversely, let us suppose that W vanishes. (5.8)' we have (5.16)

.

W\

H"oko = -^-SYHoo - -^—V

Then by

[(2n - l)Hok -(n-2)Hko ]

Whence on multiplying the two sides by Vj, (2n - 1) Hok -(n - 2) Hko = (n+1) F"2 Hoo vk On substituting this expression in (5.16), we see that (M, g) is isotropic with K= 2.

n-\

- F Ho0

Let us suppose now that H'jk| = K (5'k gji - 8'i gjk),

(K = constant)

On putting this expression in (5.7), we note that W= 0. Conversely suppose that W = 0. implications:

We the have the following

W 'jki = 0, => W 'okl = 0 <=> W joko

=W\=0

Projective Vector Fields

209 l

The last relation proves that (M, g) is isotropic, therefore Wis i

*

zero. Thus the difference W - W is zero. We have therefore by (5.4) and (5.7) (5.17)

5',(vr 5j Hkr) - 5ik(vr5jHlr) = (n-iySjCHu - Hlk)

Let us multiply this relation by vk; from the fact that n > 2, we have (n-2)v'vk8jHlk = 0,

vk8jHlk = 0

Thus (5.17) gives us

(M, g) being isotropic, on making explicit this relation, by (5.14), we obtain gjid ' k K + V|5 j k K - g jk d 'iK - v k d ji K = 0

let us multiply the two sides by g*1; we get (n *2): (n-2) 8 kK - vk gj'5 j,K = 0 => 8 \K = 0 Thus (M, g) has sectional curvature constant. Theorem [3]. Let (M, g) be Finslerian manifold (dim M > 2) ; in i

order that the projective tensor W ( respectively the restricted * projective tensor W ) is zero it is necessary and sufficient that (M, g) be isotropic (respectively has constant curvature in the Berwald connection) From the above theorem it follows :

210

Initiation to Global Finslerian Geometry

* Corollary [3]. - For the restrictedprojective invariants Wand G (dim M> 2) to vanish, it is necessary and sufficient that (M, g) be a Riemannian manifold with constant sectional curvature. 6. Case where Ricci directional curvature satisfies certain conditions The Ricci directional curvature at a point z e V(M) is defined as the scalar C(x, u) = F" 2 H,JVV = F^RjjvV, where (Ry = R1^)- Now C is homogenous of degree zero in v and therefore determines a scalar function on the unitary tangent fibre bundle W(M). Let us suppose that the Ricci tensor Hy satisfies (6.1)

AH(v,v) = 0,

where H(v, v) = HJJVV. If X e P0(M), on taking the Lie derivative of the two sides of (6.1) we get D0L(X)H00 - 4 Hoo v|/ = 0,

Hoo = H(v, v),

Let: (6.2) whence

4 Do DoV + — HOoXj/= 0,

(6.3) F"4 A (W A V) - F"4 A v|/. A V +

C(x, u) (i|//F)2 = 0, n— 1

where u = v/F. Let us assume M is compact without boundary; then the first term of (6.3) is a divergence ; by integration on W(M) we obtain : (6.4)

= - ^ -

f C(x,u)(M//F)2ri(g)

Yl — I

"

W{M)

Projective Vector Fields

211

If C(x ; u) < 0, then D0\j/ = 0 and by (7.3) we get \\J = 0 ; thus, M being compact X is an isometry. If C(x, u) = 0, then Do i|/ = 0, from the expression of \\i it follows Do Do Do Xo - 0, M being compact we conclude then V0V0X0 = 0 = \\i, therefore X is an isometry. Theorem. If a compact Finslerian manifold without boundary satisfies the condition (6.1)(D$ H(v, v) = 0)and such that C(x, u) <0on the unitary fibre bundle then P0(M)=I0(M)[3]. Let us suppose that the Ricci directional curvature C(X, u) is constant, non positive, on the unitary fibre bundle W(M); in this case it satisfies the condition (6.1) of the above theorem, we have Corollary. On a compact Finslerian manifold without boundary with Ricci directional curvature non positive constant, the largest connected group of projective of transformations coincides with the largest connected group ofisometries.[3] 7. The complete case. A. Let X be a vector field leaving invariant the 1-form of splitting Vv; we then have (7.1)

A A X + R(X,v)v = 0.

Let g be a geodesic parametrized according its arc length s. consider along the geodesic the function (7.2)

u = Xj ^~ = g(X, u), as

(u = dx/ds),

We

212 212

to Global Global Finslerian Geometry Initiation to

where u is the unitary tangent vector to the geodesic considered. From the differential equation of the geodesies: Vw' ds

du' ds

* r';jk (x,u)u j u k = 0.

We deduce ds

ds d2[i

and by (7.1) we have —j-= 0. ds Thus (7.3)

n = - ^ |s=o s+ u0 = (VjXi u V ^ s + g(X, u)0.

let us suppose that (M, g) is geodesically complete then s can take every value from 0 to + QC if the vector X is of bounded length, so is the function n, \\ju\\ < \\X\\, By (7.3) it follows that one has necessarily : VjXi i^u1 = 0 X is therefore an infinitesimal isometry. We have thus established a generalization of a result of Hano[19]. Theorem. On a geodesically complete Finslerian manifold every vector field of bounded length leaving invariant the 1-form Vv of the splitting is an isometry(see Chapter III) [3]. B.let us suppose the Ricci directional curvature to be constant. Then we have by vertical derivation (7.4)

Hji + Hij + v d • H] r = 2 C gji,

Projective Vector Fields

213

where C is a constant. Let X e Po(M); on taking the Lie derivative by X of the two sides of the (7.4) we obtain (7.5)

(n-1) [Dj v|/,+ D, i|/j+ Do d] \|/,] + 2 C tj, = 0 On multiplying (7.5) by v* and v1

where t = L(X)g. successively (7.6)

Vv v|/+ -2^- Vv g(X, v) = 0.

Let us suppose C - constant < 0, and put K2=

1-/T

Y

From (7.6) we obtain by derivation (7.7)

V;VOH/-K2F"V

=0

We then conclude along the geodesic g (7.8)

- ^ ds

-K 2 ¥ = =0.

Now the solution of (7.8) is of the form (7.9)

¥

=Ach(Ks + B),

where A and B are constants along g. (7.6) we obtain ds

whence (7.10)

^

K> ds

On the other hand from

214

Global Finslerian Geometry Initiation to Global

where D is constant along g. Now from (7.2) we have ||//| < \X\. Let us suppose X to have bounded length ; from (7.10) it follows that the length of X can be bounded only if A = 0. From (7.9) we have T = 0, therefore X leaves invariant the 1 -form of splitting. After the preceding theorem X is an isometry. (7.6) we conclude along the geodesic g :

as2

If C= 0, by

= 2A => u = As 2 + Bs + D,

where A, B and D are constants on g. Similarly \i can be bounded only if A and B vanish, whence \\i is zero , and we are led to the preceding case. Theorem.- If (M, g) is geodesically complete Finslerian manifold with directional Ricci curvature non positive constant, then every projective vector field of bounded length is an isometry[3]. 8. Case where the Ricci directional curvature is a strictly positive constant. Let us suppose H(u, u) = C, a positive constant; then the function T satisfies (7.6), whence by derivation —

1 n-\

—

where u is the horizontal vector field over u ( V i u = 0, p( u ) = u,

M=i). Let us put: -

w_j

,

~

i

i

o

, /

-—j"

Let g be a geodesic of M ; we put along g

,

uJ

Projective Vector Fields

215

ds where s is the arc length ; by (9.1) T satisfies on g the differential equation of the second order : (8.2)

ds

In the following we suppose that the fundamental function F is symmetric (F(x, kv) = |A,|F(x, v), VA, e R) and we define a distance function on M by putting dx d(A, B) = inf c e C(A, B) L(c), L(c) = J L(x, — )dt a

where C(A, B), (A, B e M) is the set of rectifiable curves joining A to B. Let us suppose (M, g) is metrically complete; by the theorem Hopf-Rinow, valid in Finslerian geometry, [31], (M, g) is then geodesically complete. In addition, we suppose that projective infinitesimal transformation X leaves the torsion trace co-vector invariant (8.3)

L ( 1 ) T * = O « • dif

= 0,

Thus / is a function defined on M. Moreover the mean directional curvature is a positive constant; therefore bounded from below by a positive constant; by the theorem of Myers of Finslerian geometry [31] (in the proof of this theorem it is the expression of the Ricci directional curvature which intervenes); therefore M is compact. By (8.2) / becomes along the geodesic g. (8.4)

/ = A cos Ks + B sin Ks + C,

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Initiation to Global Finslerian Geometry

where A, B, and C are constants on g. The function / attains its absolute maximum and minimum on M. Let us suppose that this maximum is attained at s = 0 and the critical value of / at s = 0 is +1 ; then we have (8.5)

df == -AK sin Ks + B Kcos Ks ds

whence (8.6)

0 = df ds

S=O

= BK and

f(0) = A + C = l .

Let P o e M, a point corresponding to s = 0 ; the function / has an absolute minimum for s — 7i/K; we denote this point by Pie M and suppose that the value of the function / at this point is equal to -1 ; therefore (8.7)

/(7t/K) = -A + C = -l

From (8.4), (8.6) and (8.7) it follows : (8.8)

/ = cos Ks. 9. The Second Variational of the Length [Id]

Let g : [0, t] —» M be a Finslerian geodesic, parametrized by curvilinear abscissa, situated in a neighborhood U with local coordinates. We define on p"'(U) the semi-basic 1-form : (9.1) y

J

c o = — d x ' = Uidx dxl

and consider the integral

(* = &.) ds}

Projective Vector Fields

217 r

(9.2)

L(g)= { co(£) = { co,

where g : [0, t] ->• W(M) is the canonical lift of g on the unitary tangent fibre bundle. Let t, be vector field on U and ^ its lift on p"'(U). We denote by y a = exp(a^) the local 1-parameter group generated by £, (a G ]-s, +s[) and by y a = exp ( a ^ ) its extension [po exp(a^) = exp(a£,) op] We denote by Y the restriction of t, to g and g a =ya • g , ga= y a- g • The length of the g a is defined by (9.3)

L(g a )= | .

co=

whence the second variational is determined by

(9.4)

J

^ f J U o = J. U£)n£)a>

here L( ^ ) denotes the Lie derivative by £. The right hand side of (9.4) must be evaluated along the lift of the geodesic g (Vu = 0). The expression under the sign of the integration becomes

(9.5)

L(i)L(£)co = i(£) di(£)dco + di(£)di(£)co

where i( ) denotes the operator of the interior product. Now (9.6) and (9.7)

dco = V - dVuj = VUJ Aco^i + Uj Q*!

where (co'j) is 1-form of the Finslerian connection and Q. the corresponding curvature 2-form. We obtain

218 (9.8)

Initiation to to Global Finslerian Geometry L(£)L(£)co = i(£) d(V-u,)dx' + V | u, V^1 - V^^Vuj + UjQijC £ )£ + V - u,d^ + d(u, V - £)

The vector field u being unitary we have i(OVu j ^ ' j - u ' u j ) V - ^ We, then, obtain along g: (9.9) L( | g = dg(u, Vf Y) + [g(Vfi Y, V. Y)-g(R(Y, u)u, Y) -||V fi g(Y,u)|| 2 ]ds Theorem [3]. If g(s) is a geodesic of the Finslerian connection parametrized according to the its arc length on [0, t] we then have along g :

(9.10)

^l a =og(V [; Y,u)|o da [g( Vfi Y, V, Y) - g(R(Y, u)u, Y) -1| V, g(Y, u)||2]ds. 10. Homeomorphie to the Sphere

Let Z be Jacobi vector field [3] along the geodesic but orthogonal to g, zero at the point s = Oand let g(t) ( s= t) a fixed point on g ; we put: (10.1)

Z(s) = ||Z(t)||Y(s),

where ||Z(t)|| is the norm of Z at the point g(t). It is clear that Y(t) is unitary.

Projective Vector Fields

219

Let be o(oc) a geodesic passing through g(t) and tangent to Y at this point (10.2)

<J(0) = g(t),

da

So a(a) determines a 1-parmater family of geodesies g a with go = g and such the Jacobi vector field is realized by ga. There exists s > 0 such that for ae]-s, s[ we have (10.3)

g a = exp (aY) g.

Let L(ga) be the length of the geodesic ga (starting from the origin s= 0) with the above hypotheses the second variation of L(ga) defined by (9.10) then becomes ,2

(10.4)

d

-

a

a=0=

j [g(VfiY, V,Y)-g(R(Y,u)u,Y]ds

where u is the horizontal vector tangent to g and Y is the Jacobi field along g orthogonal to u (10.5)

V fi V fi Y + R(Y,u)u = 0,

g(Y, u) = 0.

On taking into account (10.1) and (10.5) the formula (10.4) becomes:

00.6)

da7

We now calculate this second variational with the help of (8.4) and (8.8). In fact if L(ga) is the length of the geodesic g a between PQ (S= 0) and a(a) we have by (8.8)

220

Initiation to Global Finslerian Geometry

(10.7)

/ ( a ( a ) = cosKL(g a )

On the other hand the function *F deduced from / by derivation must satisfy an equation of the type (8.2) where at s we substitute the arc length a along the geodesic a(a). We can choose / in the form : (10.8)

/ (o(a)) = Ai(t) cos Ka+ B,(t) sin Ka.

Equating (10.7) to (10.8) we have (10.9)

cos KL(ga) - A, (t) cos Ka + Bi (t) sin Ka .

A derivation with respect to a gives us : (10.10) -K sin KL(ga) ^

= -K A,(t) sin Ka + Bj Kcos Ka

and for a = 0 we obtain Bi = 0. Thus on putting a= 0 in (10.9) we have Ai(t) = cos Kt. The relation (10.10) then becomes sin KL(ga). -&L- = cos Kt .sin Ka. A second derivation with respect to a then gives K cos K L ( g a ) ( ^ ; )2 . + sin K L ( g a / ^ f g ) whence for a= 0 :

= K. cos Ka,

Projective Vector Fields

221

72

(10.11)

da2

la=0

Equating (10.6) to (10.11) and by integration we obtain : (10.12)

||Z(s)|| = 1/K ||Z(s)||'s=o • sin Ks.

From this relation it follows that the Jacobi field along the geodesic g, vanishing at Po = g(0), vanishes again at the point Pi = g(7t/K). Let g a be the 1-parameter variational of the geodesic g, passing through Po, the points which at distance 7i/K from Po, denoted ga(7t/K) describing a curve y whose tangent vector at a point is the Jacobi vector field at this point. Now, after (10.12), the Jacobi vector field vanishes at ga(u/K). So the curve y becomes the identity at the point Pi. On reversing the roles Po and Pi we conclude that Po and Pi are the only critical points of f. On supposing the in addition that M is simply connected we deduce using Milnor [30] that M is homeomorphic to an n-sphere. Theorem [3]. Let (M, g) be complete, simply connected Finslerian manifold of dimension n with Ricci directional curvature constant strictly positive et let X be an infinitesimal projective transformation leaving the torsion co-vector invariant. Then M is compact, X defines 1-parameter global group of restricted projective transformations and M is homeomorphic to a sphere.

222

CHAPTER VIII CONFORMAL VECTOR FIELDS ON THE UNITARY TANGENT FIBRE BUNDLE Abstract. Let X be a vector field over M and exp(tX)the local 1-parameter group generated by X and exp(tJT) its lift to V(M). We call X a conformal vector field or a conformal infinitesimal transformation if there exists a function cp on M such that the Lie derivative of the metric tensor g is equal to 2 (p.g. To every field of co-vectors Y(z) e A^W) is associated a covariant symmetric 2tensor x(Y). We calculate its square by making intervene explicitly the curvature of the space. This formula helps us characterize the infinitesimal conformal transformations when M is compact without boundary. In paragraph 4 we establish a formula giving the square of the vertical part of the lift of conformal vector field on V(M) in terms of the flag curvature and we show that there exists a non-trivial conformal vector field only if the integral of the quadratic form (R(X, u)u, X) is positive. Next we take up the case when the scalar curvature H = g*k H jk where 2 H jk = c^Hiv, v)/<9vJ 2, and M compact, the largest connected group Co(M) of infinitesimal conformal transformations coincides with the largest connected group of isometry Io(M), thus generalizing the Riemannian case. We deal with case where the 1-form X = Xj(z)dx' corresponding to the vector field is semiclosed and obtain in the conformal case the corresponding equations. Let u be a semi-closed 1 -form whose co-differential 8a is independent of the direction and let A a be the horizontal Laplacian and let X(y), y e W(M) a the function such that A a = X(y) a. We give an estimate of ^.(y) as a function of proper value of the flag curvature. The rest of the chapter is devoted to the lift to W(M) of the conformal vector field when its dual 1-form is semi-closed.

Conformal Vector Fields

223

1. The Co-differential of a 2-form. Let u : M —» W(M) a unitary vector field and co = Ujdx' the corresponding 1-form. We denote by (dco)""1 the (n-l) th exterior power of dco, by r\ the volume element of W(M) where (1.1)

(w-1)!

W e suppose M to be compact and without boundary. W e denote by 5 the co-differential operator, formal adjoint of d, in the global scalar product defined over W ( M ) . If 7i i = a;(z) dx 1 and %2 = bjVu1 (bjv1 = 0) are respectively the horizontal 1-form and the vertical 1-form over W(M) w e have (see chapter III, §7) (1.2)

5TH = - (V j aj - ajVoT j ) = - g ij Djaj

(1.3)

87t2 =

i

i

ij

^

S

Svl

where (T1) is the torsion trace vector. Vk and V, are the components of the covariant derivations in the Finslerian connection and Dj is the covariant derivation in the Berwald connection. Let Ap q(W) be the space of differential forms of type (p, q) over W(M)., that is to say the forms containing p times the horizontal form dx and q times the vertical form (3 =Vu. We denote by AP(W) the space of p-forms of type (p, 0) and by H the restriction of A pq to AP(W) (by abuse of language, the projection). If C,(z) e Ai(W), then dC, is the sum of the forms of type (2,0) and of type (1,1) on W(M). Let us put (1.4)

C^Y^dx1

224

Initiation to Global Finslerian Geometry

The restriction of the differential dC, to A2CW) will be denoted d C,

(1.5)

a = d C, = Hd; = - (Dj Yj - DjYj )(z)dxjAdxj

We now propose to calculate the horizontal part of the codifferential of d <;. To do it, let 71 = £ j (z)dxj e Ai(W). Then the restriction of the co-differential of d C, to Ai(W) will be a 1-form denoted by 8 d C, such that (1.6)

<S d C,,n> =

We denote by ( , ) the local scalar product at z and by < , > the global scalar product on W(M) ([1]). Then we have (a, d 7t)(z) = | g ' V (DkCi - D,Q0 (DiYj - DjYO(z)

gV =g lk D k [g il 0(D 1 Y J -D J Y j )](z) - glk Dk i C, (DiYj - DjY0(z) - g ' V Ci Dk (DiYj - DjYiXz)

(1-7)

Now (1.8)

- glk Dkg"1 = -2vrVrTJ" = - 2V0Tjli

where Vo — vrVr. In virtue of the symmetry property of the torsion tensor T, the second term of the right hand side is zero. We therefore have (1.9)

(a, d 7i)(z)=-gik Dk ( D . Y J - D J Y O ^ (z)+Div on W(M)

M being assumed compact without boundary, on integrating (1.9) on W(M), and taking into account (1.6) we obtain (1.10)

(SdQj

= (Sa)j = - glk Dk(DjYj -DjYi

Conformal Vector Fields

225

We have thus an analogous expression in the Riemannian case. 2. A Lemma To the 1-form Q= Yj (z) dx1 £ Ai (W) we associate a covariant symmetric 2-tensor x defined by

(2.1)

1 2 Xj j(Y) = DjYj + Dj Yj + - D o (5- Yj + d*Y()+- Tgn 2

,

j

n

d) = — - , Vo = v'Vj and T is a function, homogeneous of degree dv' zero in v. We choose Mr1 on W(M) such that the trace of x is zero : (g, x) z = 0 . We then have

The square of the tensor x becomes (2.3)

i xij Xij = (x, x)z = x1J D,Yj + | T lj D05,Yj

Now the first term of the right hand side becomes: (2.4) x1J DiYj=gV Tki DiYj=g lkD,(gjlTk,Yj)-g ik D ig jl xkl Yj-glk Now (2.5)

D igk , = - 2 Do Tin

On taking into account of (1.10) and (2.5) we obtain

226 (2.6)

Initiation to Global Finslerian Geometry -glk Dj xk, = ( £ < x ) , - ^ g l k DiD0( dk Y, + d,Y k )--D, y 2 w ik + -H/D o T l -2g D i D l Y k

where Ti is the torsion trace co-vector. On using the Ricci identity ( see Chapter I) the last term becomes -2 glk Dj D, Yk = -2glk(D,DiYk - Yr Hrkll - d r Y k H r 0ll ) (2.7).

= -2D,(glk D,Yk)+ 4D0 T,lk D,Yk - 2 [YrHrkll + dr Yk Hrol,]gkl

-2 gik D,D, Yk= 2D, V|/+2D0 Tf x r f D A A Yk - -D0T,i|/ -2[Y r H r kli +e r Y k H r oli ]g ik where H is the curvature of the Berwald connection. On putting (2.7) in (2.6) we get - g i k DiT k l = P, + 2D 0 T, ik T lk

(2.8)

where (2.9) P, = {8 a ), + 2 (1 - - ) D,v|/ - 2( Yr Hrkli + d;Yk H r oli ) gik n ik + g [D, D o d, Y k - | Di D o ( d-k Y, + On the other hand, the last term of the right hand side of (2.3) can be put in the form (2.10)

X

-

x1J Do (d , Yj) = | Do (xIJa, Yj) - X- Do T1J d;. Yj.

Thus taking into account (2.8), (2.9) and (2.4), the relation (2.3) becomes:

Conformal Vector Fields

227

(2.11) (x, x) (z) = gik Dj (xk,Y') + | Do (x kl 5, Y,) + P, Y1 - ^ Doxkld, Y, We remark that the first two terms of the right hand side are divergences on W(M). We have : Lemma 1. For every 1-form £, = Yt(z) dx' on W(M) we have the formula (2.11). 3. A Characterisation of Conformal infinitesimal transformations when the manifold is compact Let X be a vector field on M and exp(uX) the 1 -parameter group of local transformations generated by X. We denote by exp(u x ) its extension to V(M) and by L( x ) the Lie derivative (see chap III). We say that X is an infinitesimal conformal Finslerian transformation if there exists a function (p on M such that (3-1)

L ( i ) g i j = 2(Pgij

To the vector field X on M we associate by duality, defined by the metric, a 1 -form on W(M) which we will denote also by X e A(W) (X — Xj(z) dx1). On using the Berwald connection (3.1) becomes (3.2)

Dj Xj + DJ XJ + Do (Si Xj) - 2

Let us take the trace of the two sides; we get

228 (3.4)

Initiation to Global Finslerian Geometry x(X)ij = D,Xj + DjX, + Do (6, Xj) + - 8(X + pv)glJ n

where 8(X +pv) is independent of the directions :

(3.5)

d;.8(X

0

(d ££) ov

Thus, for X to be an infinitesimal conformal transformation it is necessary and sufficient that T(X) = 0 and (3.5) holds. After (2.11) the square of x(X) becomes (3.6) (T(X) T(X)) (Z) = ( ; , X) + Div dependent on x on W(M) where (3.7)

Div dependent on x = glk Dj (x kl X1) + - D0(xkl dk X,)

In virtue of (2.8) and (2.11) we have (3.8) ;, = P, - T,ik D ox ik=-[gik DiXk,+ D 0(T,ik x ik)+ D 0 (T,lk)x ik] Once again let, taking into account (2.9) (3.9) ) - 2[X r H r k ,,+ dr X k H r oli ]g ik - T, lk D o x ik with

a = - (DjXj - Dj Xj) dx1 Adxj

Now let us suppose M compact without boundary. On integrating (3.6) on W(M) we obtain : (3.10)

<x(X), x(X)> = < , X>

Conformal Vector Fields

229

Let us suppose that x = 0; from (3.8) it follows that C, = 0; conversely suppose that C, = 0; after (3.10) we have T = 0. Therefore X is an infinitesimal conformal transformation. Theorem. In order that a vector field X on a compact Finslerian manifold without boundary defines an infinitesimal conformal transformation it is necessary and sufficient that ^ = 0 and S(X+ pv) be independent of the direction : (S a), + 2(1 - ^ ) D , 5(X + pv) + gik(D,D00;Xk (3.11)

=2i/j,XJ+TfD0Tik

and

dl S(X + pv) = 0

D ^

where H$ is defined by (3.12)

^ J , = gklHjkll + 2T' jr H r oll

In case X is an infinitesimal isometry we have C, = 0, and 8(X + pv) = 0. Corollary. In order that a vector field X on a compact Finslerian manifold without boundary defines an infinitesimal isometry it is necessary and sufficient that C, = 0, and S(X + pv) = 0. 4. Curvature and Infinitesimal Conformal Transformations in the compact case A. Lemma 2. Let X be a vector field on M, we then have (4.1)

H (X, X) - n(H(X, u)u, X) = Div on M

For the proof (see chapter VII §4, lemma 2, relation (4.2), (4.3) and (4.8))

230

Initiation Global Finslerian Finslerian Geometry Geometry Initiation to Global

Lemma 3. Let X be a vector field on M. We have (4.2) F"2(X, HijokgikXo-Hrojo d't Xj X 0 )=(H(X,u)u, X))+Div on W(M) where X 0 = (X, v). Proof. In virtue of Bianchi identity we have V2 d) XjHVoXo = F gJk 3' (XiH'oko X o F" 3 ) + F"2 (X, U)ok g lk X 0 ) -(H(X, u)u, X))

Now the first term of the right hand side is a divergence. Hence the lemma Lemma 4 Let X be a conformal vector field on M and let y/=S(X+pv) be a function defined on M where p = (X, T*), we have (4.3)

(X, T*) D08(X + pv) = Div on W(M)

where T* is the torsion trace vector. Proof. Let Y be a vertical l-form defined by its components: (4.4)

Y i = F'X.Do V - Ui(X, v) Do \|/ F"2

Do v|/ = v1 D, y

(4.5)

Fgij d) Y i= 2(X, T*) Do\|/ + (X,di|/) - n(X, v)D0

2

where (X, d\\)) = XlD\\\i. We now calculate the last two terms of the right hand side. To this effect, X being conformal, we have (4.6)

Dj Xj + Dj Xi + D o 8; Xj + - v|/gij = 0 n

Conformal Vector Fields

231

On multiplying the two sides by v1 and v" successively we get \|/ = - nD0 (X, v)F"2 Thus the last term of the right hand side of (4.8) becomes: (4.7)

-n(X, v) Do y.F"2 = -nD0 ( F"2 (X, v)i|/) + n F"2 Do (X, v) y = Div on W(M) - \\i2

It remains to calculate (X, d\\i). We have \\i independent of the direction: (4.8)

(X, di|/) = X1 Dj i|/= g'J Dj (Xj.v|/) - gij Dj Xj i|/

= Div on W(M) + V + Do (X, T*)v|/ = Div on W(M) + D o [(X, T*)v|/] - (X, T*) Do\|/ + v|/2 = Div on W(M) - (X,T*) Do\|/ + v|/2 Taking into account (4.7) and (4.8) the relation (4.5) becomes : F gij d, 7 j = (X, T*) Do\|/ + Div on W(M) Now, after (1.3), the left hand side is a divergence on W(M); hence the lemma. Remark. To the vector field X on M we associate the 1 -form X on W(M) defined by (4.9) where (4.10)

X = Xj (z) dxj + F"1 X j Dvj Xi = F 1 (D0Xj - Vi Do (X, v) F'2)

After (1.2) and (1.3), the co differential of X is

232

Initiation to Global Finslerian Geometry

On making explicit the last term, after (4.10) we obtain : (4.11) -8X = 2 [g'j Di Xj + D o (X, T*) - ^ 2

D

°^2' t

Let us suppose now that X is an infinitesimal conformal transformation. We have \|/ = 5(X +pv) = - n

°y

'

}

F Thus 5 X = 8(X+pv) = i|/ B. Let us suppose that X is conformal. Therefore it satisfies (3.11). Let us multiply the two sides of this relation by v1 . Then we get: (4.12) -g ik Di(D k X 0 -D 0 X k ) +2 (1-1/n) D o 5X+2 D 0 D 0 (X, T*)=2# J0 X J On the other hand from (4.9) it follows, on multiplying the two sides by v1 and on exchanging j and k (4.13)

(D k X 0 + D 0 X k ) + - 5 X v k = 0 n

On multiplying the two sides of (4.12) by F"2(X, v) and on using (4.13) we obtain: (4.14) -F"2 (X, v) glk DiDk (X, v) + ( 1 - - )(X, v) F"2 D05 X n + F"2 (X, v) D0D0 (X, T*) = F"2 (X,v) H}0 Xj We have successively

Conformal Vector Fields

233

(4.15) -F"2 (X, v) glk DiDk(X,v)=- F"2 glk Dj [Dk(X, v)(X, v)] + F- 2 g l k Di(X,v)D k (X,v) = Div on W(M) + F"2 glk D, (X, v) Dk (X, v)] (4.16) (l-2/n)F" 2 (X,v)D 0 5 X = (l-2/n)D0[(X, v)5X F"2] + - l/(l-2/n) (8X,

8X)

and, after lemma 4, (4.17) F"2 (X, v)D0D0(X, T*)=D0[F"2(X, v) D0(X, T*)]+D0(X, T * ) - 8 X n = Div on W(M) - - (X, T*)D05 X n = Div on W(M). Thus on putting (4.15) (4.16) and (4.17) in (4.14) and on using lemma 3 we obtain: (4.18) (D-X, D f i X ) + ^ ( l - ^ ) ( 5 X , =(H(X, u)u, X) +Div on W(M). where u is a horizontal vector field over u, (p u - u), M is compact and without boundary; on integrating the above relation on W(M) we obtain :

(4.19) + - ( 1 - - ) < 5 l , 5 l > = n n where <, > denotes the global scalar product over W(M). If < 0, then 8 X = 0, and DsX=0; by vertical derivation we obtain

234

Initiation to Global Finslerian Geometry

On the other hand, X being conformal and 8 X = 0, it then follows

Theorem. Let (M, g) be an n-dimensional compact Finslerian manifold without boundary, and X an infinitesimal conformal transformation. If the sectional curvature (H(X, u)u, X) is nonpositive everywhere, then X is an infinitesimal isometry and has covariant derivation of horizontal type zero in the Finslerian and in the Berwald connection[2]. 5. Case when M compact with scalar curvature H constant Let X a conformal vector field on M ; it satisfies (3.1). The Lie derivation by X of the coefficients of the Finslerian connection becomes (see [1]) (5.1)

On multiplying (5.1) successively by v* and yk we get:

(5.2) L(X) f 1Ok= (5.3)

L(X) f i00=

From (5.2) we obtain by vertical derivation

Conformal Vector Fields

235

L( X )Gijk=5ij cpk + 5'kcpj-g Jk (p1 - vk d-j cp1 +2VJ T irk <pr + F2 5, T irk cpr (5.4) Now the Lie derivative of curvature tensor of the Berwald connection becomes (5.5) L( X )H ijk,=DkL( X )Giji-D,L( X )G Sjk+G JjkrL( X )Gr,-G ijlrL( X )Grk On contracting i and k and on multiplying the two sides v* and v1 we obtain (5.6)

L(X) H(v, v) = 2 Di L ( l ) G1 - D o

where we have put H(v, v) = Hy vV. On using (5.2) and (5.3) the above relation becomes (5.7)

L ( X ) H(v, v) - (2-n)D0(p0 - F2 [D^l+ D 0 (T j (p;]

where o denotes the multiplication contracted by v. Let us put

J

2

dvJdvk

/ / j k is a symmetric tensor of order 2, homogeneous of degree zero inv. We put (5.9)

p= Dj (p! + D o (T'cp,) =g1JDl(Pj + 2 q>iD0 T1 + Do (T'cpO

On taking into account the identity (4.2) of the relation (5.7) we obtain by vertical derivation (5.10) L ( l ) ( H k 0 + Hok) - 2(2-n)D0
236

Initiation to Global Finslerian Geometry

(5.11) L(X)H3k

= (2-n) Dj(pk - pg j k - vk Pj - Vj

Pk-

I F2 d)

k

p

whence, taking into account (5.9): (5.12) g j k L(X)# j k =2(l-n)g j k DjCpk^mp.DoT'-nDo (T'cpO-^F2 g^S,

l P

Let us suppose now that the scalar curvature is a non positive constant (5.13)

H=g>kHik

= constant < 0

By Lie derivation of the two sides of (5.13) and, on taking into account (5.12), we get (5.14) Acp

^

^-y,

«-l

«-l

Dor +1 ^ - D 0 c r ^ y i - ^ F V a-. t P 2 n-\

4

«-l

where Acp is the Laplacian on W(M) defined by (3.3),

i i D0Tj + \ \ n-\ \ n-\ \ 22

D ^ ' ) n-\

We are thus led to calculate the terms T1 cpi cp0 and cp cpi D 0 T' Lemma 5. Let Y and Z are two vertical 1-forms defined by (5.16) Y k = - F T ijk cp'cp1, z*

Z k = - FT J cpjcpk - - u k TJcpj 9o z* z*

Conformal Vector Fields

237

When u = v/F, we have the formula (5.17)

6Z - 5Y + F 2 Qy cpV = ^=2- Tj
where Qy is the Ricci tensor corresponding to the third curvature tensor of the Finslerian connection : (5.18)

Proof. Consider the expression (5.19)

- F 2 TVTii
Now

grk9f (Tjik) = dr (T'd -drgA = 3,T j + 2TrTjir

Tjik

Thus on taking into account the expression of Qy, (5.18) becomes (5.20) -5Y + F2 Q, cpV = \ g'" 3) (FTj W&) - \

TJ9J9O

On using the co-differential of Z according to the formula (1.3) and on putting it in the above relation we find the lemma. Lemma 6. Let (M, g) a Finslerian manifold of dimension n such that (5.21)

T = g1J(DiD0Tj+9,.D0DJj) = 0

Then (pcp^T1 is a divergence on W(M).

238

Initiation to Global Finslerian Geometry

Proof. cpcpiDoT1 = i Dj (p2D0Tj = i gij Dj (cp2D0Tj) - i (p2gij DiD0Tj In virtue of (5.21) we have (P(p,D0Ti=igijDi((p2D0Tj)+ | gij dr(q>2DoDoTi) After (1.2) and (1.3) each term of the right hand side is a divergence on W(M). Let us suppose that M is compact without boundary. On integrating (5.15) on W(M) on using the preceding lemmas we obtain

(5.22)

f

qy «PVT, = —

JV(M)

I

cp2r,

fl — 1 *V(M)

where we have put " F2Q,j (n-\)(n-2) Let us suppose that qy is positive definite. If H is constant, non positive from (5.22) it follows that cp = 0 for H = constant < 0. This is clear and for H = 0, cpi = 0, so cp is constant. Now from (3.3) 9 is a divergence and M is compact, hence cp = 0, and we have Theorem. Let (M, g) be a compact Finselerian manifold of dimension n> 2 without boundary. Let us suppose that the scalar curvature H defined by (5.13) be a non-positive constant and z defined by (5.21) vanishes everywhere. If the quadratic form qy defined by (5.23) is positive definite everywhere on W(M) then the largest connected group of infinitesimal conformal transformations CO(M) coincides with the largest connected group of isometries Io(M) (5.23)

qu

= glJ +

Conformal Vector Fields

239

6. Case when X = Xj(z) dx1 is semi-closed. A. Let X be a vector field on M and let X = Xj(z) dx' be the corresponding dual 1-form. Its differential is (6.1)

dX = - (DjXj - DjXj) dx1 A dxj - 8) Xjdx1 A V v"

we say that the 1-form X is semi-closed if the horizontal part of dX vanishes: (6.2)

DjXj = DjXj

On multiplying the two sides by v1: (6.3) Whence by vertical derivation: (6.4)

DjXj + D o d' Xj = Dj Xi

On taking into account (6.2) we have (6.5)

Do 8] Xj = 0

Multiplying the two sides by g'j and contracting (6.6)

D 0 (X,T*) = 0 ,

(X,T*) = (XiTi)

Let us put Z, - 2(1 - - ) Di 8X - 2 H ji Xj with 8' 8X = 0 n where H is defined by (3.12). If X is semi-closed, the expression C,, on taking into account (6.5) and (6.6) reduces to (6.7)

(6.8)

Ci = Zi-TV k D 0 T i k

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Initiation to Global Finslerian Geometry

whence from (36) and (3.9) (6.9)

(T(X), T(X))

(6.10)

= (C, X) + Div dependent on x on W(M) = (Z, X) + Div dependent on x on W(M)

Now by the above theorem, infinitesimal conformal transformations are characterized by C, = 0. and so by (6.8), Z = 0. Conversely if Z = 0 from (6.9) by integration on W(M) we have x = 0. Thus Corollary. On a Finslerian compact manifold without boundary in order that a vector field whose dual 1-form is semi-closed defines an infinitesimal conformal transformation it is necessary and sufficient that one has (1--)D,8X = i/jiX J n

(6.10)

5*§X = 0

Let us rewrite the formula (6.9) by letting flag curvature intervene directly. Using the lemma 2

(6.11) ( - (1--) AX-R(X,u)u,X)4 " n

n

(T(X),X(X))

in = Div on W(M)

where A X is the horizontal Laplacian d 8 X. Let us take a frame at a point x = py e M such that u = en and let us put Rap = Rnanp where R«p is symmetric and suppose that R«p XaX^ is defined everywhere on W(M) (of rank (n-1)). Then on a Finslerian compact manifold without boundary there exist non-vanishing semi-closed infinitesimal conformal transformations only if R^p X a X p is positive definite (a, (3 = 1, ...n-1). At the point y e W(M) let X\(y) is the least proper value of the operator R«p and put X]= min Xi(y) for ye W(M). Let X (y) be a function such that A cr= d 8a =A,(y)a where a is a semi-closed 1 -form corresponding

Conformal Vector Fields

241 241

to a vector field on M and A the horizontal Laplacian. Then the formula (6.11) becomes (6.9)

( - (l--)?i(y)G-R(au)u,a) - 1 - (x(a),x(a)) n n 2n = Div on W(M)

if

- (l--)X(y)
D o (X,T*) = 0

where T* is the torsion trace vector. From (6.13) we have (6.15)

F > = D0X0

Thus from (6 .13) it follows that the lift of X on W(M) is horizontal and satisfies (6.14). Conversely from (6.13) and (6.15) we obtain, by vertical derivation,

242 / f

1 /~\

Initiation to Global Finslerian Geometry T^2

(D.lO) r

i

i

i

T~\

"^* ~\7

f\

tpij + VjCpj + VjCpi + Uo <7( Xj — U, (pi

yJly

-,

dv'

(pij

U

:

us

dv'dvJ

Let us multiply both the sides of the above relation by g'J and contract. Now cp is homogeneous of degree zero in v. So taking into account (6.14) we obtain

FV dv'dv

J

o

By a reasoning identical to (see chapter VI §2) we show that (p is independent of the direction. So from (6.15) and (6.13) we obtain DjXj = gij cp = D> Xj

X semi-closed and conformal: Proposition. Let(M, g) be a Finslerian manifold. If the lift ofXon W(M) is horizontal and satisfies V0(X, T*) = 0 Then X is conformal and its dual with respect the Finslerian metric is semi-closed and conversely if X is conformal and semi-closed then the lift ofXon W(M) is horizontal and satisfies (6.14} Corollary. Let X be a vector field on a Riemannian manifold. If the lift of X on the sphere bundle S(M) is horizontal then the 1form dual to X is closed and conformal. Conversely if the lift on the sphere bundle of M of a conformal vector field whose dual 1form is closed is horizontal.

243

References [1] H.Akbar-Zadeh. Les Espaces de Finsler et certaines de leurs generalisations.Ann.Ec.Norm.Sup. 3 e Serie 80. (1963) 1-79 [la] H. Akbar-Zadeh. Sur les isometries infinitesimales d'une variete finslerienne compacte. C.R. Acad.Sci. Paris t. 278 (18 mars 1974) Serie A-871 [lb] H. Akbar-Zadeh. Remarques sur les isometries infinitesimales d'une variete finslerienne compacte. C.R. Acad. Sci. Paris t. 284 (2& fevrier 1977) Serie A. 451 [lc] H. Akbar-Zadeh, Sur une Connexion euclidienne d'espace d'elements lineaires . C.R Acad.Sc. Paris t.245 (1957) Serie A pp 26-28 [Id] H ; Akbar-Zadeh, Sur quelques theoremes issus du calcul des variations . C.R. Acad. Sc. Paris t.264, Serie A (1967) pp 517-519 [2] H.Akbar-Zadeh. Transformations infinitesimales conformes des varietes finsleriennes compactes. Ann.Polon.Math XXXVI (1979)213-229 [3] H.Akbar-Zadeh. Champ de vecteurs projectifs sur le fibre unitaire. J Math pures et appl. 65 (1986) p.47-79. [4] H.Akbar-Zadeh, Sur les espaces de Finsler a courbures sectionelles constantes, Acad Royale Belgique Bull. de.Sci. 5eme serie LXXIV (1988-10) 281-322 [5] H.Akbar-Zadeh, Generalized Einstein Manifolds, Journal of Geometry and Physics, 17 (1995) 342-380 [6] H.Akbar-Zadeh, Geometry of Generalized Einstein Manifolds, C.R. Acad. Sci. Paris, Serie I 339 (2004) [7] H.Akbar-Zadeh, Sur les sous-varietes des varietes finsleriennes. C.R. Acad. Sci. Paris t. 266 (1968) pp 146-148 [8] L.Berwald. Paralleliibertragung. In allgemeinen. Raumen (Atti congresso intern.Matem. Bologna 1928 IV p. 263-270). [9] N. Bourbaki. Topologie generale Livre III Hermann Paris 1955. [10] F. Brickell A Theorem on Homogeneous Functions. J London Math Society 42. (1967) pp.325-329.

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[11] H. Busemann On Normal Coordinates in Finsler Spaces Math Annalen 129(1955) [12] E. Cartan. Sur les espaces de Finsler, C.R.Acad. Sc. 1.196, (1933)p.582-586 [13] E. Cartan. Les espaces de Finsler. Paris, Hermann. (1934) [14] E. Cartan. CEuvres Completes. Partie III vol 2. Geometrie Differentielle, Gauthier -Villars (1955) p. 1393 [15] A. Diecke, Uber die Finsler-Raume mit Aj = 0 Arch. Math 51953) p.45-51 [16] C. Ehresmann, les Connexions infinitesimales dans un espace fibre differentiable, Colloque de Topologie, Bruxelles 1950, pp 2955. [17] P. Finsler, Uber Kurven und Flachen in allgemeinen Raumen, Dissertation Gottingen 1918 [18] M. Haimovici, Varietes totalement extremales et varietes totalement geodesiques dans les espaces de Finsler, Ann. Sci. Univ. Jassy I -25 (1939) pp.559-644 [19] J. Hano, On Affine transformations of a Riemannian manifold, Nagoya Math Journal vol 9 pp 98-109. [20] D.Hilbert, Die Grundlagen der Physik, Nachr.Akad.Wiss. Gottingen, (1915), 395-407 [21] A. Kawaguchi. On the theory of non-linear connections, II Tensor N.S. 6(1956)pp 165-199 [22] H. Kawaguchi. On the Finsler spaces with vanishing second curvature tensor II Tensor N.S. 26, (1972) pp. 250-254 [23] S. Kobayashi and K; Nomizu, Foundation of Differential Geometry, 1963, Interscience Publishers [24] S. Kobayashi. Transformation groups in Differential Geometry, Springer - Verlag 1972 [25] N.Koiso. On the second derivation of the total scalar curvature, Osaka Journal of Math, 16 (1979) 413-421 [26] B. Kostant, Holonomy and the Lie Algebra of infinitesimal motions of a Riemann manifold, Transactions of Amer. Math. Society t. 80 , 1955 pp 528-542

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[27] A. Lichnerowicz, Theorie Globale des connexions et des groupes d'Holonomie, Editioni Cremonese Roma (1954) [28] A. Lichnerowicz. Geometrie des groupes de Transformations (Dunod Paris, 1958) [29] M. Matsumoto. Foundations of Finsler Geometry and Special Finsler Spaces 1986. Kaiseisha Press, Japan [30] J. Milnor. Morse Theory, Ann of Math Studies 51. (1963 ) Princeton University Press, Princeton. [31] F. Moalla, Espaces de Finsler Complets (C.R. Acad. Sc. Paris, 1964, t.258, n° 8 p. 2251 etn° 10 p . 2734. [32] Y. Muto. On Einstein Metrics. Journal of Differential Geometry 9 (1974) 521 -530 [33] M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere. Journal Math. Society of Japan 4 (1962) 333-340 [34] B. Riemann Uber die Hypothesen welche der Geometrie zugrunde liegen. Habilitationsvortrag 1854. Ges. Math. Werke, 272-287. Leipzig 1892, Reproduced by Dover Publications 1953 [35] H. Rund, The Differential Geometry of Finsler Spaces, Springer - Verlag 1959 [36] N. Steenrod, The Topology of Fibre Bundles, Princeton University Press, 1951 [37] K. Yamaguchi, On Infinitesimal Projective Transformations, Hokkaido. Math. Journal 1974 (3). P.262-270

246

INDEX A Absolute differential 6, 6,11,25,26, differential 11, 25, 26, 27, 30, 54, 55, 77 1,23,48,89,122, abstract 1, 23, 48, 89, 122, 143, 191, 191, 222 affine vector fields 48 5,6,24,26,48-9,52, algebra 5, 6, 24, 26, 48–9, 52, 61-2,76,78,79,81-3,86,88 61–2, 76, 78, 79, 81–3, 86, 88 almost Euclidean connections 44 B Berwald connections 46, 71, 71, 90, 92, 95, 97, 102, 104, 136, 143, 146, 150, 151, 153–4, 156, 157, 158, 161,162,163,164,173, 161, 162, 163, 164, 173, 179,186,187,198,201, 179, 186, 187, 198, 201, 205, 207, 209, 223, 226, 234-5 227, 234–5 manifolds 122, 132, 135 Bianchi 22-3, 42, 92, identities 20, 22–3, 93, 122, 123, 124, 128, 133,151,152,153,158, 133, 151, 152, 153, 158, 198,230 198, 230 Busemann 164

C Cartan 23, 46, 47, 122, 123 51, 55, 66 coframes 15, 17, 51, coherence 7, 8, 13, 25, 26 compact Finslerian manifolds 48, 70, 74, 75, 89, 95, 97, 109,117,121,122,123, 109, 117, 121, 122, 123, 125,127, 138,141,175, 125, 127, 138, 141, 175, 197,211,229,234,241 197, 211, 229, 234, 241 case 69, 99, 191, 191, 229 complete 167,169,171,173 manifolds 167, 169, 171, 173 case 211 conformal vector fields 222, 222,230,234, 230, 234, 242 222,228, transformations 222, 228, 229, 232, 234, 238, 240 connected 3, 48, 49, 69, 78, 79, 80, 81, 81, 86–8, 86-8, 89, 95, 143, 172, 172, 173, 173, 191, 191, 202, 211,221,222,238 211, 221, 222, 238 5,6,23,24,30,32, connections 5, 6, 23, 24, 30, 32, 40,44,46,48,55,98,101, 40, 44, 46, 48, 55, 98, 101, 102,177,179,180,182 102, 177, 179, 180, 182 coordinates 3–4, 3-4, 16, 18, 32, 34, 35, 37, 40, 84, 128, 163, 164, 164, 178, 178, 198, 143, 163, 205, 216

Index covariant derivation 1, 12, 19, 21, 42, 1,12,19,21,42, 44, 70, 71, 74, 75, 76, 80, 90, 101, 101, 131, 131, 145, 146, 148, 152, 174, 176, 148,152,174, 179, 184, 187, 195, 199, 179,184,187,195,199, 201, 223, 234 derivative 11, 20, 40, 48, 11,20,40,48, 66, 77, 78, 84, 86, 87, 88, 102, 103, 122, 132, 88,102,103, 122,132, 165, 174, 175, 204, 174, 205 curvature tensor 13, 14, 16, 17, 20, 21,23,27,40,41,45, 21, 23, 27, 40, 41, 45, 78, 80, 89, 92, 95, 102, 122,123,133,136,137, 122, 123, 133, 136, 137, 140,142,143, 138, 140, 142, 143, 146, 151, 152–3, 152-3, 149, 150, 151, 156,157,159,163,164, 156, 157, 159, 163, 164, 167,168,173,174,189, 167, 168, 173, 174, 189, 198,200,201,205,207, 198, 200, 201, 205, 207, 235, 237 infinitesimal conformal 89, infinitesimal 142, 222, 229, 232, 238, 240 transformations 197, 197,229 transformations 229 projective transformations 197 transformations D 27-30, 33, degree 5, 10, 23, 27–30, 34,35,46,51,96,99,108, 34, 35, 46, 51, 96, 99, 108, 114, 134, 134, 144, 144, 147, 112, 114, 150, 154, 154, 163, 163, 202, 206, 210, 225, 235, 242

247 differentiable differentiable fibre bundles 1,1,2 2 3,35 manifolds 3, 35 functions 89, 89,93,101,118 functions 93, 101, 118 divergence formulas 48,64,66,67,71, 48, 64, 66, 67, 71, 124, 125 124, E Einstein 122,138,141,142 manifolds 122, 138, 141, 142 generalized Einstein 98,105, manifolds 89, 98, 105, 109,112,113,191 109, 112, 113, 191 equation 4, 22, 27, 31, 31, 44, 46, 147, 149, 149, 161–2, 161-2, 81, 95, 147, 163, 170, 170, 171, 171, 172, 172, 173, 163, 174, 192, 192, 193, 193, 194, 194, 195, 174, 200, 212, 215, 220 exterior differential forms 9 differential F Fibre bundle linear frames 5, 5,135 135 V(M)andW(M) 2-4,5,9, V(M) and W(M) 2–4, 5, 9, 10,12,21,23,28,74,89, 10, 12, 21, 23, 28, 74, 89, 98,122,135,154,182, 98, 122, 135, 154, 182, 197,210,211,217 197, 210, 211, 217 Finslerian connection 36, 40, 42, 45, 135, 145, 145,156 67, 69, 135, 156 175, 215 geometry 23, 46, 175, manifolds 23, 33, 84, 109, 122,125,135,138,143, 122, 125, 135, 138, 143, 144, 151, 151, 153, 153, 157 144,

248 23,40, curvature tensors 23, 40, 122,136,137, 138, 102, 122, 136, 137, 138, 151, 156–7, 156-7, 143, 149, 151, 168, 174, 198, 237 submanifolds 125, 143, 143, submanifolds 125, 175,182, 175, 182, 186, 188 form 4, 9 2,32 frames 2, 32 G 181 geodesic 135, 181 H homogeneous 5, 23, 35, 46, 49,99,108,112,114,134, 49, 99, 108, 112, 114, 134, 144, 147, 150, 154, 163, 192, 202, 206, 225, 235, 242 homeomorphie to the sphere 218 I 1,18,20,22,23,42, identities 1, 18, 20, 22, 23, 42, 122,152,153 122, 152, 153 infinitesimal infinitesimal transformation 48, 49, 51, 51, transformation 52, 53, 56, 61, 67, 68, 81, 75, 76, 77, 78, 79, 81, 192, 83, 84, 85, 86, 192, 194, 202, 215, 222, 227,241 227, 241 isometries 48, 67, 69, 70, 86, 88, 201, 211, 238 isotropic manifolds 127, 143, 150, 164

Index Finslerian manifolds 143, 144,154, 157,162,164, 144, 154, 157, 162, 164, 165 151-3, 155, 167, 167, 168, case 151–3, 189 L Laplace operator 89, 90, 93, 95, 96, 97, 222, 236, 240, 241 Landsberg 122, 123, 123, 125, manifolds 122, 129,131,132,135,174, 129, 131, 132, 135, 174, 202, 205 lemmas 81, 81,90,92,93,98,99, 90, 92, 93, 98, 99, 104, 112, 112, 113, 113, 123, 103, 104, 131, 194, 194, 195, 195, 197, 130, 131, 225, 227, 229, 230, 236 Lichnerowicz 89 length 23–4, 23–4,27,33–4,44, 27, 33–4, 44, 170, 171, 171,211-12, 162, 170, 211–12, 214-15, 216–17, 216-17, 218–20 218-20 214–15, Lie 5-6,24,26,48-9, algebra 5–6, 24, 26, 48–9, 52, 61–2, 61-2, 76, 78, 79, 81-2, 83, 86, 88 81–2, 48,49,51-2,53, derivative 48, 49, 51–2, 53, 56,57,58-9,61-3,67-8, 56, 57, 58–9, 61–3, 67–8, 77-8, 84, 84,192,194,210, 77–8, 192, 194, 210, 213, 217, 222, 227, 235 linear connection 1, 5, 6, 10, 12, 13, 14, 14, 16, 16, 17–18, 17-18, 20, 13, 21, 23, 25, 26, 36, 37, 21, 39, 40, 42, 48, 54, 56, 61, 75, 76, 77, 84 58, 61,

Index connection of directions 6, 17,48 11, 17, 48 1,5, connection of vectors 1, 5, 6,7,8,9,11,13,48 6, 7, 8, 9, 11, 13, 48 element 1 M manifolds 23, 33, 84, 89, 95, 105, 122, 122, 123, 123, 125, 98, 105, 127, 129, 129, 131, 131, 132, 132, 135, 143, 144, 144, 150, 150, 153, 138, 143, 162, 164, 164, 167, 167, 169, 157, 162, 171, 173, 227 map canonical 2, 9, 10, 17, 18 induced 2 minima fibration 123, 125, 125, 127, fibration 129, 130 Minkowskian manifolds 162 O open 3, 163 operator 167,168, 169 Dl 167, 168, 169 orthonormal frames 23, 24, 25, 26–7, 26-7, frames 32,99 32, 99 P Pfaffian Pfaffian 8,11,19,32, derivatives 8, 11, 19, 32, 37, 57, 99, 135, 191 plane axioms Finslerian geometry 175, 175,182 182

249 34,144,238, positive definite 34, 144, 238, 240, 241 projective transformation 191, 191, 193, transformation 194,196,197,199,201, 194, 196, 197, 199, 201, 202, 204, 205, 206, 208, 221 191,197,201, vector fields 191, 197, 201, 214 R Ricci 1, 18, 18, 20 identities 1, directional curvature 103, 109,143,157,191,210, 109, 143, 157, 191, 210, 211,212,214,215,221 211, 212, 214, 215, 221 70,157,204,205, curvature 70, 157, 204, 205, 214, 222 tensor 48, 74, 75, 95, 103, 104,124,141, 151,152, 104, 124, 141, 151, 152, 198, 210, 237 198, Riemann 167 S scalar curvature 89, 89,101,138, 101, 138, 142, 143, 143, 150, 150, 155, 155, 161, 142, 162, 172, 172, 173, 173, 222, 234, 162, 236,238 236, 238 Schur theorem 186, 186, 188 generalized Schur 143, 153 theorem 143, Second variational 105, 109, integral I(gtt) 89, 105, 110,112,117,121,138, 110, 112, 117, 121, 138, 141, 142 141,

250 semi-closed 222, 239, 240, 241, 242 simply connected compact Finslerian manifolds 95, 143, 143, 173,191,221 173, 191, 221 Steenrod 2 T tensors affine 4 affine large sense 4,5,11,19,48, 4, 5, 11, 19, 48, 51,53,55,56,57,62,77 51, 53, 55, 56, 57, 62, 77 restricted sense 4, 5, 23 curvature 13, 13, 14, 14, 40, 151 151 13-14, 16–17, 16-17, torsion 1, 13–14, 31, 32, 37, 20, 21, 23, 31, 40,45,48,54,55,84,85, 40, 45, 48, 54, 55, 84, 85, 93,102,112,118-19, 93, 102, 112, 118–19, 122-3, 141, 143, 122–3, 138, 141, 148,168,173, 174, 145, 148, 168, 173, 174, 175,176,177,191, 175, 176, 177, 191, 200, 222-3, 226, 202, 215, 222–3, 227, 230, 241 tensorial 5, 5,14,35,51,52,53,58 14, 35, 51, 52, 53, 58 18, 23, 26, 36, 39, theorems 18, 45, 47, 67, 68, 69, 70, 74, 81, 83, 86, 88, 76, 77, 79, 81, 95,109,112,117,121, 89, 95, 109, 112, 117, 121,

Index 125, 127, 127, 129, 129, 135, 135, 137, 125, 141, 142, 142, 153, 153, 156, 156, 157, 141, 163, 164, 164, 165, 165, 171, 162, 163, 175, 182, 182, 186, 186, 188, 173, 175, 190, 199, 199, 205, 209, 211, 212,214,218,221,229, 212, 214, 218, 221, 229, 234,238 234, 238 122, 123, totally geodesic 122, 135,137,143,181,182–4, 135, 137, 143, 181, 182–4, 186,202 186, 202 52,90,100-1,106,110, trace 52, 90, 100–1, 106, 110, 112-17,119,123,125, 112–17, 119, 123, 125, 128,138,141-2,168,191, 128, 138, 141–2, 168, 191, 201-2, 215, 222, 223, 200, 201–2, 225-6, 225–6, 227, 230, 241 V vectorfields vector fields

48,191,201,222 48, 191, 201, 222

W 66-7, without boundary 48, 66–7, 69, 70, 74, 75, 95, 96, 97, 105, 106, 106, 109, 109, 117, 100, 105, 121, 122, 122, 123, 123, 125, 125, 127, 121, 132,135,139,141,158, 132, 135, 139, 141, 158, 165, 166, 166, 197, 197, 199, 164, 165, 201, 204, 205, 210, 211, 222, 223, 224, 228, 229, 233, 234, 238, 240, 241