Initiation to Global Finslenan Geometry
NorthHolland 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
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VOLUME 68
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Initiation to Global Finslerian Geometry
H. AkbarZadeh Director of Research at C.N.R.S. Paris France
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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. 2955 [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 Coframes 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 1parameter 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 nonnegative 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 ∇iAj2 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 Codifferential of a 2form 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 semiclosed
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 nonzero 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 nonzero 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 nonzero vectors of the origin x by the positive collinearity: two nonzero 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 rz=y .It is clear that W(M) is endowed with the structure of a differentiable fibre bundle over M, whose dimension is (2nl) and whose fibres are homeomorphic to the ball Sn1 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 Coframes. 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 coframe 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 coframes 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 rform 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 rforms with scalar values of V(M). Such a form is a tensorial rform 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 rform 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 1differential 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 1differential 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 coframe of the vector space 0 z tangent to V(M) at z ; in order that the set (a1, 61) forms a coframe of 0 z it is necessary and sufficient that the system of nforms 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 coframe 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 coframe 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 coframe (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 ) 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 coframes (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 1forms over V(M). Thus any 1form 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 2form 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) = X2C jj (zi), Loj = 0
In a general manner, for an rform on V(M) to be the inverse image of an rform 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 rform will be identified with an rform 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 coframes 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 coframes (a k , 9 ). In the following, we are thus led to reason in terms of the coframe (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 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 coframes 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= (fVr'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 iT^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 = VkT',  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 0form with vector values XZU = (X1); the relation ddX' = 0 becomes  (dkdid,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 coframe (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=XrP1rklVrX1Prokl+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~rJ/i R r okV'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^ " hrJn  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'jimViP j k m +S r klPjrm(P jkrProlmPjIrProkm)+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 YVfXp[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 semimetric connection and we give a characterization of Berwald connection. We show that there exists an infinity of torsionfree 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 1form 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) skewsymmetric matrix which we denote again by (Oyv = (fflij),
(oOij + COji = 0 )
its elements are differential 1forms of y. have R> = R»Cvv 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 1form 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 skewsymmetric 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 coframe (\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'kl'lk) By (3.5) it is clear that the B1 define on V(M) a 1form with vector values. This 1form being of degree zero and satisfying (3.1) can be identified to a 1form 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 1form with values in the vector space tangent to 7t"'(x) at y. Conversely the y' form a system of generators for these 1forms , 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 1forms 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)
TCQa' + 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 1form 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 1form 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)
YijkYikj = 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 = 
akgij +  (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 (53)
T l j k =akgij
(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 + 5ijLxj5iL = 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(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) Rjjk  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,
(92)
^ 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 torsionfree 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 torsionfree connection of directions is necessarily regular. For the map \i becomes the identity map. On the other hand its curvature 2form on W(M) decomposes into a 2form of the type (2, 0), that is to say two times horizontal and a 2form 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 torsionfree 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 torsionfree 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 semimetric connection which they call Chern connection. Now S. S. Chern tried to obtain by the method of local equivalence a set of semiEuclidean 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 114136). He establishes the following theorem
Finslerian Manifolds
47
Theorem. There is a unique set of local 1forms 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 582586. 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 1forms and for the vertical 1forms on W(M) ([1],[2]). Next we study the 1parameter 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)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 qform of type R(G) (linear representation of G in a vector space ), L(X)A is also a tensorial qform of the same type. 2. If co is a 1form of connection with values in the Lie algebra GL(n, R) of adjoint type, L(X)co is tensorial 1form of adjoint type. Let Y be a vector field in the large sense, and A a semibasic 1form on V(M), that is to say it depends only on the 1form 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 )=£ (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 un Va bn = 0 = V% bn So (7.6) 5b = F(Vrbi + bjT Hj ) where 8 is the codifferential (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 1form (7.1). Similarly, if (7.8)
a = ai(z)dx j ,
zeV(M)
is a horizontal 1form on W(M) we obtain (see [1] pp 6870) (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 1form. 8. Infinitesimal Isometries [la], [lb] A. Let X be an infinitesimal transformation on M and Hz the horizontal subspace 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 1form 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.
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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)
= J (R(X, u)u, X) r\ W(M)
where < , > and ( , ) denote respectively the global and local scalar product on W(M).
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Global Finslerian Geometry Initiation to Global
Let us suppose that the integral of the right hand side of (8.6) is nonpositive. 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 nonpositive, 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 nontrivial 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 antisymmetric endomorphism AX of Tpz defined by (9.1) To this endomorphism is associated a 2form (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(ViXjVJXi)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
72
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'X1 + 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
74
Initiation to Global Finslerian Geometry
We have (9.10)
PijXVoX*
+^X 1 (V i XjV 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 )  = q>i
90
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)
(j and ,D,<j>k)=  g ^ D . t p C p ^ ) ] +g lk g jl D i (p J (D k (p,D,(p k ) = 
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*) > (nl)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, (nl)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 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 nsphere. Similarly on supposing that H*(cp*, cp*) defined by (1.8) is positive and satisfies (1.27)
H*(cp*, cp*)>(nl)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
(n4)
Geometry of Generalized Einstein Manifolds
97
Thus the study of the quadratic form 0(
(tJg1' trace t)d) ' *¥ = Div on W(M)
where t(u, u) =tjj u1 u* and d) =—dvJ Proof Let Y be a covector field on W(M) components (1.3)
defined by its
i ^ Y j  U j Y o F  1 where Yj = F1 M/toj, (f 0 = v J 7j = 0)
where o denotes the contracted multiplication by v. Y defines a vertical 1form; 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 covector 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(tjltrace t g*1) d) } vj/ + F2 g"1 dr (trace t d;\\i) Also, (1.7) FgJ' 5 Z, = F gj' d'j Z,  (n1) t'o d] v/ =F2(tjlgjl trace t)5* } v/(n2)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 1parameter family of Finslerian connections is represented ([1]) (2.1)
oo'jIpKu). = 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 1parameter 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 1form of 1parameter 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 F2 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 subfamily 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 vrt(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 rt ) ' =  A jk tjk TI + Div on W(M)
where we have put: (3.7) ^jk=^jkgjk(^+ H)
n2
F g
j k
g S ; ; y F d y n2
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 nondecomposable. 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 = (nl)b, n2
(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 n2 n2 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^2vjgkO+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 n2 «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
n2
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 av/)  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 (318) 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
2n 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 + — ^ — 5vv/ n n(n2) n(n2) where (3.29)
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)+Ai) (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 \\> n2 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 (n2)
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)] = (n1) [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/") = (n2) tj0 8'j v/' + Div on W(M) =(n2) F ^d'j (F"1tkoi)/I)+(n2) [t(u, u)trace t] \\i'+Di\ on W(M) = (n2)[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=HC n
Now \/ = x — H, we have after (4.21) (4.23)
— — F2 tlj d] • i/'  (\/ + # ) ' trace t w2 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' n2 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 (51)
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(2n)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(n2)F "2 (pDocpo  n(n2) (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   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(nl)(n2) (Acp  —*— //)  2n F2 V" Tj q>V + n(n6) Vyty0 n\ + Div on W(M) On direct calculation we obtain F 2 V* Tj 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 Xi = 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 Nonnegative 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 skewsymmetric 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 =
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 04)
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 2tensor 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 nonnegative curvature in the large sense if the scalar product of the flag curvature by a symmetric tensor of order two (R , a) is nonnegative. Theorem 1. Let (M, g) be a compact Finslerian manifold without boundary.
If the symmetric tensor R is such that (a,R\
is
everywhere nonnegative, 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
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Initiation to Global Finslerian Geometry
Let us put (23)
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