Initiation to global Finslerian geometry

Cover......Page 1
Series......Page 2
Title page......Page 3
Date-line......Page 4
Preface......Page 5
Introduction......Page 6
CONTENTS......Page 8
1. Fibre Bundles $V(M)$ and $W(M)$......Page 15
2. Frames and Co-frames......Page 16
3. Tensors and Tensor forms......Page 18
4. Linear connections......Page 19
5. Absolute differential in a linear connection. Regular linear connection......Page 20
6. Exterior differential forms......Page 23
1. Torsion tensors......Page 27
2. Curvature tensors......Page 28
8. Particular case of a linear connection of directions. Conditions of reduction......Page 31
9. Ricci identities......Page 32
10. Bianchi identities......Page 34
11. Torsion and Curvature defined by a covariant derivation......Page 35
1. Metric manifolds......Page 37
2. Euclidean connections......Page 38
3. The system of generators on $W$......Page 41
4. Special connections......Page 44
5. Case of orthonormal frames and local coordinates for the class of special connections......Page 46
6. Finslerian manifolds......Page 47
7. Finslerian connections......Page 50
8. Curvature tensors of the Finslerian connection......Page 54
9. Almost Euclidean connections......Page 58
Chapter III Isometries and affine vector fields on the unitary tangent fibre bundle......Page 62
1. Local group of 1-parameter local transformations and Lie derivative......Page 63
2. Local invariant sections......Page 67
3. Introduction of a regular linear connection......Page 68
4. The Lie derivative of a tensor in the large sense......Page 71
5. The Lie derivative of the coefficients of a regular linear connection......Page 72
6. Fundamental formula......Page 75
7. Divergence formulas......Page 78
8. Infinitesimal isometries, the compact case......Page 81
9. Ricci curvatures and Infinitesimal isometries......Page 84
10. Infinitesimal affine transformations......Page 89
11. Affine infinitesimal transformations and Covariant Derivations......Page 90
12. The group $Kz(L)$......Page 92
13. Transitive algebra of affine infinitesimal transformations......Page 93
14. The Lie Algebra $L$......Page 95
15. The case of Finslerian manifolds......Page 98
16. Case of infinitesimal isometries......Page 100
1. The Laplacian defined on the unitary tangent fibre bundle and the Finslerian curvature......Page 103
2. Case of a manifold with constant sectional curvature......Page 109
1. Fundamental lemma, Compact case......Page 112
2. Variations of scalar curvatures......Page 115
3. Generalized Einstein manifolds......Page 119
4. Second variationals of the integral $I(g_t)$......Page 126
5. Case of a conformal infinitesimal deformation......Page 132
I. Properties of compact Finslerian manifolds of non-negative curvature......Page 136
1. Landsberg manifolds......Page 137
2. Finslerian manifolds with minima fibration......Page 139
3. Case of isotropic manifolds......Page 141
4. Calculation of $(\\delta A)^2$ when $(M, g)$ is a manifold with minima fibration......Page 143
5. Case when $(M, g)$ is a Landsberg manifold. The calculation of $||\\nabla_i A_j||^2$......Page 145
6. Case of compact Berwald manifolds......Page 146
7. Finslerian manifolds whose fibres are totally geodesic or minima......Page 149
1. The first variational of $I(g_t)$......Page 152
2. Second variational......Page 155
Chapter VI Finslerian manifolds of constant sectional curvatures......Page 157
1. Finslerian manifolds......Page 158
2. Indicatrices......Page 161
3. Isotropic manifolds......Page 164
4. Properties of curvature tensors in the isotropic case......Page 165
A. Case of Berwald connection......Page 167
B. Case of Finslerian connection......Page 170
2. Necessary and sufficient conditions for an isotropic Finslerian manifold to be of constant sectional curvature......Page 171
3. Locally Minkowskian manifolds......Page 176
4. Compact isotropic manifolds with strictly negative curvature......Page 178
1. Operator $D^1$ - the isotropic case......Page 181
2. Complete manifolds with strictly negative constant sectional curvature......Page 183
3. Complete manifolds with strictly positive constant sectional curvature......Page 185
4. Complete manifolds with zero sectional curvature......Page 187
1. Finslerian submanifolds......Page 189
2. Induced and intrinsic connections of Berwald......Page 193
3. Totally geodesic submanifolds......Page 195
4. The plane axioms......Page 196
1. Infinitesimal proj ecti ve transformations......Page 205
2. Other characterizations of infinitesimal projective transformations......Page 208
3. Curvature and infinitesimal projective transformations......Page 211
4. Restricted projective vector fields......Page 215
5. Projective invariants......Page 219
6. Case when Ricci directional curvature satisfies certain conditions......Page 224
7. The complete case......Page 225
8. Case where the Ricci directional curvature is a strictly positive constant......Page 228
9. The second variational of the length......Page 230
10. Homeomorphie to the sphere......Page 232
Chapter VIII Conformal vector fields on the unitary tangent fibre bundle......Page 236
1. The Co-differential of a 2-form......Page 237
2. A Lemma......Page 239
3. A characterization of conformal infinitesimal transformations......Page 241
4. Curvature and Infinitesimal Transformation in the compact case......Page 243
5. Case when $M$ is compact with scalar curvature $H$ is constant......Page 248
6. Case when $X=X_i(z) dx^i$ is semi-closed......Page 253
References......Page 257
Index......Page 260