Curvature and Homology (eBook)

Front Cover 1
Curvature and Homology 4
Copyright Page 5
Contents 10
Preface 8
Notation Index 14
Introduction 16
CHAPTER I. RIEMANNIAN MANIFOLDS 20
1.1 Differentiable manifolds 20
1.2 Tensors 24
1.3 Tensor bundles 28
1.4 Differential forms 31
1.5 Submanifolds 36
1.6 Integration of differential forms 38
1.7 Affine connections 42
1.8 Bundle of frames 46
1.9 Riemannian geometry 49
1.10 Sectional curvature 54
1.11 Geodesic coordinates 59
Exercises 60
CHAPTER II. TOPOLOGY OF DIFFERENTIABLE MANI- FOLDS 75
2.1 Complexes 75
2.2 Singular homology 79
2.3 Stokes’ theorem 81
2.4 De Rham cohomology 82
2.5 Periods 83
2.6 Decomposition theorem for compact Riemann surfaces 84
2.7 The star isomorphism 87
2.8 Harmonic forms. The operators d and . 90
2.9 Orthogonality relations 92
2.10 Decomposition theorem for compact Riemannian manifolds 94
2.11 Fundamental theorem 95
2.12 Explicit expressions for d, d and . 96
Exercises 97
CHAPTER III. CURVATURE AND HOMOLOGY OF RIEMANNIAN MANIFOLDS 101
3.1 Some contributions of S. Bochner 101
3.2 Curvature and betti numbers 104
3.3 Derivations in a graded algebra 114
3.4 Infinitesimal transformations 117
3.5 The derivation .(X) 120
3.6 Lie transformation groups 122
3.7 Conformal transformations 125
3.8 Conformal transformations (continued) 131
3.9 Conformally flat manifolds 134
3.10 Affline collineations 138
3.11 Projective transformations 140
Exercises 143
CHAPTER IV. COMPACT LIE GROUPS 151
4.1 The Grassman algebra of a Lie group 151
4.2 Invariant differential forms 153
4.3 Local geometry of a compact semi-simple Lie group 155
4.4 Harmonic forms on a compact semi-simple Lie group 158
4.5 Curvature and betti numbers of a compact semi-simple Lie group G 160
4.6 Determination of the betti numbers of the simple Lie groups 162
Exercises 164
CHAPTER V. COMPLEX MANIFOLDS 165
5.1 Complex manifolds 166
5.2 Almost complex manifolds 169
5.3 Local hermitian geometry 177
5.4 The operators L and A 187
5.5 Kaehler manifolds 192
5.6 Topology of a Kaehler manifold 194
5.7 Effective forms on an hermitian manifold 198
5.8 Holomorphic maps. Induced structures 201
5.9 Examples of Kaehler manifolds 203
Exercises 208
CHAPTER VI. CURVATURE AND HOMOLOGY OF KAEHLER MANIFOLDS 216
6.1 Holomorphic curvature 218
6.2 The effect of positive Ricci curvature 224
6.3 Deviation from constant holomorphic curvature 225
6.4 Kaehler-Einstein spaces 227
6.5 Holomorphic tensor fields 229
6.6 Complex parallelisable manifolds 232
6.7 Zero curvature 234
6.8 Compact complex parallelisable manifolds 236
6.9 A topological characterization of compact complex parallelisable manifolds 239
6.10 d”-cohomology 240
6.11 Complex imbedding 242
6.12 Euler characteristic 246
6.13 The effect of sufficiently many holomorphic differentials 249
6.14 The vanishing theorems of Kodaira 251
Exercises 256
CHAPTER VII. GROUPS OF TRANSFORMATIONS OF KAEHLER AND ALMOST KAEHLER MANIFOLDS 263
7.1 Infinitesimal holomorphic transformations 265
7.2 Groups of holomorphic transformations 271
7.3 Kaehler manifolds with constant Ricci scalar curvature 274
7.4 A theorem on transitive groups of holomorphic transformations 277
7.5 Infinitesimal conformal transformations. Automorphisms 278
7.6 Conformal maps of manifolds with constant scalar curvature 282
7.7 Infinitesimal transformations of non-compact manifolds 284
Exercises 285
APPENDIX A. ADE RHAM'S THEOREMS 289
A.1 The 1-dimensional case 289
A.2 Cohomology 290
A.3 Homology 294
A.4 The groups Hp(M, .q) 296
A.5 The groups Hp(M, Sq) 297
A.6 Poincaré's lemma 299
A.7 Singular homology of a starshaped region in Rn 300
A.8 Inner products 302
A.9 De Rham's isomorphism theorem for simple coverings 303
A.10 De Rham's isomorphism theorem 308
A.11 De Rham's existence theorems 310
APPENDIX B. THE CUP PRODUCT 312
B.1 The cup product 312
B.2 The ring isomorphism 313
APPENDIX C. THE HODGE EXISTENCE THEOREM 315
Decomposition theorem 315
APPENDIX D. PARTITION OF UNITY 320
References 322
Author Index 326
Subject Index 328