Curvature and homology:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Academic Press
1970
|
| Edition: | 2. print. |
| Series: | Pure and applied mathematics
11 |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005376213&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | XVII, 315 S. |
Staff View
MARC
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| 100 | 1 | |a Goldberg, Samuel I. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Curvature and homology |c Samuel I. Goldberg |
| 250 | |a 2. print. | ||
| 264 | 1 | |a New York u.a. |b Academic Press |c 1970 | |
| 300 | |a XVII, 315 S. | ||
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Record in the Search Index
| _version_ | 1819361720477417472 |
|---|---|
| adam_text | CONTENTS
Preface vii
Notation Index xii
Introduction xv
Chapter I
RIEMANNIAN MANIFOLDS l
/./ Differentiable manifolds 1
1.2 Tensors 5
1.3 Tensor bundles 9
1.4 Differential forms 12
1.5 Submanifolds 17
1.6 Integration of differential forms 19
1.7 Affine connections 23
1.8 Bundle of frames 27
1.9 Riemannian geometry 30
1.10 Sectional curvature 35
1.11 Geodesic coordinates 40
Exercises 41
Chapter II
TOPOLOGY OF DIFFERENTIABLE MANI¬
FOLDS 56
2.1 Complexes 56
2.2 Singular homology 60
2.3 Stokes theorem 62
2.4 De Rham cohomology 63
2.5 Periods 64
2.6 Decomposition theorem for compact Riemann surfaces 65
2.7 The star isomorphism 68
2.8 Harmonic forms. The operators 8 and A 71
2.9 Orthogonality relations 73
2.10 Decomposition theorem for compact Riemannian manifolds 75
2.11 Fundamental theorem 76
2.12 Explicit expressions for d, 8 and A 77
Exercises 78
Chapter III
CURVATURE AND HOMOLOGY OF RIE¬
MANNIAN MANIFOLDS 82
3.1 Some contributions of S. Bochner 82
3.2 Curvature and betti numbers 85
3.3 Derivations in a graded algebra 95
ix
X CONTENTS
3.4 Infinitesimal transformations 98
3.5 The derivation 6{X) 101
3.6 Lie transformation groups 103
3.7 Conformal transformations 106
3.8 Conformal transformations (continued) 112
3.9 Conformally flat manifolds 115
3.10 Affine collineations 119
3.11 Projective transformations 121
Exercises 124
Chapter IV
COMPACT LIE GROUPS 132
4.1 The Grassman algebra of a Lie group 132
4.2 Invariant differential forms 134
4.3 Local geometry of a compact semi simple Lie group 136
4.4 Harmonic forms on a compact semi simple Lie group 139
4.5 Curvature and betti numbers of a compact semi simple Lie group G . . 141
4.6 Determination of the betti numbers of the simple Lie groups 143
Exercises 145
Chapter V
COMPLEX MANIFOLDS 146
5.1 Complex manifolds 147
5.2 Almost complex manifolds 150
5.3 Local hermitian geometry 158
5.4 The operators L and A 168
5.5 Kaehler manifolds 173
5.6 Topology of a Kaehler manifold 175
5.7 Effective forms on an hermitian manifold 179
5.8 Holomorphic maps. Induced structures 182
5.9 Examples of Kaehler manifolds 184
Exercises 189
Chapter VI
CURVATURE AND HOMOLOGY OF
KAEHLER MANIFOLDS 197
6.1 Holomorphic curvature 199
6.2 The effect of positive Ricci curvature 205
6.3 Deviation from constant holomorphic curvature 206
6.4 Kaehler Einstein spaces 208
6.5 Holomorphic tensor fields 210
6.6 Complex parallelisable manifolds 213
6.7 Zero curvature 215
6.8 Compact complex parallelisable manifolds 217
6.9 A topological characterization of compact complex parallelisable manifolds 220
6.70 d cohomology 221
6.11 Complex imbedding 223
6.12 Euler characteristic 227
CONTENTS xi
6.13 The effect of sufficiently many holomorphic differentials 230
6.14 The vanishing theorems of Kodaira 232
Exercises 237
Chapter VII
GROUPS OF TRANSFORMATIONS OF
KAEHLER AND ALMOST KAEHLER
MANIFOLDS 244
7.1 Infinitesimal holomorphic transformations 246
7.2 Groups of holomorphic transformations 252
7.3 Kaehler manifolds with constant Ricci scalar curvature 255
7.4 A theorem on transitive groups of holomorphic transformations .... 258
7.5 Infinitesimal conformal transformations. Automorphisms 259
7.6 Conformal maps of manifolds with constant scalar curvature 263
7.7 Infinitesimal transformations of non compact manifolds 265
Exercises 266
Appendix A
DE RHAM S THEOREMS 270
A.I The 1 dimensional case 270
A.2 Cohomology 271
A.3 Homology 275
A.4 The groups H (M, A ) 277
A.5 The groups H,(M, St) 278
A.6 Poincare s lemma 280
A.7 Singular homology of a starshaped region in Rn 281
A.8 Inner products 283
A.9 De Rham s isomorphism theorem for simple coverings 284
.4.10 De Rham s isomorphism theorem 289
A.11 De Rham s existence theorems 291
Appendix B
THE CUP PRODUCT 293
B.I The cup product 293
B.2 The ring isomorphism 294
Appendix C
THE HODGE EXISTENCE THEOREM 296
Decomposition theorem 296
Appendix D
PARTITION OF UNITY 301
References 303
Author Index 307
Subject Index 309
|
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| author | Goldberg, Samuel I. |
| author_facet | Goldberg, Samuel I. |
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| author_sort | Goldberg, Samuel I. |
| author_variant | s i g si sig |
| building | Verbundindex |
| bvnumber | BV008147837 |
| classification_rvk | SK 350 |
| ctrlnum | (OCoLC)256284050 (DE-599)BVBBV008147837 |
| discipline | Mathematik |
| edition | 2. print. |
| format | Book |
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| id | DE-604.BV008147837 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T09:17:26Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005376213 |
| oclc_num | 256284050 |
| open_access_boolean | |
| owner | DE-20 DE-29T |
| owner_facet | DE-20 DE-29T |
| physical | XVII, 315 S. |
| publishDate | 1970 |
| publishDateSearch | 1970 |
| publishDateSort | 1970 |
| publisher | Academic Press |
| record_format | marc |
| series | Pure and applied mathematics |
| series2 | Pure and applied mathematics |
| spellingShingle | Goldberg, Samuel I. Curvature and homology Pure and applied mathematics Krümmung - Homologie Homologietheorie (DE-588)4141714-8 gnd Homologie (DE-588)4141951-0 gnd Kurve (DE-588)4033824-1 gnd Krümmung (DE-588)4128765-4 gnd |
| subject_GND | (DE-588)4141714-8 (DE-588)4141951-0 (DE-588)4033824-1 (DE-588)4128765-4 |
| title | Curvature and homology |
| title_auth | Curvature and homology |
| title_exact_search | Curvature and homology |
| title_full | Curvature and homology Samuel I. Goldberg |
| title_fullStr | Curvature and homology Samuel I. Goldberg |
| title_full_unstemmed | Curvature and homology Samuel I. Goldberg |
| title_short | Curvature and homology |
| title_sort | curvature and homology |
| topic | Krümmung - Homologie Homologietheorie (DE-588)4141714-8 gnd Homologie (DE-588)4141951-0 gnd Kurve (DE-588)4033824-1 gnd Krümmung (DE-588)4128765-4 gnd |
| topic_facet | Krümmung - Homologie Homologietheorie Homologie Kurve Krümmung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005376213&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV010177228 |
| work_keys_str_mv | AT goldbergsamueli curvatureandhomology |