Invariance theory, the heat equation, and the Atiyah Singer index theorem:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton u.a.
CRC Press
1995
|
| Edition: | 2. ed. |
| Series: | Studies in advanced mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 516 S. |
| ISBN: | 0849378745 |
Staff View
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| 245 | 1 | 0 | |a Invariance theory, the heat equation, and the Atiyah Singer index theorem |c Peter B. Gilkey |
| 246 | 1 | 3 | |a Invariance theory, the heat equation, and the Atiyah-Singer index theorem |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Boca Raton u.a. |b CRC Press |c 1995 | |
| 300 | |a IX, 516 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 0 | |a Studies in advanced mathematics | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
1 Pseudo differential operators 1
1.0 Introduction 1
1.1 Fourier transform and Sobolev spaces 2
1.1.1 Convolution product 3
1.1.2 Fourier transform 4
1.1.3 Sobolev spaces 6
1.1.4 Duality and interpolation 9
1.2 Pseudo Differential Operators on Rm 10
1.2.1 Continuity properties 11
1.2.2 Equivalence of symbols 14
1.2.3 A wider class of symbols 15
1.2.4 Adjoints and compositions 17
1.2.5 Operators defined by kernels 20
1.2.6 Pseudo locality 23
1.2.7 Completeness 24
1.3 Pseudo differential operators on manifolds 26
1.3.1 Ellipticity 26
1.3.2 Change of coordinates 28
1.3.3 Operators on manifolds 31
1.3.4 Sobolev spaces on manifolds 33
1.3.5 Extension to vector bundles 35
1.4 Index of Fredholm Operators 36
1.4.2 Compact operators 37
1.4.2 Fredholm operators 38
1.4.4 Compositions and adjoints 39
1.4.5 Index of Fredholm operators 40
1.4.5 Properties of the index 40
1.4.6 Elliptic pseudo differential operators 42
1.5 Elliptic complexes 43
1.5.1 Hodge decomposition theorem 44
1.5.2 de Rham complex 46
1.6 Spectral theory 48
1.6.1 Self adjoint compact operators 50
1.6.2 Self adjoint elliptic operators 51
iii
iv Contents
1.6.3 Bounding the spectrum from below 53
1.6.4 Heat equation 54
1.6.5 Trace and kernel 55
1.7.0 Heat equation and index theory 56
1.7 The heat equation 56
1.7.1 Dependence on a complex parameter 57
1.7.2 Spectral theory 61
1.7.3 Heat equation 64
1.8 Local index formula 65
1.8.1 Asymptotic expansions 67
1.8.2 Index theory 72 i
1.9 Variational formulas 74
1.9.1 Generalized heat equation asymptotics 74
1.9.2 Properties of the trace 76
1.9.3 Conformal geometry 79
1.10 Lefschetz fixed point theorems 82
1.10.1 Generalized Lefschetz number 83
1.10.2 Equivariant asymptotics 85
1.10.4 Isolated fixed points 89
1.11.0 Heat asymptotics and Lefschetz number 90
1.11 Elliptic boundary value problems 90
1.11.1 Notational conventions 91
1.11.2 Operators of Dirac and Laplace type 94
1.11.3 Spectral theory 98 :
1.11.4 Heat equation 99
1.11.5 Index theory of operators of Dirac type 101
1.11.6 Non local boundary conditions 103
1.12 The Zeta function 105
1.12.1 Notational conventions 105
1.12.2 Zeta function and heat equation 107
1.12.3 Zeta function of powers 109
1.12.4 Positive semi definite operators 112
1.12.5 Eigenvalue growth estimates 113
1.13 The Eta function 114
1.13.1 Eta invariant and spectral asymmetry 115
2 Characteristic classes 121
2.0 Introduction 121
2.1 Characteristic classes of complex bundles 122
2.1.1 Notational conventions 122
2.1.2 Chern Weil homomorphism 127
2.1.3 Functorial constructions 130
2.1.4 Chern classes 131
2.1.5 Chern character 132
Contents v
2.2 Characteristic classes of real bundles 136
2.2.1 Generating functions 137
2.2.2 Euler class 139
2.2.3 Directional covariant derivative 142
2.3 Complex projective space 145
2.3.1 Holomorphic manifolds 146
2.3.2 Fiber metrics and connections 148
2.3.3 Complex projective space 149
2.3.4 Characteristic classes of complex projective space... 154
2.3.5 Dual basis to the characteristic forms 155
2.3.6 Todd class and Hirzebruch L polynomial 158
2.4 Invariance theory 160
2.4.1 Notational conventions 160
2.4.2 Dimensional analysis 162
2.4.3 Invariants of the orthogonal group 165
2.4.4 Diffeomorphism invariance 167
2.4.5 Diagonalization Lemma 173
2.5 The Gauss Bonnet theorem 175
2.5.1 The restriction map 176
2.5.2 The proof of the Gauss Bonnet theorem 178
2.5.3 Next term in the heat equation 180
2.5.4 Shuffle formulas 181
2.6 Invariance theory and Pontrjagin classes 183
2.7 Gauss Bonnet for manifolds with boundary 190
2.7.1 Boundary conditions 190
2.7.2 Associated boundary conditions 192
2.7.3 de Rham Hodge theorem 193
2.7.4 Heat equation asymptotics 195
2.7.5 Invariance theory 196
2.7.6 The Gauss Bonnet theorem for manifolds with
boundary 201
2.7.7 Doubling the manifold 202
2.8 Boundary characteristic classes 203
2.9 Singer s question 209
2.9.1 Invariance theory 209
2.9.2 Singer s question 211
2.9.3 Form valued invariants 212
3 The index theorem 215
3.0 Introduction 215
3.1 Clifford modules 216
3.1.1 Notational conventions 216
3.1.2 Homotopy groups of the orthogonal group 218
3.1.3 Clifford modules 220
vi Contents
3.1.4 Clifford modules on manifolds 221
3.1.5 Decomposing compatible connections 223
3.2 Hirzebruch signature formula 225
3.2.1 The Levi Civita connection on differential forms .... 225
3.2.2 Twisted signature complex 226
3.2.3 Product formulas 227
3.2.4 Invariants of the heat equation 228
3.2.5 Hirzebruch signature formula 230
3.2.6 Applications of the signature formula 231
3.2.7 Generalized signature formula 232
3.3 Spinors 233
3.3.1 Two dimensional spinors 234
3.3.2 Stiefel Whitney classes 235
3.3.3 Spin bundle 241
3.3.4 The spin and exterior bundles 243
3.3.5 Characteristic classes 244
3.4 The spin complex 245
3.4.1 Twisted spin complex 245
3.4.2 Product manifolds 246
3.4.3 Invariants of the heat equation 247
3.4.4 Spin, de Rham, and signature complexes 247
3.4.5 Index theorem for spin complex 248
3.4.6 Twisted de Rham complex 249
3.4.7 Yang Mills complex 249
3.4.8 Geometrical index theorem 251
3.5 The Riemann Roch theorem 252
3.5.1 Almost complex manifolds 252
3.5.2 The arithmetic genus 256
3.5.3 Holomorphic manifolds 257
3.5.4 Relations with holomorphic and Kaehler geometry.. 258
3.5.5 The spinc complex 260
3.6 K theory 263
3.6.1 K theory 263
3.6.2 Chern isomorphism 264
3.6.3 Classifying spaces 265
3.6.4 Bott periodicity 266
3.6.5 Suspension and clutching data 267
3.6.6 Orientations 268
3.6.7 External tensor product 270
3.6.8 Integration along the fibers 272
3.7 The Atiyah Singer index theorem 273
3.7.1 Extending the index to K theory 274
3.7.2 Even dimensional manifolds 275
Contents vii
3.7.3 Cohomology and K theory extensions 276
3.7.4 Odd dimensional manifolds 278
3.7.5 The real Todd genus 280
3.8 The regularity at s=0 of the eta function 284
3.8.1 Analytic facts 285
3.8.2 Interpretation in K theory 285
3.8.3 Odd dimensional manifolds 287
3.8.4 Even dimensional manifolds 288
3.9 Lefschetz fixed point formulas 292
3.9.1 Isolated fixed points 293
3.9.2 de Rham complex 295
3.9.3 Equivariant invariants 300
3.10 Index theorem for manifolds with boundary 301
3.10.1 The induced structures on the boundary 302
3.10.2 Non local boundary conditions 304
3.10.3 Index theorem for product structures 305
3.10.4 Asymptotic expansions for non product structures .. 306
3.10.5 The transgression 306
3.10.6 Spinors 308
3.10.7 Compatible elliptic complexes of Dirac type 311
3.10.8 Twisted signature complex 313
3.11 The eta invariant of locally flat bundles 314
3.11.1 Flat structures on bundles 315
3.11.2 Relative eta invariant 316
3.11.3 Secondary characteristic classes 319
3.11.4 Index theorem on trivial bundles 322
3.11.5 Relative eta invariant 323
4 Spectral geometry 327
4.0 Introduction 327
4.1 Operators of Laplace type 329
4.1.1 Spectrum of flat tori 330
4.1.2 Local geometry of operators of Laplace type 331
4.1.3 Vanishing theorems 333
4.1.4 Formulas for the heat equation asymptotics 334
4.1.5 The form valued Laplacian 338
4.1.6 A recursion relation 340
4.1.7 Leading terms in the asymptotics 341
4.1.8 Variational formulas: Table 4.1 344
4.2 Isospectral manifolds 345
4.2.1 Geometry of the spectrum 346
4.2.2 Isospectral non isometric manifolds 348
4.2.3 Compactness results 349
4.2.4 Spherical space forms 349
viii Contents
4.2.5 Isospectral non isometric metacyclic spherical space
forms 350
4.2.6 Spherical harmonics 353
4.2.7 Isospectral non isometric lens spaces 356
4.3 Non minimal operators 358
4.4 Operators of Dirac type 366
4.4.1 Local formulas 367
4.4.2 Reconstruction of the divergence terms 372
4.4.3 Non vanishing of the invariants 374
4.5 Manifolds with boundary 375
4.5.1 Boundary conditions 375
4.5.2 Dirichlet and Neumann boundary conditions 376
4.5.3 Mixed boundary conditions 381
4.5.4 Absolute boundary conditions 382
4.6 Other asymptotic formulas 383
4.6.1 Asymptotics of operators of Dirac type 384
4.6.2 Non minimal operators with absolute boundary
conditions 385
4.6.3 Heat asymptotics on small geodesic balls 389
4.6.4 Operators of Laplace type 390
4.6.5 Heat content asymptotics of non minimal operators. 393
4.7 The eta invariant of spherical space forms 394
4.7.1 Properties of the eta function 394
4.7.2 The Hurwicz zeta function 398
4.7.3 The square root of the normalized spherical Laplacian 398
4.7.4 The eta invariant on real projective space 402
4.7.5 Equivariant zeta function 405
4.7.6 Equivariant eta function 407
4.7.7 Eta invariant of spherical space forms 410
4.7.8 K theory of spherical space forms 411
4.7.9 Metrics of positive scalar curvature 413
5 Bibliographic information 419
5.0 Acknowledgement 419
5.1 Introduction 419
5.2 Historical summary 421
5.2.1 The formation of index theory 421
5.2.2 The general Atiyah Singer index theorem 424
5.2.3 The heat equation method 425
5.2.4 Index theory on open manifolds 428
5.2.5 Index theory on singular spaces 429
5.2.6 if homology and operator if theory 430
5.2.7 Index theory and physics 432
5.2.8 Other topics 433
Contents ix
5.3 List of references 435
Notation 509
Index 511
|
| any_adam_object | 1 |
| author | Gilkey, Peter B. 1946- |
| author_GND | (DE-588)1024266850 |
| author_facet | Gilkey, Peter B. 1946- |
| author_role | aut |
| author_sort | Gilkey, Peter B. 1946- |
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| building | Verbundindex |
| bvnumber | BV010119444 |
| classification_rvk | SK 260 SK 350 SK 370 SK 540 |
| classification_tum | MAT 355f MAT 474f |
| ctrlnum | (OCoLC)439649895 (DE-599)BVBBV010119444 |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV010119444 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:46:51Z |
| institution | BVB |
| isbn | 0849378745 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006719665 |
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| physical | IX, 516 S. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | CRC Press |
| record_format | marc |
| series2 | Studies in advanced mathematics |
| spelling | Gilkey, Peter B. 1946- Verfasser (DE-588)1024266850 aut Invariance theory, the heat equation, and the Atiyah Singer index theorem Peter B. Gilkey Invariance theory, the heat equation, and the Atiyah-Singer index theorem 2. ed. Boca Raton u.a. CRC Press 1995 IX, 516 S. txt rdacontent n rdamedia nc rdacarrier Studies in advanced mathematics Analiza - Parcialne diferencialne enačbe na mnogoterostih Geometrija Topologija Indextheorem (DE-588)4140055-0 gnd rswk-swf Wärmeleitungsgleichung (DE-588)4188859-5 gnd rswk-swf Invariantentheorie (DE-588)4162209-1 gnd rswk-swf Indextheorem (DE-588)4140055-0 s DE-604 Invariantentheorie (DE-588)4162209-1 s Wärmeleitungsgleichung (DE-588)4188859-5 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006719665&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gilkey, Peter B. 1946- Invariance theory, the heat equation, and the Atiyah Singer index theorem Analiza - Parcialne diferencialne enačbe na mnogoterostih Geometrija Topologija Indextheorem (DE-588)4140055-0 gnd Wärmeleitungsgleichung (DE-588)4188859-5 gnd Invariantentheorie (DE-588)4162209-1 gnd |
| subject_GND | (DE-588)4140055-0 (DE-588)4188859-5 (DE-588)4162209-1 |
| title | Invariance theory, the heat equation, and the Atiyah Singer index theorem |
| title_alt | Invariance theory, the heat equation, and the Atiyah-Singer index theorem |
| title_auth | Invariance theory, the heat equation, and the Atiyah Singer index theorem |
| title_exact_search | Invariance theory, the heat equation, and the Atiyah Singer index theorem |
| title_full | Invariance theory, the heat equation, and the Atiyah Singer index theorem Peter B. Gilkey |
| title_fullStr | Invariance theory, the heat equation, and the Atiyah Singer index theorem Peter B. Gilkey |
| title_full_unstemmed | Invariance theory, the heat equation, and the Atiyah Singer index theorem Peter B. Gilkey |
| title_short | Invariance theory, the heat equation, and the Atiyah Singer index theorem |
| title_sort | invariance theory the heat equation and the atiyah singer index theorem |
| topic | Analiza - Parcialne diferencialne enačbe na mnogoterostih Geometrija Topologija Indextheorem (DE-588)4140055-0 gnd Wärmeleitungsgleichung (DE-588)4188859-5 gnd Invariantentheorie (DE-588)4162209-1 gnd |
| topic_facet | Analiza - Parcialne diferencialne enačbe na mnogoterostih Geometrija Topologija Indextheorem Wärmeleitungsgleichung Invariantentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006719665&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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