The Atiyah Patodi Singer index theorem:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Wellesley, Mass.
Peters
1993
|
| Schriftenreihe: | Research notes in mathematics
4 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | X, 377 S. graph. Darst. |
| ISBN: | 1568810024 |
Internformat
MARC
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| 100 | 1 | |a Melrose, Richard B. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a The Atiyah Patodi Singer index theorem |c Richard B. Melrose |
| 246 | 1 | 3 | |a The Atiyah-Patodi-Singer index theorem |
| 264 | 1 | |a Wellesley, Mass. |b Peters |c 1993 | |
| 300 | |a X, 377 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Research notes in mathematics |v 4 | |
| 650 | 4 | |a Atiyah-Singer, Théorème d' | |
| 650 | 7 | |a Globale analyse |2 gtt | |
| 650 | 4 | |a Atiyah-Singer index theorem | |
| 650 | 0 | 7 | |a Indextheorem |0 (DE-588)4140055-0 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Indextheorem |0 (DE-588)4140055-0 |D s |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-006049937 | ||
Datensatz im Suchindex
| _version_ | 1804123543213965312 |
|---|---|
| adam_text | CONTENTS
Preface xi
List of Figures xiii
Introduction and the proof 1
1. The Atiyah Singer index theorem 2
2. The Atiyah Patodi Singer index theorem 2
3. Boundary conditions versus 6 geometry 4
4. Preliminaries to the proof 5
5. The proof 9
6. Weighting 10
7. Outline 12
Chapter 1. Ordinary differential operators 15
1.1. Operators and coordinates 15
1.2. Index 17
1.3. General statement 21
1.4. Kernels 22
Chapter 2. Exact 6 geometry 25
2.1. Manifolds 25
2.2. The 6 tangent bundle 27
2.3. Exact 6 metrics 30
2.4. Differential operators 32
2.5. Levi Civita Connection 35
2.6. Riemann curvature tensor 39
2.7. Associated bundles 43
2.8. Covariant differentiation 44
2.9. Christoffel symbols 47
2.10. Warped products 50
2.11. Curvature formulae 51
2.12. Orientation 53
2.13. 6 connections 55
2.14. Characteristic classes 60
2.15. Hermitian bundles 63
2.16. de Rham cohomology 65
2.17. 6 characteristic classes 66
viii Contents
Chapter 3. Spin structures 67
3.1. Euclidean Dirac operator 67
3.2. Clifford algebra 69
3.3. Periodicity 71
3.4. Clifford bundle 75
3.5. Clifford modules 76
3.6. Clifford bundle of bTX 79
3.7. Spin group 80
3.8. Spin representations 84
3.9. Spin structures 86
3.10. Clifford connections 88
3.11. Twisted Dirac operators 90
3.12. Spin structure for a 6 metric 90
3.13. Boundary behaviour 91
3.14. Dirac operators of warped products 93
Chapter 4. Small 6 calculus 95
4.1. Inward pointing spherical normal bundle 95
4.2. The 6 stretched product 98
4.3. Submanifolds of Xj 99
4.4. Lifting vector fields 101
4.5. Densities 105
4.6. The space of pseudodifferential operators 108
4.7. Distributions 110
4.8. Kernels of b differential operators 113
4.9. The small space of b pseudodifferential operators 114
4.10. Symbol map 115
4.11. Elementary mapping properties 117
4.12. Asymptotic completeness 121
4.13. Small parametrix 122
4.14. Non compactness 124
4.15. Indicial operator 125
4.16. General coefficients 130
4.17. Examples 132
4.18. Trace class operators 134
4.19. The 6 integral 140
4.20. The 6 trace functional 142
Chapter 5. Full calculus 144
5.1. Mellin transform 144
5.2. Inversion of the indicial family 146
Contents ii
5.3. Analytic Fredholm theory 150
5.4. Conjugation by powers 152
5.5. Commutator identity for the 6 trace 153
5.6. Invertibility of the indicial operator 156
5.7. Kernel of the inverse of the indicial operator 160
5.8. Index formula for invariant operators 161
5.9. Composition in the small calculus 164
5.10. Polyhomogeneous conormal distributions 166
5.11. Mellin transform and polyhomogeneity 171
5.12. Boundary terms 173
5.13. True parametrix 174
5.14. Finitely residual terms 176
5.15. Boundedness on Sobolev spaces 178
5.16. Calculus with bounds 180
5.17. Fredholm properties 183
5.18. Extended index sets 185
5.19. Formal solutions 187
5.20. Finer parametrix 189
5.21. Composition with boundary terms 190
5.22. Residual terms 196
5.23. Composition in general 199
5.24. General bundles and summary 201
5.25. Parametrices and null space 204
5.26. Generalized inverse 206
Chapter 6. Relative index, cohomology and resolvent . . . 208
6.1. Boundary pairing 208
6.2. Relative index formula 213
6.3. Riemann Roch for surfaces 216
6.4. Hodge theory 220
6.5. Extended L2 null space of Dirac operators 233
6.6. Resolvent family 234
6.7. Analytic continuation of the resolvent 238
6.8. Poles of the resolvent 242
6.9. Spectral theory 246
6.10. Scattering matrices 248
Chapter 7. Heat calculus 252
7.1. Heat space 252
7.2. Standard heat calculus 261
7.3. Heat kernel 263
7.4. 6 heat space 273
x Contents
7.5. 6 heat calculus 276
7.6. Bundle coefficients 281
7.7. Long time behaviour, Predholm case 283
7.8. Long time behaviour, non Fredholm case 291
Chapter 8. Local index theorem 295
8.1. Simple rescaling 295
8.2. Rescaling bundles 297
8.3. Rescaling and connections 301
8.4. Getzler s rescaling 305
8.5. Rescaled trace 309
8.6. Rescaled heat calculus 312
8.7. Rescaled normal operator 313
8.8. Lichnerowicz formula 317
8.9. Mehler s formula 322
8.10. Local index formula 326
8.11. The 6 case 328
8.12. Graded Hermitian Clifford modules 329
8.13. The eta integrand 332
8.14. The modified eta invariant 334
8.15. Variation of eta 336
8.16. Spectral flow 338
8.17. The circle 341
Chapter 9. Proof revisited and applications 343
9.1. The APS theorem 343
9.2. Euler characteristic 347
9.3. Signature formula 351
9.4. Flat coefficient bundles 356
9.5. Zeta function 357
9.6. Analytic torsion 362
9.7. The 6 eta invariant 362
Bibliography 364
Index of Notations 368
Author Index 371
Index 372
|
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| id | DE-604.BV009127462 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:31:26Z |
| institution | BVB |
| isbn | 1568810024 |
| language | English |
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| oclc_num | 246533723 |
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| physical | X, 377 S. graph. Darst. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | Peters |
| record_format | marc |
| series | Research notes in mathematics |
| series2 | Research notes in mathematics |
| spelling | Melrose, Richard B. Verfasser aut The Atiyah Patodi Singer index theorem Richard B. Melrose The Atiyah-Patodi-Singer index theorem Wellesley, Mass. Peters 1993 X, 377 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Research notes in mathematics 4 Atiyah-Singer, Théorème d' Globale analyse gtt Atiyah-Singer index theorem Indextheorem (DE-588)4140055-0 gnd rswk-swf Indextheorem (DE-588)4140055-0 s DE-604 Research notes in mathematics 4 (DE-604)BV005530429 4 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006049937&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Melrose, Richard B. The Atiyah Patodi Singer index theorem Research notes in mathematics Atiyah-Singer, Théorème d' Globale analyse gtt Atiyah-Singer index theorem Indextheorem (DE-588)4140055-0 gnd |
| subject_GND | (DE-588)4140055-0 |
| title | The Atiyah Patodi Singer index theorem |
| title_alt | The Atiyah-Patodi-Singer index theorem |
| title_auth | The Atiyah Patodi Singer index theorem |
| title_exact_search | The Atiyah Patodi Singer index theorem |
| title_full | The Atiyah Patodi Singer index theorem Richard B. Melrose |
| title_fullStr | The Atiyah Patodi Singer index theorem Richard B. Melrose |
| title_full_unstemmed | The Atiyah Patodi Singer index theorem Richard B. Melrose |
| title_short | The Atiyah Patodi Singer index theorem |
| title_sort | the atiyah patodi singer index theorem |
| topic | Atiyah-Singer, Théorème d' Globale analyse gtt Atiyah-Singer index theorem Indextheorem (DE-588)4140055-0 gnd |
| topic_facet | Atiyah-Singer, Théorème d' Globale analyse Atiyah-Singer index theorem Indextheorem |
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