| Description |
1 online resource (392 pages) |
| Series |
Research Notes in Mathematics Ser. ; v. 4 |
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Research Notes in Mathematics Ser
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| Contents |
Cover; Half Title; Title Page; Copyright Page; Dedication; Contents; Preface; List of Figures; Introduction and the proof; 1. The Atiyah-Singer index theorem; 2. The Atiyah-Patodi-Singer index theorem; 3. Boundary conditions versus b-geometry; 4. Preliminaries to the proof; 5. The proof; 6. Weighting; 7. Outline; Chapter 1. Ordinary differential operators; 1.1. Operators and coordinates; 1.2. Index; 1.3. General statement; 1.4. Kernels; Chapter 2. Exact b-geometry; 2.1. Manifolds; 2.2. The b-tangent bundle; 2.3. Exact b -metrics; 2.4. Differential operators; 2.5. Levi-Civita Connection |
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2.6. Riemann curvature tensor2.7. Associated bundles; 2.8. Covariant differentiation; 2.9. Christoffel symbols; 2.10. Warped products; 2.11. Curvature formulae; 2.12. Orientation; 2.13. b-connections; 2.14. Characteristic classes; 2.15. Hermitian bundles; 2.16. de Rham cohomology; 2.17. b -characteristic classes; Chapter 3. Spin structures; 3.1. Euclidean Dirac operator; 3.2. Clifford algebra; 3.3. Periodicity; 3.4. Clifford bundle; 3.5. Clifford modules; 3.6. Clifford bundle of bTX; 3.7. Spin group; 3.8. Spin representations; 3.9. Spin structures; 3.10. Clifford connections |
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3.11. Twisted Dirac operators3.12. Spin structure for a b -metric; 3.13. Boundary behaviour; 3.14. Dirac operators of warped products; Chapter 4. Small b-calculus; 4.1. Inward-pointing spherical normal bundle; 4.2. The b-stretched product; 4.3. Submanifolds of Xb2; 4.4. Lifting vector fields; 4.5. Densities; 4.6. The space of pseudodifferential operators; 4.7. Distributions; 4.8. Kernels of b-differential operators; 4.9. The small space of b-pseudodifferential operators; 4.10. Symbol map; 4.11. Elementary mapping properties; 4.12. Asymptotic completeness; 4.13. Small parametrix |
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4.14. Non-compactness4.15. Indicial operator; 4.16. General coefficients; 4.17. Examples; 4.18. Trace class operators; 4.19. The b-integral; 4.20. The b-trace functional; Chapter 5. Full calculus; 5.1. Mellin transform; 5.2. Inversion of the indicial family; 5.3. Analytic Fredholm theory; 5.4. Conjugation by powers; 5.5. Commutator identity for the b-trace; 5.6. Invertibility of the indicial operator; 5.7. Kernel of the inverse of the indicial operator; 5.8. Index formula for invariant operators; 5.9. Composition in the small calculus; 5.10. Polyhomogeneous conormal distributions |
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5.11. Mellin transform and polyhomogeneity5.12. Boundary terms; 5.13. True parametrix; 5.14. Finitely residual terms; 5.15. Boundedness on Sobolev spaces; 5.16. Calculus with bounds; 5.17. Fredholm properties; 5.18. Extended index sets; 5.19. Formal solutions; 5.20. Finer parametrix; 5.21. Composition with boundary terms; 5.22. Residual terms; 5.23. Composition in general; 5.24. General bundles and summary; 5.25. Parametrices and null space; 5.26. Generalized inverse; Chapter 6. Relative index, cohomology and resolvent; 6.1. Boundary pairing; 6.2. Relative index formula |
| Notes |
6.3. Riemann-Roch for surfaces |
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Print version record |
| Form |
Electronic book
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| ISBN |
9781439864609 |
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1439864608 |
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