Elliptic operators, topology and asymptotic methods

書誌事項

Elliptic operators, topology and asymptotic methods

John Roe

(Pitman research notes in mathematics series, 395)

Longman, 1998

2nd ed

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注記

First published 1988

Bibliography: p. 203-206

Includes index

内容説明・目次

内容説明

This Research Note gives an introduction to the circle of ideas surrounding the `heat equation proof' of the Atiyah-Singer index theorem. Asymptotic expansions for the solutions of partial differential equations on compact manifolds are used to obtain topological information, by means of a `supersymmetric' cancellation of eigenspaces. The analysis is worked out in the context of Dirac operators on Clifford bundles. The work includes proofs of the Hodge theorem; eigenvalue estimates; the Lefschetz theorem; the index theorem; and the Morse inequalities. Examples illustrate the general theory, and several recent results are included. This new edition has been revised to streamline some of the analysis and to give better coverage of important examples and applications. Readership: The book is aimed at researchers and graduate students with a background in differential geometry and functional analysis.

目次

Introduction Chapter 1 Resume of Riemannian geometry Chapter 2 Connections, curvature, and characteristic class Chapter 3 Clifford algebras and Dirac operators Chapter 4 The Spin groups Chapter 5 Analytic properties of Dirac operators Chapter 6 Hodge theory Chapter 7 The heat and wave equations Chapter 8 Traces and eigenvalue asymptotics Chapter 9 Some non-compact manifolds Chapter 10 The Lefschetz formula Chapter 11 The index problem Chapter 12 The Getzler calculus and the local index theorem Chapter 13 Applications of the index theorem Chapter 14 Witten's approach to Morse theory Chapter 15 Atiyah's GAMMA-index theorem References

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