Invariance Theory : The Heat Equation and the Atiyah-Singer Index Theorem

ISBN-13: 9780849378744 / Angielski / Twarda / 1994 / 536 str.

Peter B. Gilkey; Gilkey B. Gilkey; Steven G. Krantz

Opis

This text examines the Atiyah-Singer theorem using the heat equation, which gives a local formula for the index of any elliptic complex. Heat equation methods are also used to discuss Lefschetz fixed point formulas, the Gauss-Bonnet theorem for a manifold with smooth boundary, and the geometrical theorem for a manifold with smooth boundary. The book presents a careful treatment of non-self-adjoint operators, asymptotics of the heat equation and variational formulas. It also introduces spectral geometry and provides a list of asymptotic formulas. The bibliography has been complied by Herbert Schroeder.

Spis treści

Pseudo-Differential Operators Introduction Fourier Transform and Sobolev Spaces Pseudo-Differential Operators on Rm Pseudo-Differential Operators on Manifolds Index of Fredholm Operators Elliptic Complexes Spectral Theory The Heat Equation Local Index Formula Variational Formulas Lefschetz Fixed Point Theorems The Zeta Function The Eta Function Characteristic Classes Introduction Characteristic Classes of Complex Bundles Characteristic Classes of Real Bundles Complex Projective Space Invariance Theory The Gauss-Bonnet Theorem Invariance Theory and Pontrjagin Classes Gauss-Bonnet for Manifolds with Boundary Boundary Characteristic Classes Singer's Question The Index Theorem Introduction Clifford Modules Hirzebruch Signature Formula Spinors The Spin Complex The Riemann-Roch Theorem K-Theory The Atiyah-Singer Index Theorem The Regularity at s = 0 of the Eta Function Lefschetz Fixed Point Formulas Index Theorem for Manifolds with Boundary The Eta Invariant of Locally Flat Bundles Spectral Geometry Introduction Operators of Laplace Type Isospectral Manifolds Non-Minimal Operators Operators of Dirac Type Manifolds with Boundary Other Asymptotic Formulas The Eta Invariant of Spherical Space Forms A Guide to the Literature Acknowledgment Introduction Bibliography Notation

O autorze

Gilkey, Peter B.