(Not in Invariance Theory
The Heat Equation and the Atiyah-Singer Index Theorem
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$239.95$215.96 - Hardback: 536 pages
- Published: December 1994
- ISBN: 978-0-8493-7874-4
- Publisher: CRC Press
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- By Peter B. Gilkey.
- Series Edited by Steven G. Krantz.
Series: Studies in Advanced Mathematics.
This book treats the Atiyah-Singer index 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 author uses invariance theory to identify the integrand of the index theorem for classical elliptic complexes with the invariants of the heat equation.
Table of Contents
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
