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

by Gilkey; Peter B.

• ISBN13: 9780849378744
• ISBN10: 0849378745
• Edition: 2nd
• Format: Hardcover
• Copyright: 1994-12-22
• Publisher: CRC Press

Summary

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