1st Edition

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

By Peter B. Gilkey Copyright 1995

    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.

    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

    Biography

    Gilkey, Peter B.