The Laplacian on a Riemannian Manifold ：An Introduction to Analysis on Manifolds ( London Mathematical Society Student Texts )

Publication subTitle ：An Introduction to Analysis on Manifolds

Publication series ：London Mathematical Society Student Texts

Author： Steven Rosenberg  

Publisher： Cambridge University Press‎

Publication year： 1997

E-ISBN: 9780511885709

P-ISBN(Paperback): 9780521463003

Subject： O186.12 Riemannian geometry

Keyword： 黎曼几何

Language： ENG

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The Laplacian on a Riemannian Manifold

Description

This text on analysis of Riemannian manifolds is a thorough introduction to topics covered in advanced research monographs on Atiyah-Singer index theory. The main theme is the study of heat flow associated to the Laplacians on differential forms. This provides a unified treatment of Hodge theory and the supersymmetric proof of the Chern-Gauss-Bonnet theorem. In particular, there is a careful treatment of the heat kernel for the Laplacian on functions. The Atiyah-Singer index theorem and its applications are developed (without complete proofs) via the heat equation method. Zeta functions for Laplacians and analytic torsion are also treated, and the recently uncovered relation between index theory and analytic torsion is laid out. The text is aimed at students who have had a first course in differentiable manifolds, and the Riemannian geometry used is developed from the beginning. There are over 100 exercises with hints.

Chapter

Cover

Title

Copyright

Contents

Introduction

1 The Laplacian on a Riemannian Manifold

1.1 Basic Examples

1.1.1 The Laplacian on S1 and R

1.1.2 Heat Flow on S1 and R

1.2 The Laplacian on a Riemannian Manifold

1.2.1 Riemannian Metrics

1.2.2 L2 Spaces of Functions and Forms

1.2.3 The Laplacian on Functions

1.3 Hodge Theory for Functions and Forms

1.3.1 Analytic Preliminaries

1.3.2 The Heat Equation Proof of the Hodge Theorem for Functions

1.3.3 The Hodge Theorem for Differential Forms

1.3.4 Regularity Results

1.4 De Rham Cohomology

1.5 The Kernel of the Laplacian on Forms

2 Elements of Differential Geometry

2.1 Curvature

2.2 The Levi-Civita Connection and Bochner Formula

2.2.1 The Levi-Civita Connection

2.2.2 Weitzenböck Formulas and Gårding's Inequality

2. 3 Geodesies

2.4. The Laplacian in Exponential Coordinates

3 The Construction of the Heat Kernel

3.1 Preliminary Results for the Heat Kernel

3.2 Construction of the Heat Kernel

3.2.1 Construction of the Parametrix

3.2.2 The Heat Kernel for Functions

3.3. The Asymptotics of the Heat Kernel

3.4 Positivity of the Heat Kernel

4 The Heat Equation Approach to the Atiyah-Singer Index Theorem

4.1 The Chern-Gauss-Bonnet Theorem

4.1.1 The Heat Equation Approach

4.1.2 Proof of the Chern-Gauss-Bonnet Theorem

4.2 The Hirzebruch Signature Theorem and the Atiyah-Singer Index Theorem

4.2.1 A Survey of Characteristic Forms

4.2.2 The Hirzebruch Signature Theorem

4.2.3 The Atiyah-Singer Index Theorem

5 Zeta Functions of Laplacians

5.1 The Zeta Function of a Laplacian

5.2 Isospectral Manifolds

5.3 Reidemeister Torsion and Analytic Torsion

5.3.1 Reidemeister Torsion

5.3.2 Analytic Torsion

5.3.3 The Families Index Theorem and Analytic Torsion

Bibliography

Index

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