The Laplacian on a Riemannian Manifold: An

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.

Introduction

vii

1 The Laplacian on a Riemannian Manifold

1

(51)

1.1 Basic Examples

2

(8)

1.1.1 The Laplacian on S1 and R

3

(2)

1.1.2 Heat Flow on S1 and R

5

(5)

1.2 The Laplacian on a Riemannian Manifold

10

(12)

1.2.1 Riemannian Metrics

10

(4)

1.2.2 L2 Spaces of Functions and Forms

14

(3)

1.2.3 The Laplacian on Functions

17

(5)

1.3 Hodge Theory for Functions and Forms

22

(17)

1.3.1 Analytic Preliminaries

22

(5)

1.3.2 The Heat Equation Proof of the Hodge Theorem for Functions

27

(6)

1.3.3 The Hodge Theorem for Differential Forms

33

(2)

1.3.4 Regularity Results

35

(4)

1.4 De Rham Cohomology

39

(7)

1.5 The Kernel of the Laplacian on Forms

46

(6)

2 Elements of Differential Geometry

52

(36)

2.1 Curvature

52

(11)

2.2 The Levi-Civita Connection and Bochner Formula

63

(14)

2.2.1 The Levi-Civita Connection

63

(4)

2.2.2 Weitzenbock Formulas and Garding's Inequality

67

(10)

2.3 Geodesics

77

(6)

2.4 The Laplacian in Exponential Coordinates

83

(5)

3 The Construction of the Heat Kernel

88

(21)

3.1 Preliminary Results for the Heat Kernel

88

(2)

3.2 Construction of the Heat Kernel

90

(9)

3.2.1 Construction of the Parametrix

90

(4)

3.2.2 The Heat Kernel for Functions

94

(5)

3.3 The Asymptotics of the Heat Kernel

99

(7)

3.4 Positivity of the Heat Kernel

106

(3)

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

109

(33)

4.1 The Chern-Gauss-Bonnet Theorem

109

(17)

4.1.1 The Heat Equation Approach

110

(4)

4.1.2 Proof of the Chern-Gauss-Bonnet Theorem

114

(12)

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

126

(16)

4.2.1 A Survey of Characteristic Forms

126

(6)

4.2.2 The Hirzebruch Signature Theorem

132

(5)

4.2.3 The Atiyah-Singer Index Theorem

137

(5)

5 Zeta Functions of Laplacians

142

(23)

5.1 The Zeta Function of a Laplacian

142

(7)

5.2 Isospectral Manifolds

149

(2)

5.3 Reidemeister Torsion and Analytic Torsion

151

(14)

5.3.1 Reidemeister Torsion

151

(1)

5.3.2 Analytic Torsion

152

(9)

5.3.3 The Families Index Theorem and Analytic Torsion

161

(4)

Bibliography

165

(5)

Index

170

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The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds

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