黎曼几何和几何分析 : 第6版

1 Riemannian Manifolds

1.1 Manifolds and Differentiable Manifolds

1.2 Tangent Spaces

1.3 Submanifolds

1.4 Riemannian Metrics

1.5 Existence of Geodesics on Compact Manifolds

1.6 The Heat Flow and the Existence of Geodesics

1.7 Existence of Geodesics on Complete Manifolds

Exercises for Chapter 1

2 Lie Groups and Vector Bundles

2.1 Vector Bundles

2.2 Integral Curves of Vector Fields.Lie Algebras

2.3 Lie Groups

2.4 Spin Structures

Exercises for Chapter 2

3 The Laplace Operator and Harmonic Differential Forms

3.1 The Laplace Operator on Functions

3.2 The Spectrum of the Laplace Operator

3.3 The Laplace Operator on Forms

3.4 Representing Cohomology Classes by Harmonic Forms

3.5 Generalizations

3.6 The Heat Flow and Harmonic Forms

Exercises for Chapter 3

4 Connections and Curvature

4.1 Connections in Vector Bundles

4.2 Metric Connections.The Yang—Mills Functional

4.3 The Levi—Civita Connection

4.4 Connections for Spin Structures and the Dirac Operator

4.5 The Bochner Method

4.6 Eigenvalue Estimates by the Method of Li—Yau

4.7 The Geometry of Submanifolds

4.8 Minimal Submanifolds

Exercises for Chapter 4

5 Geodesics and Jacobi Fields

5.1 First and second Variation of Arc Length and Energy

5.2 Jacobi Fields

5.3 Conjugate Points and Distance Minimizing Geodesics

5.4 Riemannian Manifolds of Constant Curvature

5.5 The Rauch Comparison Theorems and Other Jacobi Field Estimates

5.6 Geometric Applications of Jacobi Field Estimates

5.7 Approximate Fundamental Solutions and Representation Formulas

5.8 The Geometry of Manifolds of Nonpositive Sectional Curvature

Exercises for Chapter 5

A Short Survey on Curvature and Topology

6 Symmetric Spaces and Kahler Manifolds

6.1 Complex Projective Space

6.2 Kahler Manifolds

6.3 The Geometry of Symmetric Spaces

6.4 Some Results about the Structure of Symmetric Spaces

6.5 The Space Sl（n，IR）／SO（n，IR）

6.6 Symmetric Spaces of Noncompact Type

Exercises for Chapter 6

7 Morse Theory and Floer Homology

7.1 Preliminaries： Aims of Morse Theory

7.2 The Palais—Smale Condition，Existence of Saddle Points

7.3 Local Analysis

7.4 Limits of Trajectories of the Gradient Flow

7.5 Floer Condition，Transversality and Z2—Cohomology

7.6 Orientations and Z—homology

7.7 Homotopies

7.8 Graph flows

7.9 Orientations

7.10 The Morse Inequalities

7.11 The Palais—Smale Condition and the Existence of Closed Geodesics

Exercises for Chapter 7

8 Harmonic Maps between Riemannian Manifolds

8.1 Definitions

8.2 Formulas for Harmonic Maps.The Bochner Technique

8.3 The Energy Integral and Weakly Harmonic Maps

8.4 Higher Regularity

8.5 Existence of Harmonic Maps for Nonpositive Curvature

8.6 Regularity of Harmonic Maps for Nonpositive Curvature

8.7 Harmonic Map Uniqueness and Applications

Exercises for Chapter 8

9 Harmonic Maps from Riemann Surfaces

9.1 Two—dimensional Harmonic Mappings

9.2 The Existence of Harmonic Maps in Two Dimensions

9.3 Regularity Results

Exercises for Chapter 9

10 Variational Problems from Quantum Field Theory

10.1 The Ginzburg—Landau Functional

10.2 The Seiberg—Witten Functional

10.3 Dirac—harmonic Maps

Exercises for Chapter 10

A Linear Elliptic Partial Differential Equations

A.1 Sobolev Spaces

A.2 Linear Elliptic Equations

A.3 Linear Parabolic Equations

B Fundamental Groups and Covering Spaces

Bibliography

Index

Riemannian geometry is characterized, and research is oriented towards and shaped by concepts (geodesics, connections, curvature, ...) and objectives, in particular to understand certain classes of (compact) Riemannian manifolds defined by curvature conditions (constant or positive or negative curvature, ...). By way of contrast, geometric analysis is a perhaps somewhat less systematic collection of techniques, for solving extremal problems naturally arising in geometry and for investigating and characterizing their solutions. It turns out that the two fields complement each other very well; geometric analysis offers tools for solving difficult problems in geometry, and Riemannian geometry stimulates progress in geometric analysis by setting ambitious goals.
　　It is the aim of this book to be a systematic and comprehensive introduction to Riemannian geometry and a representative introduction to the methods of geometric analysis. It attempts a synthesis of geometric and analytic methods in the study of Riemannian manifolds.
　　The present work is the sixth edition of my textbook on Riemannian geometry and geometric analysis. It has developed on the basis of several graduate courses I taught at the Ruhr~University Bochum and the University of Leipzig. The main new feature of the present edition is a systematic presentation of the spectrum of the Laplace operator and its relation with the geometry of the underlying Riemannian marufold. Naturally, I have also included several smaller additions and minor corrections (for which I am grateful to several readers). Moreover, the organization of the chapters has been systematically rearranged.