本书是一部值得一读的研究生教材(全英文版)，内容主要涉及黎曼几何基本定理的研究，如霍奇定理、Rauch比较定理、Lyusternik和Fet定理调和映射的存在性等，书中还有当代数学研究领域中的最热门论题，有些内容则是首次出现在教科书中。本书各章均附有习题。

1. Foundational Material

　1.1 Manifolds and Differentiable Manifolds

　1.2 Tangent Spaces

　1.3 Submanifolds

　1.4 Riemannian Metrics

　1.5 Vector Bundles

　1.6 Integral Curves of Vector Fields. Lie Algebras

　1.7 Lie Groups

　1.8 Spin Structures

　Exercises for Chapter 1

2. De Rham Cohomology and Harmonic Differential Forms

　2.1 The Laplace Operator

　2.2 Representing Co homology Classes by Harmonic Forms

　2.3 Generalizations

　Exercises for Chapter 2

3. Parallel Transport, Connections, and Covariant Derivatives

　3.1 Connections in Vector Bundles

　3.2 Metric Connections. The Yang-Mills Functional

　3.3 The Levi-Civita Connection

　3.4 Connections for Spin Structures and the Dirac Operator

　3.5 The Bochner Method

　3.6 The Geometry of Submanifolds. Minimal Submanifolds

　Exercises for Chapter 3

4. Geodesics and Jacobi Fields

　4.1 1st and 2nd Variation of Arc Length and Energy

　4.2 Jacobi Fields

　4.3 Conjugate Points and Distance Minimizing Geodesics

　4.4 Riemannian Manifolds of Constant Curvature

　4.5 The Rauch Comparison Theorems and Other Jacobi Field Estimates

　4.6 Geometric Applications of Jacobi Field Estimates

　4.7 Approximate Fundamental Solutions and Representation Formulae

　4.8 The Geometry of Manifolds of Nonpositive Sectional Curvature

　 Exercises for Chapter 4

　A Short Survey on Curvature and Topology

5. Symmetric Spaces and Kahler Manifolds

　5.1 Complex Projective Space

　5.2 Kahler Manifolds

　5.3 The Geometry of Symmetric Spaces

　5.4 Some Results about the Structure of Symmetric Spaces

　5.5 The Space SI(n,R)/SO(n,R)

　5.6 Symmetric Spaces of Noncompact Type as Examples of Nonpositively Curved Riemannian Manifolds

　 Exercises for Chapter 5

6. Morse Theory and Floer Homology

　6.1 Preliminaries: Aims of Morse Theory

　6.2 Compactness: The Palais-Smale Condition and the Existence of Saddle Points

　6.3 Local Analysis: Nondegeneracy of Critical Points, Morse Lemma, Stable and Unstable Manifolds

　6.4 Limits of Trajectories of the Gradient Flow

　6.5 The Morse-Smale-Floer Condition: Transversality and Z2-Cohomology

　6.6 Orientations and Z-homology

　6.7 Homotopies

　6.8 Graph flows

　6.9 Orientations

　6.10 The Morse Inequalities

　6.11 The Palais-Smale Condition and the Existence of Closed Geodesics

 Exercises for Chapter 6

7. Variational Problems from Quantum Field Theory

　7.1 The Ginzburg-Landau Functional

　7.2 The Seiberg-Witten Functional

　Exercises for Chapter 7

8. Harmonic Maps

 8.1 Definitions

 8.2 Twodimensional Harmonic Mappings and Holomorphic Quadratic Differentials

 8.3 The Existence of Harmonic Maps in Two Dimensions

 8.4 Definition and Lower Semicontinuity of the Energy Integral

 8.5 Weakly Harmonic Maps. Regularity Questions

 8.6 Higher Regularity

 8.7 Formulae for Harmonic Maps. The Bochner Technique

 8.8 Harmonic Maps into Manifolds of Nonpositive Sectional Curvature: Existence

 8.9 Harmonic Maps into Manifolds of Nonpositive Sectional Curvature: Regularity

 8.10 Harmonic Maps into Manifolds of Nonpositive Sectional Curvature: Uniqueness and Other properties

 Exercises for Chapter 8

Appendix A: Linear Elliptic Partial Differential Equation

 A.1 Sobolev Spaces

 A.2 Existence and Regularity Theory for Solutions of Linear Elliptic Equations

Appendix B: Fundamental Groups and Covering Spaces.

Bibliography

Index