本书是一部值得一读的研究生教材,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*热门论题,有些内容则是**出现在教科书中。本书各章有习题。
目次: 基本理论; 德拉姆上同调和调和微分形式;并行传输、联络和共变导数;测地学和雅**场;对称空间和Kahler流形;莫斯理论和Floer同调;量子场论中的变分问题;调和映射。
读者对象:数学和理论物理专业的研究生、教师和科研人员。

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 Cohomology 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 Coonections 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
6.Morse Theory and Floer Homology
7.Variational Problems from Quantum Field Theory
8.Harmonic Maps
Appendix A:Linear Elliptic Partial Differential Equation
Appendix B:Fundamental Groups and Covering Spaces
Index