本书介绍黎曼几何中的重要技巧和定理，为满足那些希望专门研究黎曼几何的学生，书中还包含大量关于较深论题的背景材料。本书还介绍了*新的研究闷题。各种练习散布全书，帮助读者深入理解书中内容。本书是为数不多的整合了黎曼几何的几何和分析两方面内容的专著之一，适合熟悉张量和斯托克斯定理等流形理论的读者，可作为研究生一学年课程的教材。

Preface
Chapter 1. Riemannian Metrics
1. Riemannian Manifolds and Maps
2. Groups and Riemannian Manifolds
3. Local Representations of Metrics
4. Doubly Warped Products
5. Exercises
Chapter 2. Curvature
1. Connections
2. The Connection in Local Coordinates
3. Curvature
4. The Fundamental Curvature Equations
5. The Equations of Riemannian Geometry
6. Some Tensor Concepts
7. Further Study
8. Exercises
Chapter 3. Examples
1. Computational Simplifications
2. Warped Products
3. Hyperbolic Space
4. Metrics on Lie Groups
5. Riemannian Submersions
6. Fhrther Study
7. Exercises
Chapter 4. Hypersurfaces
1. The Gauss Map
2. Existence of Hypersurfaces
3. The Gauss-Bonnet Theorem
4. Further Study
5. Exercises
Chapter 5. Geodesics and Distance
1. Mixed Partials
2. Geodesics
3. The Metric Structure of a Riemannian Manifold
4. First Variat of Energy
5. The Exponential Map
6. Why Short Geodesics Are Segments
7. Local Geometry in Constant Curvature
8. Completeness
9. Characterization of Segments
10. Riemannian Isometries
11. Further Study
12. Exercises
Chapter 6. Sectional Curvature Comparison I
1. The Connection Along Curves
2. Second Variation of Energy
3. Nonpositive Sectional Curvature
4. Positive Curvature
5. Basic Comparison Estimates
6. More on Positive Curvature
7. Further Study
8. Exercises
Chapter 7. The Bochner Technique
1. Killing Fields
2. Hodge Theory
3. Harmonic Forms
4. Clifford Multiplication on Forms
5. The Curvature Tensor
6. Further Study
7. Exercises
Chapter 8. Symmetric Spaces and Holonomy
1. Symmetric Spaces
2. Examples of Symmetric Spaces
3. Holonomy
4. Curvature and Holonomy
5. Further Study
6. Exercises
Chapter 9. Ricci Curvature Comparison
1. Volume Comparison
2. Fundamental Groups and Ricci Curvature
3. Manifolds of Nonnegative Ricci Curvature
4. Further Study
5. Exercises
Chapter 10. Convergence
1. Gromov-Hausdorff Convergence
2. HSlder Spaces and Schauder Estimates
3. Norms and Convergence of Manifolds
4. Geometric Applications
5. Harmonic Norms and Ricci curvature
6. Further Studv
7. Exercises
Chapter 11. Sectional Curvature Comparison 2
Appendix. De Rham Cohomology
Bibliography
Index