这本精心编写的教材介绍了微分几何的美妙思想和结果。前半部分介绍了曲线和曲面的几何，为一般理论提供了大量的动机和直觉。第二部分研究一般流形的几何，特别强调了联络和曲率。文中配有许多图表和示例。读者阅读本书前需要先学习本科的数学分析和线性代数。新版做了很多改进，包括更多的图表和习题，并为很多选定习题提供了解答。

Preface to the English Edition
Preface to the German Edition
Chapter 1. Notations and Prerequisites from Analysis
Chapter 2. Curves in IRn
2A Frenet curves in IRn
2B Plane curves and space curves
2C Relations between the curvature and the torsion
2D The Frenet equations and the fundamental theorem of thelocal theory of curves
2E Curves in Minkowski space IR31
2F The global theory of curves
Exercises
Chapter 3. The Local Theory of Surfaces
3A Surface elements and the first fundamental form
3B The Gauss map and the curvature of surfaces
3C Surfaces of rotation and ruled surfaces
3D Minimal surfaces
3E Surfaces in Minkowski space IR31
3F Hypersurfaces in IRn+1
Exercises
Chapter 4. The Intrinsic Geometry of Surfaces
4A The covariant derivative
4B Parallel displacement and geodesics
4C The Gauss equation and the Theorema Egregium
4D The fundamental theorem of the local theory of surfaces
4E The Gaussian curvature in special parameters
4F The Gauss-Bonnet Theorem
4G Selected topics in the global theory of surfaces
Exercises
Chapter 5. Riemannian Manifolds
5A The notion of a manifold
5B The tangent space
5C Riemannian metrics
5D The Riemannian connection
Chapter 6. The Curvature Tensor
6A Tensors
6B The sectional curvature
6C The Ricci tensor and the Einstein tensor
Chapter 7. Spaces of Constant Curvature
7A Hyperbolic space
7B Geodesics and Jacobi fields
7C The space form problem
7D Three-dimensional Euclidean and spherical space forms
Exercises
Chapter 8. Einstein Spaces
8A The variation of the Hilbert-Einstein functional
8B The Einstein field equations
8C Homogenous Einstein spaces
8D The decomposition of the curvature tensor
8E The Weyl tensor
8F Duality for four-manifolds and Petrov typesExercises
Solutions to selected exercises
Bibliography
List of notation
Index