这本精心编写的教材介绍了微分几何的美妙思想和结果。前半部分涵盖了曲线和曲面的几何，它们为一般理论提供了很多动力和直觉。第二部分研究一般流形的几何，特别强调联络和曲率。书中附有许多图表和示例。阅读本书之前需要先学习本科的数学分析和线性代数。新版做了很多修订，包括更多的图表和习题，并新增了很多精选习题的解答。 这个新版本是一个提升改进的版本，而上一版已经是关于微分几何和黎曼几何的优秀入门教材了。除了各种修订，作者还新增了许多问题的解答，以使本书更适合课堂使用。 —Colin Adams, Williams College Kühnel 的这本关于微分几何的书是对该主题的极好和有用的介绍。 ……关于微分几何有很多不同的观点，也有很多通往其概念的路径。本书提供了一个出色的、令人兴奋且优美的基础，可以用来探索这个深刻而基础的数学主题。 —Louis Kauffman, University of Illinois at Chicago

前辅文
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 the
  local theory of curves
  2E Curves in Minkowski space IR
  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 IR
  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 types
  Exercises
Solutions to selected exercises
Bibliography
List of notation
Index