本书是一部经典的微分几何教材，初版于20世纪60年代，被全世界许多大学用于课堂教学。畅销40多年，产生了深远影响。新版延续了第1版善于化难为易，贴近实用而又不失严格的风格，主要对一些主题进行了扩展。更加强调拓扑性质、测地线的性质、向量场的奇异性等。更为重要的是，修订版增加了计算机建模的内容，提供了Mathematica和Maple程序。此外，还增加了相应的计算机习题，补充了奇数号码习题的答案，更便于教学。

本书介绍曲线和曲面几何的入门知识，主要内容包括欧氏空间上的积分、帧场、欧氏几何、曲面积分、形状算子、曲面几何、黎曼几何、曲面上的球面结构等。修订版扩展了一些主题，更加强调拓扑性质、测地线的性质、向量场的奇异性等。更为重要的是，修订版增加了计算机建模的内容，提供了Mathematica和Maple程序。此外，还增加了相应的计算机习题，补充了奇数号码习题的答案，更便于教学。

本书适合作为高等院校本科生相关课程的教材，也适合作为相关专业研究生和科研人员的参考书。

1. Calculus on Euclidean Space

 1.1. Euclidean Space

 1.2. Tangent Vectors

 1.3. Directional Derivatives

 1.4. Curves in R3

 1.5. 1-Forms

 1.6. Differential Forms

 1.7. Mappings

 1.8. Summary

2. Frame Fields

 2.1. Dot Product

 2.2. Curves

 2.3. The Frenet Formulas

 2.4. Arbitrary-speed Curves

 2.5. Covariant Derivatives

 2.6. Frame Fields

 2.7. Connection Forms

 2.8. The Structural Equations

3.Euclidean Geometry

 3.1. Isometries of R3

 3.2. The Tangent Map of an Isometry

 3.3. Orientation

 3.4. Euclidean Geometry

 3.5. Congruence of Curves

 3.6. Summary

4.Calculus on a SUrface

 4.1. Surfaces in R3

 4.2. Patch Computations

 4.3. Differentiable Functions and Tangent Vectors

 4.4. Differential Forms on a Surface

 4.5. Mappings of Surfaces

 4.6. Integration of Forms

 4.7. Topological Properties of Surfaces

 4.8. Manifcllds

 4.9. Summary

5.Shape Operators

 5.1. The Shape Operator of M c R3

 5.2. Normal Curvature

 5.3. Gaussian Curvature

 5.4. Computational Techniques

 5.5. The Implicit Case

 5.6. Special Curves in a Surface

 5.7. Surfaces of Revolution

 5.8. Summary

6.Geometry Of Sudaces in R3

 6.1. The Fundamental Equations

 6.2. Form Computations

 6.3. Some Global Theorems

 6.4. Isometries and Local Isometries

 6.5. Intrinsic Geometry of Surfaces in R3

 6.6. Orthogonal Coordinates

 6.7. Integration and Orientation

 6.8. Total Curvature

 6.9. Congruence of Surfaces

 6.10. Summary

7.Riemannian Geometry

 7.1. Geometric Surfaces

 7.2. Gaussian Curvature

 7.3. Covariant Derivative

 7.4. Geodesics

 7.5. Clairaut Parametrizations

 7.6. The Gauss.Bonnet Theorem

 7.7. Applications of Gauss。Bonnet

 7.8. Summary

8.GIobaI Structure of Suffaces

 8.1. Length.Minimizing Properties of Geodesics

 8.2. Complete Surfaces

 8.3. Curvature and Conjugate Points

 8.4. Covering Surfaces

 8.5. Mappings That Preserve Inner Products

 8.6. Surfaces of Constant Curvature

 8.7. Theorems of Bonnet and Hadamard

 8.8. Summary

Appendix：Computer Formulas

Bibliography

Answers to Odd-Numbered Exercises

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