《微分几何基础》是由(美国)尼尔 (Barrett O'neill)编写，人民邮电出版社出版的一本书籍。

作者：(美国)尼尔 (Barrett O'neill)

Barrett O'Neill，加州大学洛杉矶分校教授。1951年在麻省理丁学院获得博士学位。他的研究方向包括：曲线和曲面几何，计算机和曲面，黎曼几何，黑洞理论等。另著有Semi-Riemannian Geometry with Applications to Relativity和The Geometry of Kerr Black Holes等书。

《微分几何基础(英文版·第2版修订版)》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。

《微分几何基础(英文版·第2版修订版)》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

2.9. Summary

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. Manifolds

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 Surfaces in R

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. Global Structure of Surfaces

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

Index