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《微分幾何基礎(英文版·第2版修訂版)》介紹麯綫和麯麵幾何的入門知識，主要內容包括歐氏空間上的積分、幀場、歐氏幾何、麯麵積分、形狀算子、麯麵幾何、黎曼幾何、麯麵上的球麵結構等。修訂版擴展瞭一些主題，更加強調拓撲性質、測地綫的性質、嚮量場的奇異性等。更為重要的是，修訂版增加瞭計算機建模的內容，提供瞭Mathematica和Maple程序。此外，還增加瞭相應的計算機習題，補充瞭奇數號碼習題的答案，更便於教學。

《微分幾何基礎(英文版·第2版修訂版)》適閤作為高等院校本科生相關課程的教材，也適閤作為相關專業研究生和科研人員的參考書。

Barrett O'Neill，加州大學洛杉磯分校教授。1951年在麻省理工學院獲得博士學位。他的研究方嚮包括：麯綫和麯麵幾何，計算機和麯麵，黎曼幾何，黑澗理論等。另著有Semi-Riemannian Geometry with Applications to Relativity和The Geometry of Kerr BlackHoles等書。

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. Summary2. 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 Equations3.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. Summary4.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. Summary5.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. Summary6.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. Summary7.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. Summary8.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. SummaryAppendix：Computer FormulasBibliographyAnswers to Odd-Numbered Exerciseslndex
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