微分几何基础

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

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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