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内容简介
《曲线与曲面的微分几何》（英文版）是一本关于曲线和曲面微分几何的导论，介绍微分几何这两个方面的局部特性与整体特性。同传统的微分几何教材不同，本书更广泛地应用初等线性代数的知识，并把重点放在基本的几何论据上。
本书是（英文版）一本关于曲线和曲面微分几何的导论，介绍微分几何这两个方面的局部特性与整体特性。同传统的微分几何教材不同，本书更广泛地应用初等线性代数的知识，并把重点放在基本的几何论据上。
为取得概念与实际材料之间的适度平衡，本书还包含大量的例子，并合理安排习题，其中包含经典微分几何的某些实际题材。
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顾客评论
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目录
目 录 Some Remarks on Using this Book vii Curves 1 Introduction I Parametrized Curves 2 Regular Curves; Arc Length 5 The Vector Product in Ra3 11 The Local Theory of Curves Parametrized by Arc Length 16 The Local Canonical Form 27 Global Properties of Plane Curves 30 Regular Surfaces 51 Introduction 51 Regular Surfaces; Inverse Images of Regular Values 52 Change of Parameters; Differential Functions on Surfaces 69 The Tangent Plane; the Differential of a Map 83 The First Fundamental Form; Area 92 Orientation of Surfaces 102 A Characterization of Compact Orientable Surfaces 109 A Geometric Definition of Area 114 Appendix: A Brief Review on Continuity and Differentiability 118 3. The Geometry of the Gauss Map 134 3-1 Introduction 134 3-2 The Definition of the Gauss Map and Its Fundamental Properties 135 3-3 The Gauss Map in Local Coordinates 153 3-4 Vector Fields 175 3-5 Ruled Surfaces and Minimal Surfaces 188 Appendix: Self-Adjoint Linear Maps and Quadratic Forms 214 4. The Intrinsic Geometry of Surfaces 217 4-1 Introduction 217 4-2 Isometries; Conformal Maps 218 4-3 The Gauss Theorem and the Equations of Compatibility 231 4-4 Parallel Transport; Geodesics 238 4-5 The Gauss-Bonnet Theorem and its Applications 264 4-6 The Exponential Map. Geodesic Polar Coordinates 283 4-7 Further Properties of Geodesics. Convex Neighborhoods 298 Appendix: Proofs of the Fundamental Theorems of The Local Theory of Curves and Surfaces 309 5. Global Differential Geometry 315 5-1 Introduction 315 5-2 The Rigidity of the Sphere 317 5-3 Complete Surfaces. Theorem of Hopf-Rinow 325 5-4 First and Second Variations of the Arc Length; Bonnet''''s Theorem 339 5-5 Jacobi Fields and Conjugate Points 357 5-6 Covering Spaces; the Theorems of Hadamard 371 5-7 Global Theorems for Curves; the Fary-Miinor Theorem 380 5-8 Surfaces of Zero Gaussian Curvature 408 5-9 Jacobi''''s Theorems 415 5-10 Abstract Surfaces; Further Generalizations 425 5-11 Hilbert''''s Theorem 446 Appendix: Point-Set Topology of Euclidean Spaces 456 Bibliography and Comments 471 Hints and Answers to Some Exercises 475 Index 497
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