Manfredo do Carmo (1928–2018)

Indeholder "Preface", "Some Remarks on Using this Book", "1. Curves", " 1.1 Introduction", " 1.2 Parametrized Curves", " 1.3 Regular Curves; Arc Length", " 1.4 The Vector Product in R^3", " 1.5 The Local Theory of Curves Parametrized by Arc Length", " 1.6 The Local Canonical Form", " 1.7 Global Properties of Plane Curves", "2. Regular Surfaces", " 2.1 Introduction", " 2.2 Regular Surfaces; Inverse Images of Regular Values", " 2.3 Change of Parameters; Differential Functions on Surfaces", " 2.4 The Tangent Plane; the Differential of a Map", " 2.5 The First Fundamental Form; Area", " 2.6 Orientation of Surfaces", " 2.7 A Characterization of Compact Orientable Surfaces", " 2.8 A Geometric Definition of Area", " Appendix: A Brief Review on Continuity and Differentiability", "3. The Geometry of the Gauss Map", " 3.1 Introduction", " 3.2 The Definition of the Gauss Map and Its Fundamental Properties", " 3.3 The Gauss Map in Local Coordinates", " 3.4 Vector Fields", " 3.5 Ruled Surfaces and Minimal Surfaces", " Appendix: Self-Adjoint Linear Maps and Quadratic Forms", "4. The Intrinsic Geometry of Surfaces", " 4.1 Introduction", " 4.2 Isometries; Conformal Maps", " 4.3 The Gauss Theorem and the Equations of Compatibility", " 4.4 Parallel Transport; Geodesics", " 4.5 The Gauss-Bonnet Theorem and its Applications", " 4.6 The Exponential Map. Geodesic Polar Coordinates", " 4.7 Further Properties of Geodesics. Convex Neighborhoods", " Appendix: Proofs of the Fundamental Theorems of The Local Theory of Curves and Surfaces", "5. Global Differential Geometry", " 5.1 Introduction", " 5.2 The Rigidity of the Sphere", " 5.3 Complete Surfaces. Theorem of Hopf-Rinow", " 5.4 First and Second Variations of the Arc Length; Bonnet's Theorem", " 5.5 Jacobi Fields and Conjugate Points", " 5.6 Covering Spaces; the Theorems of Hadamard", " 5.7 Global Theorems for Curves; the Fary-Milnor Theorem", " 5.8 Surfaces of Zero Gaussian Curvature", " 5.9 Jacobi's Theorems", " 5.10 Abstract Surfaces; Further Generalizations", " 5.11 Hilbert's Theorem", " Appendix: Point-Set Topology of Euclidean Spaces", "Bibliography and Comments", "Hints and Answers to Some Exercises", "Index".

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