Differential geometry of curves and surfaces / Manfredo P. do Carmo.
Idioma: Inglés Lenguaje original: Portugués Editor: Upper Saddle River, N.J. : Prentice-Hall, c1976Descripción: viii, 503 p. : il. ; 24 cmISBN: 0132125897Otra clasificación: 53-02Preface v Some Remarks on Using this Book vii 1. Curves [1] 1-1 Introduction [1] 1-2 Parametrized Curves [2] 1-3 Regular Curves; Arc Length [5] 1-4 The Vector Product in R3 [11] 1-5 The Local Theory of Curves Parametrized by Arc Length [16] 1-6 The Local Canonical Form [27] 1-7 Global Properties of Plane Curves [30] 2. Regular Surfaces [51] 2- 1 Introduction [51] 2-2 Regular Surfaces; Inverse Images of Regular Values [52] 2-3 Change of Parameters; Differential Functions on Surfaces [69] 2-4 The Tangent Plane; the Differential of a Map [83] 2-5 The First Fundamental Form; Area [92] 2-6 Orientation of Surfaces [102] 2-7 A Characterization of Compact Orientable Surfaces [109] 2-8 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-Milnor 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]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 53 C287 (Browse shelf) | Available | A-7976 |
"A free translation, with additional material, of a book and a set of notes, both published originally in Portuguese."
Includes bibliographical references and index.
Preface v --
Some Remarks on Using this Book vii --
1. Curves [1] --
1-1 Introduction [1] --
1-2 Parametrized Curves [2] --
1-3 Regular Curves; Arc Length [5] --
1-4 The Vector Product in R3 [11] --
1-5 The Local Theory of Curves Parametrized by Arc Length [16] --
1-6 The Local Canonical Form [27] --
1-7 Global Properties of Plane Curves [30] --
2. Regular Surfaces [51] --
2- 1 Introduction [51] --
2-2 Regular Surfaces; Inverse Images of Regular Values [52] --
2-3 Change of Parameters; Differential Functions on Surfaces [69] --
2-4 The Tangent Plane; the Differential of a Map [83] --
2-5 The First Fundamental Form; Area [92] --
2-6 Orientation of Surfaces [102] --
2-7 A Characterization of Compact Orientable Surfaces [109] --
2-8 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-Milnor 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] --
MR, 52 #15253
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