Notes on differential geometry / by Noel J. Hicks.

Por: Hicks, Noel JSeries Van Nostrand mathematical studies: no. 3.Editor: Princeton, N.J. : Van Nostrand, c1965Descripción: vi, 183 p. : il. ; 21 cmTema(s): Geometry, DifferentialOtra clasificación: 53-01
Contenidos:
Chapter 1. MANIFOLDS [1]
1.1 Manifolds [1]
1.2 Smooth functions [3]
1.3 Vectors and vector fields [5]
1.4 The Jacobian of a map [9]
1.5 Curves and integral curves [10]
1.6 Submanifolds [13]
Chapter 2. HYPERSURFACES OF Rn [18]
2.1 The standard connexion on Rn [18]
2.2 The sphere map and the Weingarten map [21]
2.3 The Gauss equation [25]
2.4 The Gauss curvature and Codazzi-Mainardi equations [28]
2.5 Examples [29]
2.6 Some applications [35]
Chapter 3. SURFACES IN R3 [38]
3.1 Smoothness and the neighborhood of a non-umbilic point [38]
3.2 Surfaces of constant curvature [40]
3.3 Parallel surfaces (normal maps) [44]
3.4 Examples (surfaces of revolution) [45]
3.5 Lines of curvature [46]
Chapter 4. TENSORS AND FORMS [49]
Chapter 5. CONNEXIONS [56]
5.1 Invariant viewpoint [56]
5.2 Cartan viewpoint [61]
5.3 Coordinate viewpoint [63]
5.4 Difference tensor of two connexions [64]
5.5 Bundle viewpoint [65]
Chapter 6. RIEMANNIAN MANIFOLDS AND SUBMANIFOLDS [69]
6.1 Length and distance [69]
6.2 Riemannian connexion and curvature [71]
6.3 Curves in Riemannian manifolds [74]
6.4 Submanifolds [75]
6.5 Hypersurfaces [77]
6.6 Cartan viewpoint and coordinate viewpoint [81]
6.7 Canonical spaces of constant curvature [83]
6.8 Existence [84]
Chapter 7. OPERATORS ON FORMS AND INTEGRATION [89]
7.1 Exterior derivative [89]
7.2 Contraction [91]
7.3 Lie derivative [92]
7.4 General covariant derivative [94]
7.5 Integration of forms and Stokes’ theorem [98]
7.6 Integration in a Riemannian manifold [101]
Chapter 8. GAUSS-BONNET THEORY AND RIGIDITY [105]
8.1 Gauss-Bonnet formula [105]
8.2 Index Theorem [111]
8.3 Gauss-Bonnet form [114]
8.4 Characteristic forms [116]
8.5 Rigidity problems [120]
Chapter 9. EXISTENCE THEORY [123]
9.1 Involutive distributions and the Frobenius theorem [123]
9.2 The fundamental existence theorem for hypersurfaces [128]
9.3 The exponential map [131]
9.4 Convex neighborhoods [134]
9.5 Special coordinate systems [136]
9.6 Isothermal coordinates and Riemann surfaces [138]
Chapter 10. TOPICS IN RIEMANNIAN GEOMETRY [142]
10.1 Jacobi fields and conjugate points [142]
10.2 First and second variation formulae [147]
10.3 Geometric interpretation of Riemannian curvature [154]
10.4 The Morse Index Theorem [157]
10.5 Completeness [163]
10.6 Manifolds with constant Riemannian curvature [167]
10.7 Manifolds without conjugate points [170]
10.8 Manifolds with non-positive curvature [172]
Bibliography [176]
Index [181]
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Libros Libros Instituto de Matemática, CONICET-UNS
Libros ordenados por tema 53 H631 (Browse shelf) Available A-2224

GEOMETRÍA II


Bibliografía: p. 176-180.

Chapter 1. MANIFOLDS [1] --
1.1 Manifolds [1] --
1.2 Smooth functions [3] --
1.3 Vectors and vector fields [5] --
1.4 The Jacobian of a map [9] --
1.5 Curves and integral curves [10] --
1.6 Submanifolds [13] --
Chapter 2. HYPERSURFACES OF Rn [18] --
2.1 The standard connexion on Rn [18] --
2.2 The sphere map and the Weingarten map [21] --
2.3 The Gauss equation [25] --
2.4 The Gauss curvature and Codazzi-Mainardi equations [28] --
2.5 Examples [29] --
2.6 Some applications [35] --
Chapter 3. SURFACES IN R3 [38] --
3.1 Smoothness and the neighborhood of a non-umbilic point [38] --
3.2 Surfaces of constant curvature [40] --
3.3 Parallel surfaces (normal maps) [44] --
3.4 Examples (surfaces of revolution) [45] --
3.5 Lines of curvature [46] --
Chapter 4. TENSORS AND FORMS [49] --
Chapter 5. CONNEXIONS [56] --
5.1 Invariant viewpoint [56] --
5.2 Cartan viewpoint [61] --
5.3 Coordinate viewpoint [63] --
5.4 Difference tensor of two connexions [64] --
5.5 Bundle viewpoint [65] --
Chapter 6. RIEMANNIAN MANIFOLDS AND SUBMANIFOLDS [69] --
6.1 Length and distance [69] --
6.2 Riemannian connexion and curvature [71] --
6.3 Curves in Riemannian manifolds [74] --
6.4 Submanifolds [75] --
6.5 Hypersurfaces [77] --
6.6 Cartan viewpoint and coordinate viewpoint [81] --
6.7 Canonical spaces of constant curvature [83] --
6.8 Existence [84] --
Chapter 7. OPERATORS ON FORMS AND INTEGRATION [89] --
7.1 Exterior derivative [89] --
7.2 Contraction [91] --
7.3 Lie derivative [92] --
7.4 General covariant derivative [94] --
7.5 Integration of forms and Stokes’ theorem [98] --
7.6 Integration in a Riemannian manifold [101] --
Chapter 8. GAUSS-BONNET THEORY AND RIGIDITY [105] --
8.1 Gauss-Bonnet formula [105] --
8.2 Index Theorem [111] --
8.3 Gauss-Bonnet form [114] --
8.4 Characteristic forms [116] --
8.5 Rigidity problems [120] --
Chapter 9. EXISTENCE THEORY [123] --
9.1 Involutive distributions and the Frobenius theorem [123] --
9.2 The fundamental existence theorem for hypersurfaces [128] --
9.3 The exponential map [131] --
9.4 Convex neighborhoods [134] --
9.5 Special coordinate systems [136] --
9.6 Isothermal coordinates and Riemann surfaces [138] --
Chapter 10. TOPICS IN RIEMANNIAN GEOMETRY [142] --
10.1 Jacobi fields and conjugate points [142] --
10.2 First and second variation formulae [147] --
10.3 Geometric interpretation of Riemannian curvature [154] --
10.4 The Morse Index Theorem [157] --
10.5 Completeness [163] --
10.6 Manifolds with constant Riemannian curvature [167] --
10.7 Manifolds without conjugate points [170] --
10.8 Manifolds with non-positive curvature [172] --
Bibliography [176] --
Index [181] --

MR, 31 #3936

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