Lectures on differential geometry / Shlomo Sternberg.
Series Prentice-Hall mathematics seriesEditor: Englewood Cliffs, N.J. : Prentice-Hall, c1964Descripción: xi, 390 p. : il. ; 24 cmTema(s): Geometry, DifferentialOtra clasificación: 58-01 (53-01)I Algebraic Preliminaries [1] 1. Tensor products of vector spaces [2] 2. The tensor algebra of a vector space [5] 3. The contravariant and symmetric algebras [10] 4. Exterior algebra [14] 5. Exterior equations [24] II Differentiable Manifolds [31] 1. Definitions [32] 2. Differential maps [39] 3. Sard’s theorem [45] 4. Partitions of unity, approximation theorems [55] 5. The tangent space [69] 6. The principal bundle [73] 7. The tensor bundles [84] 8. Vector fields and Lie derivatives [89] III Integral Calculus on Manifolds [97] 1. Ths operator d [99] 2. Chains and integration [104] 3. Integration of densities [111] 4. 0 and n-dimensional cohomology, degree [120] 5. Frobenius' theorem [130] 4. Darboux's theorem [137] 7. Hamiltonian structures [141] The Calculus of Variations [148] 1. Legendre transformations [150] 2. Necessary conditions [153] 3. Conservation laws [170] 4. Sufficient conditions [174] 5. Conjugate and focal points, Jacobi's condition [182] 6. The Riemannian case [199] 7. Completeness [207] 8. Isometries [212] Lie Groups [214] 1. Definitions [214] 2. The invariant forms and the Lie algebra [217] 3. Normal coordinates, exponential map [222] 4. Closed subgroups [228] 5. Invariant metrics [231] 6. Forms with values In a vector space [234] VI Differential Geometry of Euclidean Space [237] 1. The equations of structure of Euclidean space [237] 2. The equations of structure of a submanifold [244] 3. The equations of structure of a Riemann manifold [246] 4. Curves in Euclidean space [252] 5. The second fundamental form [264] 6. Surfaces [279] VII The Geometry of G-Structures [293] 1. Principal and associated bundles, connections [294] 2. G-structures [309] 3. Prolongations [331] 4. Structures of finite type [339] 5. Connections on G-structures [351] 6. The spray of a linear connection [361] appendix I Two Existence Theorems [369] appendix II Outline of Theory of Integration on E [374] References [385] Index [387]
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58 Si617-2 Lecture notes on elementary topology and geometry / | 58 So687 Soliton mathematics / | 58 So718 Singularidades de aplicações diferenciáveis / | 58 St839 Lectures on differential geometry / | 58 St927 Plateau's problem and the calculus of variations / | 58 T396 Theory of limit cycles / | 58 T452 Structural stability and morphogenesis : |
Bibliografía: p. 385-386.
I Algebraic Preliminaries [1] --
1. Tensor products of vector spaces [2] --
2. The tensor algebra of a vector space [5] --
3. The contravariant and symmetric algebras [10] --
4. Exterior algebra [14] --
5. Exterior equations [24] --
II Differentiable Manifolds [31] --
1. Definitions [32] --
2. Differential maps [39] --
3. Sard’s theorem [45] --
4. Partitions of unity, approximation theorems [55] --
5. The tangent space [69] --
6. The principal bundle [73] --
7. The tensor bundles [84] --
8. Vector fields and Lie derivatives [89] --
III Integral Calculus on Manifolds [97] --
1. Ths operator d [99] --
2. Chains and integration [104] --
3. Integration of densities [111] --
4. 0 and n-dimensional cohomology, degree [120] --
5. Frobenius' theorem [130] --
4. Darboux's theorem [137] --
7. Hamiltonian structures [141] --
The Calculus of Variations [148] --
1. Legendre transformations [150] --
2. Necessary conditions [153] --
3. Conservation laws [170] --
4. Sufficient conditions [174] --
5. Conjugate and focal points, Jacobi's condition [182] --
6. The Riemannian case [199] --
7. Completeness [207] --
8. Isometries [212] --
Lie Groups [214] --
1. Definitions [214] --
2. The invariant forms and the Lie algebra [217] --
3. Normal coordinates, exponential map [222] --
4. Closed subgroups [228] --
5. Invariant metrics [231] --
6. Forms with values In a vector space [234] --
VI Differential Geometry of Euclidean Space [237] --
1. The equations of structure of Euclidean space [237] --
2. The equations of structure of a submanifold [244] --
3. The equations of structure of a Riemann manifold [246] --
4. Curves in Euclidean space [252] --
5. The second fundamental form [264] --
6. Surfaces [279] --
VII The Geometry of G-Structures [293] --
1. Principal and associated bundles, connections [294] --
2. G-structures [309] --
3. Prolongations [331] --
4. Structures of finite type [339] --
5. Connections on G-structures [351] --
6. The spray of a linear connection [361] --
appendix I Two Existence Theorems [369] --
appendix II Outline of Theory of Integration on E [374] --
References [385] --
Index [387] --
MR, 33 #1797
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