An introduction to differentiable manifolds and Riemannian geometry / William M. Boothby.
Series Pure and applied mathematics (Academic Press): 63.Editor: New York : Academic Press, 1975Descripción: xiv, 424 p. : il. ; 24 cmISBN: 0121160505Tema(s): Differentiable manifolds | Riemannian manifoldsOtra clasificación: 58-01 (22Exx 53C20)Contents Preface I. Introduction to Manifolds 1. Preliminary Comments on Rn [1] 2. Rn" and Euclidean Space [4] 3. Topological Manifolds [6] 4. Further Examples of Manifolds. Cutting and Pasting [11] 5. Abstract Manifolds. Some Examples 14 Notes [18] II. Functions of Several Variables and Mappings 1. Differentiability for Functions of Several Variables [20] 2. Differentiability of Mappings and Jacobians [25] 3. The Space of Tangent Vectors at a Point of Rn [29] 4. Another Definition of Ta(Rn) [32] 5. Vector Fields on Open Subsets of Rn [37] 6. The Inverse Function Theorem [41] 7. The Rank of a Mapping [46] Notes [50] III. Differentiable Manifolds and Submanifolds 1. The Definition of a Differentiable Manifold [52] 2. Further Examples [60] 3. Differentiable Functions and Mappings [65] 4. Rank of a Mapping. Immersions [69] 5. Submanifolds [75] 6. Lie Groups [81] 7. The Action of a Lie Group on a Manifold. Transformation Groups [89] 8. The Action of a Discrete Group on a Manifold [95] 9. Covering Manifolds [100] Notes [104] IV. Vector Fields on a Manifold 1. The Tangent Space at a Point of a Manifold [106] 2. Vector Fields [115] 3. One-Parameter and Local One-Parameter Groups Acting on a Manifold [122] 4. The Existence Theorem for Ordinary Differential Equations [130] 5. Some Examples of One-Parameter Groups Acting on a Manifold [138] 6. One-Parameter Subgroups of Lie Groups [145] 7. The Lie Algebra of Vector Fields on a Manifold [149] 8. Frobenius’ Theorem [156] 9. Homogeneous Spaces [164] Notes [171] Appendix Partial Proof of Theorem 4.1 [172] V. Tensors and Tensor Fields on Manifolds 1. Tangent Covectors [175] Covectors on Manifolds [176] Covector Fields and Mappings [178] 2. Bilinear Forms. The Riemannian Metric [181] 3. Riemannian Manifolds as Metric Spaces [185] 4. Partitions of Unity [191] Some Applications of the Partition of Unity [193] 5. Tensor Fields [197] Tensors on a Vector Space [197] Tensor Fields [199] Mappings and Covariant Tensors [200] The Symmetrizing and Alternating Transformations [201] 6. Multiplication of Tensors [204] Multiplication of Tensors on a Vector Space [205] Multiplication of Tensor Fields [206] Exterior Multiplication of Alternating Tensors [207] The Exterior Algebra on Manifolds [211] 7. Orientation of Manifolds and the Volume Element [213] 8. Exterior Differentiation [217] An Application to Frobenius’ Theorem [221] Notes [225] VI. Integration on Manifolds 1. Integration in Rn. Domains of Integration [227] Basic Properties of the Riemann Integral [228] 2. A Generalization to Manifolds [233] Integration on Riemannian Manifolds [237] 3. Integration on Lie Groups [241] 4. Manifolds with Boundary [248] 5. Stokes’s Theorem for Manifolds with Boundary [256] 6. Homotopy of Mappings. The Fundamental Group [263] Homotopy of Paths and Loops. The Fundamental Group [265] 7. Some Applications of Differential Forms. The de Rham Groups [271] The Homotopy Operator [274] 8. Some Further Applications of de Rham Groups [278] The de Rham Groups of Lie Groups [282] 9. Covering Spaces and the Fundamental Group 286 Notes [292] VII. Differentiation on Riemannian Manifolds 1. Differentiation of Vector Fields along Curves in Rn [294] The Geometry of Space Curves [297] Curvature of Plane Curves [301] 2. Differentiation of Vector Fields on Submanifolds of Rn [303] Formulas for Covariant Derivatives [308] VXp,Y and Differentiation of Vector Fields [310] 3. Differentiation on Riemannian Manifolds [313] Constant Vector Fields and Parallel Displacement [319] 4. Addenda to the Theory of Differentiation on a Manifold [321] The Curvature Tensor [321] The Riemannian Connection and Exterior Differential Forms [324] 5. Geodesic Curves on Riemannian Manifolds [326] 6. The Tangent Bundle and Exponential Mapping. Normal Coordinates [331] 7. Some Further Properties of Geodesics [338] 8. Symmetric Riemannian Manifolds [347] 9. Some Examples [353] Notes [360] VIII. Curvature 1. The Geometry of Surfaces in E3 [362] The Principal Curvatures at a Point of a Surface [366] 2. The Gaussian and Mean Curvatures of a Surface [370] The Theorema Egregium of Gauss [373] 3. Basic Properties of the Riemann Curvature Tensor [378] 4. The Curvature Forms and the Equations of Structure [385] 5. Differentiation of Covariant Tensor Fields [391] 6. Manifolds of Constant Curvature [399] Spaces of Positive Curvature [402] Spaces of Zero Curvature [404] Spaces of Constant Negative Curvature [405] Notes [410] References [413] Index [417]
| 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 | 58 B725 (Browse shelf) | Available | A-4537 |
Bibliografía: p. 413-416.
Contents --
Preface --
I. Introduction to Manifolds --
1. Preliminary Comments on Rn [1] --
2. Rn" and Euclidean Space [4] --
3. Topological Manifolds [6] --
4. Further Examples of Manifolds. Cutting and Pasting [11] --
5. Abstract Manifolds. Some Examples 14 Notes [18] --
II. Functions of Several Variables and Mappings --
1. Differentiability for Functions of Several Variables [20] --
2. Differentiability of Mappings and Jacobians [25] --
3. The Space of Tangent Vectors at a Point of Rn [29] --
4. Another Definition of Ta(Rn) [32] --
5. Vector Fields on Open Subsets of Rn [37] --
6. The Inverse Function Theorem [41] --
7. The Rank of a Mapping [46] --
Notes [50] --
III. Differentiable Manifolds and Submanifolds --
1. The Definition of a Differentiable Manifold [52] --
2. Further Examples [60] --
3. Differentiable Functions and Mappings [65] --
4. Rank of a Mapping. Immersions [69] --
5. Submanifolds [75] --
6. Lie Groups [81] --
7. The Action of a Lie Group on a Manifold. Transformation Groups [89] --
8. The Action of a Discrete Group on a Manifold [95] --
9. Covering Manifolds [100] --
Notes [104] --
IV. Vector Fields on a Manifold --
1. The Tangent Space at a Point of a Manifold [106] --
2. Vector Fields [115] --
3. One-Parameter and Local One-Parameter Groups Acting on a Manifold [122] --
4. The Existence Theorem for Ordinary Differential Equations [130] --
5. Some Examples of One-Parameter Groups Acting on a Manifold [138] --
6. One-Parameter Subgroups of Lie Groups [145] --
7. The Lie Algebra of Vector Fields on a Manifold [149] --
8. Frobenius’ Theorem [156] --
9. Homogeneous Spaces [164] --
Notes [171] --
Appendix Partial Proof of Theorem 4.1 [172] --
V. Tensors and Tensor Fields on Manifolds --
1. Tangent Covectors [175] --
Covectors on Manifolds [176] --
Covector Fields and Mappings [178] --
2. Bilinear Forms. The Riemannian Metric [181] --
3. Riemannian Manifolds as Metric Spaces [185] --
4. Partitions of Unity [191] --
Some Applications of the Partition of Unity [193] --
5. Tensor Fields [197] --
Tensors on a Vector Space [197] --
Tensor Fields [199] --
Mappings and Covariant Tensors [200] --
The Symmetrizing and Alternating Transformations [201] --
6. Multiplication of Tensors [204] --
Multiplication of Tensors on a Vector Space [205] --
Multiplication of Tensor Fields [206] --
Exterior Multiplication of Alternating Tensors [207] --
The Exterior Algebra on Manifolds [211] --
7. Orientation of Manifolds and the Volume Element [213] --
8. Exterior Differentiation [217] --
An Application to Frobenius’ Theorem [221] --
Notes [225] --
VI. Integration on Manifolds --
1. Integration in Rn. Domains of Integration [227] --
Basic Properties of the Riemann Integral [228] --
2. A Generalization to Manifolds [233] --
Integration on Riemannian Manifolds [237] --
3. Integration on Lie Groups [241] --
4. Manifolds with Boundary [248] --
5. Stokes’s Theorem for Manifolds with Boundary [256] --
6. Homotopy of Mappings. The Fundamental Group [263] --
Homotopy of Paths and Loops. The Fundamental Group [265] --
7. Some Applications of Differential Forms. The de Rham Groups [271] --
The Homotopy Operator [274] --
8. Some Further Applications of de Rham Groups [278] --
The de Rham Groups of Lie Groups [282] --
9. Covering Spaces and the Fundamental Group 286 Notes [292] --
VII. Differentiation on Riemannian Manifolds --
1. Differentiation of Vector Fields along Curves in Rn [294] --
The Geometry of Space Curves [297] --
Curvature of Plane Curves [301] --
2. Differentiation of Vector Fields on Submanifolds of Rn [303] --
Formulas for Covariant Derivatives [308] --
VXp,Y and Differentiation of Vector Fields [310] --
3. Differentiation on Riemannian Manifolds [313] --
Constant Vector Fields and Parallel Displacement [319] --
4. Addenda to the Theory of Differentiation on a Manifold [321] --
The Curvature Tensor [321] --
The Riemannian Connection and Exterior Differential Forms [324] --
5. Geodesic Curves on Riemannian Manifolds [326] --
6. The Tangent Bundle and Exponential Mapping. Normal Coordinates [331] --
7. Some Further Properties of Geodesics [338] --
8. Symmetric Riemannian Manifolds [347] --
9. Some Examples [353] --
Notes [360] --
VIII. Curvature --
1. The Geometry of Surfaces in E3 [362] --
The Principal Curvatures at a Point of a Surface [366] --
2. The Gaussian and Mean Curvatures of a Surface [370] --
The Theorema Egregium of Gauss [373] --
3. Basic Properties of the Riemann Curvature Tensor [378] --
4. The Curvature Forms and the Equations of Structure [385] --
5. Differentiation of Covariant Tensor Fields [391] --
6. Manifolds of Constant Curvature [399] --
Spaces of Positive Curvature [402] --
Spaces of Zero Curvature [404] --
Spaces of Constant Negative Curvature [405] --
Notes [410] --
References [413] --
Index [417] --
MR, 54 #13956
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