## 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 |
---|---|---|---|---|---|---|---|---|

Libros | 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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