An introduction to differentiable manifolds and Riemannian geometry / William M. Boothby.
Series Pure and applied mathematics (Academic Press): 120.Editor: Orlando : Academic Press, 1986Edición: 2nd edDescripción: xvi, 430 p. : il. ; 24 cmISBN: 0121160521; 012116053X (pbk.)Tema(s): Differentiable manifolds | Riemannian manifoldsOtra clasificación: 58-01 (22Exx 53C20)Contents Preface to the Second Edition xi Preface to the First Edition xiii I. Introduction to Manifolds 1. Preliminary Comments on Rn [1] 2. and Euclidean Space [4] 1 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 R" [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 [96] 9. Covering Manifolds [101] Notes [104] IV. Vector Fields on a Manifold 1. The Tangent Space at a Point of a Manifold [107] 2. Vector Fields [116] 3. One-Parameter and Local One-Parameter Groups Acting on a Manifold [12] 4. The Existence Theorem for Ordinary Differential Equations [131] 5. Some Examples of One-Parameter Groups Acting on a Manifold [139] 6. One-Parameter Subgroups of Lie Groups [147] 7. The Lie Algebra of Vector Fields on a Manifold [151] 8. Frobenius’ Theorem [158] 9. Homogeneous Spaces [165] Appendix Partial Proof of Theorem 4.1 [172] Notes [174] V. Tensors and Tensor Fields on Manifolds 1. Tangent Covectors [177] Covectors on Manifolds 178 Covector Fields and Mappings [180] 2. Bilinear Forms. The Riemannian Metric [183] 3. Riemannian Manifolds as Metric Spaces [187] 4. Partitions of Unity [193] Some Applications of the Partition of Unity [195] 5. Tensor Fields [199] Tensors on a Vector Space [199] Tensor Fields 201 Mappings and Covariant Tensors [202] The Symmetrizing and Alternating Transformations [203] 6. Multiplication of Tensors [206] Multiplication of Tensors on a Vector Space 206 Multiplication of Tensor Fields [208] Exterior Multiplication of Alternating Tensors 209 The Exterior Algebra on Manifolds [213] 7. Orientation of Manifolds and the Volume Element [215] 8. Exterior Differentiation [219] An Application to Frobenius’ Theorem [223] Notes [227] VI. Integration on Manifolds 1. Integration in Rn. Domains of Integration [230] Basic Properties of the Riemann Integral [231] 2. A Generalization to Manifolds [236] Integration on Riemannian Manifolds [240] 3. Integration on Lie Groups [244] 4. Manifolds with Boundary [251] 5. Stokes’s Theorem for Manifolds with Boundary [259] 6. Homotopy of Mappings. The Fundamental Group [266] Homotopy of Paths and Loops. The Fundamental Group [268] 7. Some Applications of Differential Forms. The de Rham Groups [274] The Homotopy Operator [277] 8. Some Further Applications of de Rham Groups [281] The de Rham Groups of Lie Groups [285] 9. Covering Spaces and the Fundamental Group 289 Notes [296] VII. Differentiation on Riemannian Manifolds 1. Differentiation of Vector Fields along Curves in Rn [298] The Geometry of Space Curves [301] Curvature of Plane Curves [305] 2. Differentiation of Vector Fields on Submanifolds of Rn [307] Formulas for Covariant Derivatives [312] VXp T and Differentiation of Vector Fields [314] 3. Differentiation on Riemannian Manifolds [317] Constant Vector Fields and Parallel Displacement [323] 4. Addenda to the Theory of Differentiation on a Manifold [325] The Curvature Tensor [325] The Riemannian Connection and Exterior Differential Forms [328] 5. Geodesic Curves on Riemannian Manifolds [330] 6. The Tangent Bundle and Exponential Mapping. Normal Coordinates [335] 7. Some Further Properties of Geodesics [342] 8. Symmetric Riemannian Manifolds [351] 9. Some Examples [357] Notes [364] VIII. Curvature 1. The Geometry of Surfaces in E3 [366] The Principal Curvatures at a Point of a Surface [370] 2. The Gaussian and Mean Curvatures of a Surface [374] The Theorema Egregium of Gauss [377] 3. Basic Properties of the Riemann Curvature Tensor [382] 4. The Curvature Forms and the Equations of Structure [389] 5. Differentiation of Covariant Tensor Fields [396] 6. Manifolds of Constant Curvature [403] Spaces of Positive Curvature [406] Spaces of Zero Curvature [408] Spaces of Constant Negative Curvature 409 Notes [414] References [417] Index [423]
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Incluye índice.
Bibliografía: p. 417-422.
Contents --
Preface to the Second Edition xi Preface to the First Edition xiii --
I. Introduction to Manifolds --
1. Preliminary Comments on Rn [1] --
2. and Euclidean Space [4] --
1 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 R" [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 [96] --
9. Covering Manifolds [101] --
Notes [104] --
IV. Vector Fields on a Manifold --
1. The Tangent Space at a Point of a Manifold [107] --
2. Vector Fields [116] --
3. One-Parameter and Local One-Parameter Groups Acting on a Manifold [12] --
4. The Existence Theorem for Ordinary Differential Equations [131] --
5. Some Examples of One-Parameter Groups Acting on a Manifold [139] --
6. One-Parameter Subgroups of Lie Groups [147] --
7. The Lie Algebra of Vector Fields on a Manifold [151] --
8. Frobenius’ Theorem [158] --
9. Homogeneous Spaces [165] --
Appendix Partial Proof of Theorem 4.1 [172] --
Notes [174] --
V. Tensors and Tensor Fields on Manifolds --
1. Tangent Covectors [177] --
Covectors on Manifolds 178 Covector Fields and Mappings [180] --
2. Bilinear Forms. The Riemannian Metric [183] --
3. Riemannian Manifolds as Metric Spaces [187] --
4. Partitions of Unity [193] --
Some Applications of the Partition of Unity [195] --
5. Tensor Fields [199] --
Tensors on a Vector Space [199] --
Tensor Fields 201 Mappings and Covariant Tensors [202] --
The Symmetrizing and Alternating Transformations [203] --
6. Multiplication of Tensors [206] --
Multiplication of Tensors on a Vector Space 206 Multiplication of Tensor Fields [208] --
Exterior Multiplication of Alternating Tensors 209 The Exterior Algebra on Manifolds [213] --
7. Orientation of Manifolds and the Volume Element [215] --
8. Exterior Differentiation [219] --
An Application to Frobenius’ Theorem [223] --
Notes [227] --
VI. Integration on Manifolds --
1. Integration in Rn. Domains of Integration [230] --
Basic Properties of the Riemann Integral [231] --
2. A Generalization to Manifolds [236] --
Integration on Riemannian Manifolds [240] --
3. Integration on Lie Groups [244] --
4. Manifolds with Boundary [251] --
5. Stokes’s Theorem for Manifolds with Boundary [259] --
6. Homotopy of Mappings. The Fundamental Group [266] --
Homotopy of Paths and Loops. The Fundamental Group [268] --
7. Some Applications of Differential Forms. The de Rham Groups [274] --
The Homotopy Operator [277] --
8. Some Further Applications of de Rham Groups [281] --
The de Rham Groups of Lie Groups [285] --
9. Covering Spaces and the Fundamental Group 289 Notes [296] --
VII. Differentiation on Riemannian Manifolds --
1. Differentiation of Vector Fields along Curves in Rn [298] --
The Geometry of Space Curves [301] --
Curvature of Plane Curves [305] --
2. Differentiation of Vector Fields on Submanifolds of Rn [307] --
Formulas for Covariant Derivatives [312] --
VXp T and Differentiation of Vector Fields [314] --
3. Differentiation on Riemannian Manifolds [317] --
Constant Vector Fields and Parallel Displacement [323] --
4. Addenda to the Theory of Differentiation on a Manifold [325] --
The Curvature Tensor [325] --
The Riemannian Connection and Exterior Differential Forms [328] --
5. Geodesic Curves on Riemannian Manifolds [330] --
6. The Tangent Bundle and Exponential Mapping. Normal Coordinates [335] --
7. Some Further Properties of Geodesics [342] --
8. Symmetric Riemannian Manifolds [351] --
9. Some Examples [357] --
Notes [364] --
VIII. Curvature --
1. The Geometry of Surfaces in E3 [366] --
The Principal Curvatures at a Point of a Surface [370] --
2. The Gaussian and Mean Curvatures of a Surface [374] --
The Theorema Egregium of Gauss [377] --
3. Basic Properties of the Riemann Curvature Tensor [382] --
4. The Curvature Forms and the Equations of Structure [389] --
5. Differentiation of Covariant Tensor Fields [396] --
6. Manifolds of Constant Curvature [403] --
Spaces of Positive Curvature [406] --
Spaces of Zero Curvature [408] --
Spaces of Constant Negative Curvature 409 Notes [414] --
References [417] --
Index [423] --
MR, 87k:58001
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