An Introduction to Differentiable Manifolds and Riemannian Geometry
2nd Edition
Published on 21. April 1986
429 pages
978-0-08-087439-5 (ISBN)
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An Introduction to Differentiable Manifolds and Riemannian Geometry
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William M. Boothby
Introduction to Differentiable Manifolds and Riemannian Geometry
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09/1997
2nd Edition
Academic Press
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An Introduction to Differentiable Manifolds and Riemannian Geometry
An Introduction to Differentiable Manifolds and Riemannian Geometry
E-Book
08/1975
1st Edition
Academic Press
€54.95
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Content
- Front Cover
- An Introduction to Differentiable Manifolds and Riemannian Geometry
- Copyright Page
- Contents
- Preface to the Second Edition
- Preface to the First Edition
- Chapter I. Introduction to Manifolds
- 1. Preliminary Comments on Rn
- 2. Rn and Euclidean Space
- 3. Topological Manifolds
- 4. Further Examples of Manifolds. Cutting and Pasting
- 5. Abstract Manifolds. Some Examples
- Notes
- Chapter II. Functions of Several Variables and Mappings
- 1. Differentiability for Functions of Several Variables
- 2. Differentiability of Mappings and Jacobians
- 3. The Space of Tangent Vectors at a Point of Rn
- 4. Another Definition of Ta(Rn)
- 5. Vector Fields on Open Subsets of Rn
- 6. The Inverse Function Theorem
- 7. The Rank of a Mapping
- Notes
- Chapter III. Differentiable Manifolds and Submanifolds
- 1. The Definition of a Differentiable Manifold
- 2. Further Examples
- 3. Differentiable Functions and Mappings
- 4. Rank of a Mapping. Immersions
- 5. Submanifolds
- 6. Lie Groups
- 7. The Action of a Lie Group on a Manifold. Transformation Groups
- 8. The Action of a Discrete Group on a Manifold
- 9. Covering Manifolds
- Notes
- Chapter IV. Vector Fields on a Manifold
- 1. The Tangent Space at a Point of a Manifold
- 2. Vector Fields
- 3. One-Parameter and Local One-Parameter Groups Acting on a Manifold
- 4. The Existence Theorem for Ordinary Differential Equations
- 5. Some Examples of One-Parameter Groups Acting on a Manifold
- 6. One-Parameter Subgroups of Lie Groups
- 7. The Lie Algebra of Vector Fields on a Manifold
- 8. Frobenius' Theorem
- 9. Homogeneous Spaces
- Appendix Partial Proof of Theorem 4.1
- Notes
- Chapter V. Tensors and Tensor Fields on Manifolds
- 1. Tangent Covectors
- 2. Bilinear Forms. The Riemannian Metric
- 3. Riemannian Manifolds as Metric Spaces
- 4. Partitions of Unity
- 5. Tensor Fields
- 6. Multiplication of Tensors
- 7. Orientation of Manifolds and the Volume Element
- 8. Exterior Differentiation
- Notes
- Chapter VI. Integration on Manifolds
- 1. Integration in Rn. Domains of Integration
- 2. A Generalization to Manifolds
- 3. Integration on Lie Groups
- 4. Manifolds with Boundary
- 5. Stokes's Theorem for Manifolds with Boundary
- 6. Homotopy of Mappings. The Fundamental Group
- 7. Some Applications of Differential Forms. The de Rham Groups
- 8. Some Further Applications of de Rham Groups
- 9. Covering Spaces and the Fundamental Group
- Notes
- Chapter VII. Differentiation on Riemannian Manifolds
- 1. Differentiation of Vector Fields along Curves in Rn
- 2. Differentiation of Vector Fields on Submanifolds of Rn
- 3. Differentiation on Riemannian Manifolds
- 4. Addenda to the Theory of Differentiation on a Manifold
- 5. Geodesic Curves on Riemannian Manifolds
- 6. The Tangent Bundle and Exponential Mapping. Normal Coordinates
- 7. Some Further Properties of Geodesics
- 8. Symmetric Riemannian Manifolds
- 9. Some Examples
- Notes
- Chapter VIII. Curvature
- 1 . The Geometry of Surfaces in E3
- 2. The Gaussian and Mean Curvatures of a Surface
- 3. Basic Properties of the Riemann Curvature Tensor
- 4. The Curvature Forms and the Equations of Structure
- 5. Differentiation of Covariant Tensor Fields
- 6. Manifolds of Constant Curvature
- Notes
- References
- Index
