An Introduction to Differentiable Manifolds and Riemannian Geometry

Name: An Introduction to Differentiable Manifolds and Riemannian Geometry
Brand: Academic Press
Price: 54.95 EUR
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Unknown Author(Herausgeber*in)

Academic Press

2. Auflage

Erschienen am 21. April 1986

429 Seiten

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Inhalt

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

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