UID:
almahu_9947366245802882
Format:
1 online resource (441 p.)
ISBN:
1-281-76322-5
,
9786611763220
,
0-08-087379-0
Series Statement:
Pure and applied mathematics
Content:
An introduction to differentiable manifolds and Riemannian geometry
Note:
Description based upon print version of record.
,
Front Cover; An Introduction to Differentiable Manifolds and Riemannian Geometry; Copyright Page; Contents; Preface; 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)
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5. Vector Fields on Open Subsets of Rn6. 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
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3. One-Parameter and Local One-Parameter Groups Acting on a Manifold4. 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; Notes; Appendix: Partial Proof of Theorem 4.1; 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
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6. Multiplication of Tensors7. Orientation of Manifolds and the Volume Element; 8. Exterior Differentiation; Notes; Chapter Vl. 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 or 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
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Chapter VII. Differentiation on Riemannian Manifolds1. 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
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3. Basic Properties of the Riemann Curvature Tensor
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English
Additional Edition:
ISBN 0-12-116050-5
Language:
English
