UID:
almahu_9948595854202882
Format:
1 online resource (447 p.)
Edition:
2nd ed.
ISBN:
1-281-76323-3
,
9786611763237
,
0-08-087439-8
Series Statement:
Pure and applied mathematics ; v. 120
Content:
An introduction to differentiable manifolds and Riemannian geometry (2nd Ed)
Note:
Description based upon print version of record.
,
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 Rn4. 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 Manifold1. 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 Spaces4. 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 GroupNotes; 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
,
English
Additional Edition:
ISBN 0-12-116052-1
Language:
English
