Geometry of Manifolds

Name: Geometry of Manifolds | Geometry of Manifolds
Brand: Academic Press
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Academic Press

Erschienen am 29. August 2011

272 Seiten

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978-0-08-087327-5 (ISBN)

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Inhalt

Front Cover
Geometry of Manifolds
Copyright Page
Contents
Preface
Chapter 1. Manifolds
1.1 Introductory Material and Notation
1.2 Definition of a Manifold
1.3 Tangent Space
1.4 Vector Fields
1.5 Submanifolds
1.6 Distributions and Integrability
Chapter 2. Lie Groups
2.1 Lie Groups
2.2 Lie Algebras
2.3 Lie Group-Lie Algebra Correspondence
2.4 Homomorphisms
2.5 Exponential Map
2.6 Representations
Chapter 3. Fibre Bundles
3.1 Transformation Groups
3.2 Principal Bundles
3.3 Associated Bundles
3.4 Reduction of the Structural Group
Chapter 4. Differential Forms
4.1 Introduction
4.2 Classical Notion of Differential Form
4.3 Grassmann Algebras
4.4. Existence of Grassmann Algebras
4.5 Differential Forms
4.6 Exterior Derivative
4.7 Action of Maps
4.8 Frobenius' Theorem
4.9. Vector-Valued Forms and Operations
4.10 Forms on Complex Manifolds
Chapter 5. Connexions
5.1 Definitions and First Properties
5.2. Parallel Translation
5.3 Curvature Form and the Structural Equation
5.4 Existence of Connexions and Connexions in Associated Bundles
5.5 Structural Equations for Horizontal Forms
5.6 Holonomy
Chapter 6. Affine Connexions
6.1 Definitions
6.2. The Structural Equations of an Affine Connexion
6.3 The Exponential Maps
6.4 Covariant Differentiation and Classical Forms
Chapter 7. Riemannian Manifolds
7.1 Definitions and First Properties
7.2 The Bundle of Frames
7.3 Riemannian Connexions
7.4 Examples and Problems
Chapter 8. Geodesics and Complete Riemannian ManifoIds
8.1 Geodesics
8.2 Complete Riemannian Manifolds
8.3 Continuous Curves
Chapter 9. Riemannian Curvature
9.1 Riemannian Curvature
9.2 Computation of the Riemannian Curvature
9.3 Continuity of the Riemannian Curvature
9.4 Rectangles and Jacobi Fields
9.5 Theorems Involving Curvature
Chapter 10. Immersions and the Second Fundamental Form
10.1 Definitions
10.2 The Connexions
10.3 Curvature
10.4 The Second Fundamental Form
10.5 Curvature and the Second Fundamental Form
10.6 The Local Gauss Map
10.7 Hessians of Normal Coordinates of N
10.8 A Formulation of the Immersion Problem
10.9 Hypersurfaces
Chapter 11. Second Variation of Arc Length
11.1 First and Second Variation of Arc Length
11.2 The Index Form
11.3 Focal Points and Conjugate Points
11.4 The Infinitesimal Deformations
11.5 The Morse Index Theorem
11.6 The Minimum Locus
11.7 Closed Geodesics
11.8 Convex Neighborhoods
11.9 Rauch's Comparison Theorem
11.10 Curvature and Volume
Appendix. Theorems on Diferential Equations
Bibliography
Subject Index

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