UID:
almahu_9947366706002882
Format:
1 online resource (287 pages)
ISBN:
1-281-76633-X
,
9786611766337
,
0-08-087327-8
Series Statement:
Pure and applied mathematics; a series of monographs and textbooks ; 15
Note:
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
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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
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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
,
English
Additional Edition:
ISBN 0-12-374565-9
Language:
English
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