Differential geometry : bundles, connections, metrics and curvature /
Bundles, connections, metrics and curvature are the lingua franca of modern differential geometry and theoretical physics. Supplying graduate students in mathematics or theoretical physics with the fundamentals of these objects, this book would suit a one-semester course on the subject of bundles an...
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| Format: | eBook |
| Language: | English |
| Language Notes: | English. |
| Published: |
Oxford :
Oxford University Press,
2011.
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| Series: | Oxford graduate texts in mathematics ;
23. |
| Subjects: | |
| Online Access: | Connect to the full text of this electronic book |
Table of Contents:
- Cover; Contents; 1 Smooth manifolds; 1.1 Smooth manifolds; 1.2 The inverse function theorem and implicit function theorem; 1.3 Submanifolds of R[sup(m)]; 1.4 Submanifolds of manifolds; 1.5 More constructions of manifolds; 1.6 More smooth manifolds: The Grassmannians; Appendix 1.1 How to prove the inverse function and implicit function theorems; Appendix 1.2 Partitions of unity; Additional reading; 2 Matrices and Lie groups; 2.1 The general linear group; 2.2 Lie groups; 2.3 Examples of Lie groups; 2.4 Some complex Lie groups; 2.5 The groups Sl(n; C), U(n) and SU(n).
- 2.6 Notation with regards to matrices and differentialsAppendix 2.1 The transition functions for the Grassmannians; Additional reading; 3 Introduction to vector bundles; 3.1 The definition; 3.2 The standard definition; 3.3 The first examples of vector bundles; 3.4 The tangent bundle; 3.5 Tangent bundle examples; 3.6 The cotangent bundle; 3.7 Bundle homomorphisms; 3.8 Sections of vector bundles; 3.9 Sections of TM and T*M; Additional reading; 4 Algebra of vector bundles; 4.1 Subbundles; 4.2 Quotient bundles; 4.3 The dual bundle; 4.4 Bundles of homomorphisms; 4.5 Tensor product bundles.
- 4.6 The direct sum4.7 Tensor powers; Additional reading; 5 Maps and vector bundles; 5.1 The pull-back construction; 5.2 Pull-backs and Grassmannians; 5.3 Pull-back of differential forms and push-forward of vector fields; 5.4 Invariant forms and vector fields on Lie groups; 5.5 The exponential map on a matrix group; 5.6 The exponential map and right/left invariance on Gl(n; C) and its subgroups; 5.7 Immersion, submersion and transversality; Additional reading; 6 Vector bundles with C[sup(n)] as fiber; 6.1 Definitions; 6.2 Comparing definitions; 6.3 Examples: The complexification.
- 6.4 Complex bundles over surfaces in R[sup(3)]6.5 The tangent bundle to a surface in R[sup(3)]; 6.6 Bundles over 4-dimensional submanifolds in R[sup(5)]; 6.7 Complex bundles over 4-dimensional manifolds; 6.8 Complex Grassmannians; 6.9 The exterior product construction; 6.10 Algebraic operations; 6.11 Pull-back; Additional reading; 7 Metrics on vector bundles; 7.1 Metrics and transition functions for real vector bundles; 7.2 Metrics and transition functions for complex vector bundles; 7.3 Metrics, algebra and maps; 7.4 Metrics on TM; Additional reading; 8 Geodesics.
- 8.1 Riemannian metrics and distance8.2 Length minimizing curves; 8.3 The existence of geodesics; 8.4 First examples; 8.5 Geodesics on SO(n); 8.6 Geodesics on U(n) and SU(n); 8.7 Geodesics and matrix groups; Appendix 8.1 The proof of the vector field theorem; Additional reading; 9 Properties of geodesics; 9.1 The maximal extension of a geodesic; 9.2 The exponential map; 9.3 Gaussian coordinates; 9.4 The proof of the geodesic theorem; Additional reading; 10 Principal bundles; 10.1 The definition; 10.2 A cocycle definition; 10.3 Principal bundles constructed from vector bundles.