Book contents
- Frontmatter
- Contents
- Preface
- 1 Linear algebra
- 2 Multilinear algebra
- 3 Differentiation on manifolds
- 4 Homotopy and de Rham cohomology
- 5 Elementary homology theory
- 6 Integration on manifolds
- 7 Vector bundles
- 8 Geometric manifolds
- 9 The degree of a smooth map
- Appendix A Mathematical background
- Appendix B The spectral theorem
- Appendix C Orientations and top-dimensional forms
- Appendix D Riemann normal coordinates
- Appendix E Holonomy of an infinitesimal loop
- Appendix F Frobenius' theorem
- Appendix G The topology of electrical circuits
- Appendix H Intrinsic and extrinsic curvature
- References
- Index
7 - Vector bundles
Published online by Cambridge University Press: 05 June 2014
- Frontmatter
- Contents
- Preface
- 1 Linear algebra
- 2 Multilinear algebra
- 3 Differentiation on manifolds
- 4 Homotopy and de Rham cohomology
- 5 Elementary homology theory
- 6 Integration on manifolds
- 7 Vector bundles
- 8 Geometric manifolds
- 9 The degree of a smooth map
- Appendix A Mathematical background
- Appendix B The spectral theorem
- Appendix C Orientations and top-dimensional forms
- Appendix D Riemann normal coordinates
- Appendix E Holonomy of an infinitesimal loop
- Appendix F Frobenius' theorem
- Appendix G The topology of electrical circuits
- Appendix H Intrinsic and extrinsic curvature
- References
- Index
Summary
The Editor is convinced that the notion of a connection in a vector bundle will soon find its way into a class on advanced calculus, as it is a fundamental notion and its applications are wide-spread. His chapter “Vector Bundles with a Connection” hopefully will show that it is basically an elementary concept.
S. S. ChernThe definitions
Let M be a differentiable manifold. Crudely put, a vector bundle is just a collection, or bundle, of vector spaces, one for each point p of M, that vary smoothly as p varies. For example, if M is a differentiable manifold and if TpM is the tangent space to M at a point p then the union TM of all the TpM as p varies over M is a vector bundle called the tangent bundle of M. Similarly, the cotangent bundle T*pM of M is just the union of all the cotangent spaces T*pM as p varies over M.
Essentially, a vector bundle over M is a space E that looks locally like the Cartesian product of M with a vector space. As we are primarily concerned with the local properties of vector bundles, we do not lose much by limiting ourselves to product bundles. But, for those who insist on knowing all the gory details, the official definition is provided here. The beginner should skim over the next two paragraphs and revisit them only as needed.
- Type
- Chapter
- Information
- Manifolds, Tensors, and FormsAn Introduction for Mathematicians and Physicists, pp. 176 - 192Publisher: Cambridge University PressPrint publication year: 2013