Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 The concept of a manifold
- 2 Vector and tensor fields
- 3 Mappings of tensors induced by mappings of manifolds
- 4 Lie derivative
- 5 Exterior algebra
- 6 Differential calculus of forms
- 7 Integral calculus of forms
- 8 Particular cases and applications of Stokes' theorem
- 9 Poincaré lemma and cohomologies
- 10 Lie groups: basic facts
- 11 Differential geometry on Lie groups
- 12 Representations of Lie groups and Lie algebras
- 13 Actions of Lie groups and Lie algebras on manifolds
- 14 Hamiltonian mechanics and symplectic manifolds
- 15 Parallel transport and linear connection on M
- 16 Field theory and the language of forms
- 17 Differential geometry on TM and T *M
- 18 Hamiltonian and Lagrangian equations
- 19 Linear connection and the frame bundle
- 20 Connection on a principal G-bundle
- 21 Gauge theories and connections
- 22 Spinor fields and the Dirac operator
- Appendix A Some relevant algebraic structures
- Appendix B Starring
- Bibliography
- Index of (frequently used) symbols
- Index
19 - Linear connection and the frame bundle
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- Introduction
- 1 The concept of a manifold
- 2 Vector and tensor fields
- 3 Mappings of tensors induced by mappings of manifolds
- 4 Lie derivative
- 5 Exterior algebra
- 6 Differential calculus of forms
- 7 Integral calculus of forms
- 8 Particular cases and applications of Stokes' theorem
- 9 Poincaré lemma and cohomologies
- 10 Lie groups: basic facts
- 11 Differential geometry on Lie groups
- 12 Representations of Lie groups and Lie algebras
- 13 Actions of Lie groups and Lie algebras on manifolds
- 14 Hamiltonian mechanics and symplectic manifolds
- 15 Parallel transport and linear connection on M
- 16 Field theory and the language of forms
- 17 Differential geometry on TM and T *M
- 18 Hamiltonian and Lagrangian equations
- 19 Linear connection and the frame bundle
- 20 Connection on a principal G-bundle
- 21 Gauge theories and connections
- 22 Spinor fields and the Dirac operator
- Appendix A Some relevant algebraic structures
- Appendix B Starring
- Bibliography
- Index of (frequently used) symbols
- Index
Summary
• In this chapter we begin with a systematic study of connections in principal bundles, which have numerous important applications in modern theoretical physics (gauge field theory representing the most prominent application). This theory deals with concepts which are fairly simple, yet not sufficiently motivated for a beginner in the field. Therefore in our presentation a whole chapter is devoted first to one particular case – the linear connection. Although this topic is already well known from Chapter 15, here we adopt a brand new approach. It turns out that the novel point of view on the good old linear connection clearly indicates a direction towards a straightforward, albeit far-reaching generalization. This results in an elegant and powerful conceptual framework unifying such seemingly different structures as those represented by linear connection and gauge fields.
Frame bundle π : LM → M
• A novel point of view on the good old linear connection on (M, ∇) consists in expressing the fundamental concepts of the theory in terms of a larger manifold, which is denoted by LM and which may be canonically associated with any manifold M. This manifold is automatically endowed with some structure (due to the way it is constructed; recall a similar situation for TM and T*M). If there is a connection on M, however, the structure becomes even richer and it affords the opportunity of the complete reformulation of the concept of connection on M in terms of the new structure on LM. (This reformulation turns out to be particularly convenient from the perspective of a generalization.)
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- Chapter
- Information
- Differential Geometry and Lie Groups for Physicists , pp. 524 - 550Publisher: Cambridge University PressPrint publication year: 2006