Differential Geometry and Lie Groups for Physicists

Differential geometry plays an increasingly important role in modern theoretical physics and applied mathematics. This 2006 textbook gives an introduction to geometrical topics useful in theoretical physics and applied mathematics, covering: manifolds, tensor fields, differential forms, connections, symplectic geometry, actions of Lie groups, bundles, spinors, and so on. Written in an informal style, the author places a strong emphasis on developing the understanding of the general theory through more than 1000 simple exercises, with complete solutions or detailed hints. The book will prepare readers for studying modern treatments of Lagrangian and Hamiltonian mechanics, electromagnetism, gauge fields, relativity and gravitation. Differential Geometry and Lie Groups for Physicists is well suited for courses in physics, mathematics and engineering for advanced undergraduate or graduate students, and can also be used for active self-study. The required mathematical background knowledge does not go beyond the level of standard introductory undergraduate mathematics courses.

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 Stoke's Theorem
9. Poincaré Lemma and cohomologies
10. Lie Groups - basic facts
11. Differential geometry of 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 Dirac operator
Appendices
Bibliography
Index