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.
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.
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