Syllabus:
Part 1 - Manifolds and smooth maps
1. Topological manifolds
2. Charts and Atlases; atlas topology
3. Smooth functions and maps, categorical properties
4. Examples, diffeomorphisms. Existence of smooth functions.
Part 2 - Tangent space, Derivative; Submanifolds
5. Tangent space as derivations
6. Basis via charts and velocity of smooth curves
7. Immersions, submersions and level sets
8. Embeddings and Submanifolds
Part 3 - Differential forms and vector bundles
9. Differential forms on R^n: wedge product and exterior derivative
10. Introduction to Multilinear and Exterior algebra
11. Pull-back of differential forms; differential forms on manifolds
12. Vector bundles and smooth sections
13. Tangent and cotangent bundles; bundles of differential forms
Part 4 - Vector fields, flows and Lie derivatives
14. The Lie algebra of vector fields
15. One-parameter groups of diffeomorphisms
16. Lie derivative and Cartan's formula
Part 5 - Lie groups and smooth actions
17. Lie groups and homomorphisms; examples
18. Subgroups, kernels and images; the closed subgroup theorem
19. The Lie algebra of a Lie group; exponential map
20. Actions of a Lie group on a manifold; homogeneous spaces
Part 6 - Integration and de Rham cohomology
21. Closed and exact forms
22. Orientation on manifolds
23. Cohomology groups of compact manifolds
24. Mayer-Vietoris long exact sequence
25. Degree theory; Euler characteristic
Recommended Bibliography:
- John M. Lee, Introduction to Smooth Manifolds, Springer GTM 218, 2nd Edition, 2013.
- Loring Tu, Introduction to Manifolds, Springer, Universitext, 2nd Edition, 2011.
- Nigel Hitchin, Differentiable Manifolds, Course C3.1b 2012, in https://people.maths.ox.ac.uk/hitchin/hitchinnotes/manifolds2012.pdf