Programa:
Part 0 - Preliminaries on Euclidean space
1. Review of Topology and calculus on R^n
2. Differential forms on R^n: wedge product and exterior derivative
3. Pull-back of differential forms, change of variables
4. Topological manifolds
Part 1 - Manifolds and smooth maps
5. Charts and Atlases; atlas topology
6. Smooth functions and maps, categorical properties
7. Examples, diffeomorphisms. Existence of smooth functions
8. Immersions, submersions and level sets
9. Embeddings and Submanifolds
Part 2 - Differential forms and integration
10. Introduction to Multilinear and Exterior algebra
11. Differential forms on manifolds; closed and exact forms
12. Orientation on manifolds, integration; Stokes theorem
Part 3 - Lie groups and smooth actions
13. Lie groups and homomorphisms; examples
14. Subgroups, kernels and images; the closed subgroup theorem
15. The Lie algebra of a Lie group; exponential map
16. Actions of a Lie group on a manifold; homogeneous spaces
Part 4 - Vector fields, flows and Lie derivatives
17. Tangent vectors as derivations; velocity of smooth curves
18. Vector bundles and smooth sections; the tangent bundle
19. The Lie algebra of vector fields
20. One-parameter groups of diffeomorphisms; flows of vector fields
21. Lie derivative and Cartan's formula
Part 5 - Topological invariants and de Rham cohomology
22. Cohomology groups of compact manifolds
23. Mayer-Vietoris long exact sequence
24. Degree theory; Euler characteristic
Recommended Bibliography:
- 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
- S. Morita, Geometry of Differential Forms, AMS, Translations of Mathematical Monographs 201 (2001).
- John M. Lee, Introduction to Smooth Manifolds, Springer GTM 218, 2nd Edition, 2013.