MATH-F-420: Differential geometry and applications

Differential geometry is a part of mathematics
dealing with smooth manifolds using ideas from 
tensor calculus and Lie algebra theory. The main tools 
of differential geometry are tensors, vector 
bundles and connections.

I will give a gentle introduction to some of these topics.

Approximate syllabus.

0. Inverse function theorem. Examples of 
submanifolds. Sard lemma and its applications.

1. Smooth manifolds, partition of unity,
Whitney embedding theorems.

2. De Rham algebra, Cartan formula, 
Lie derivative, Stokes' theorem, de Rham cohomology,
Poincare lemma, applications to topology.

3. Commutator of vector fields. 
Frobenius theorem. 

It would be helpful if the students know
point-set topology and basics of topology 
on manifolds.

Useful books: 
Milnor, J.: Topology from the Differentiable Viewpoint.
Laurent Schwartz: Cours d' Analyse.