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.