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.
The students might be interested in the notes I gave to
a course of differential geometry in Moscow
http://bogomolov-lab.ru/KURSY/GEOM-2013/
(English) but I won't follow these notes
to the point.
Approximate syllabus.
1. Smooth manifolds, partition of unit,
Whitney embedding theorems.
2. Sheaves and germs of functions.
Smooth manifolds as ringed spaces.
3. Derivations on the ring of smooth functions;
vector fields as derivations. Vector bundles;
equivalence of different definitions of vector
bundles. Serre-Swan theorem.
4. De Rham algebra, Cartan formula,
Lie derivative, Stokes' theorem, de Rham cohomology,
Poincare lemma, applications to topology.
5. Commutator of vector fields.
Frobenius theorem.
It would be helpful if the students know
point-set topology and basics of topology
on manifolds, but if there is a need, I can
give remedial lectures and problem sets for
these subjects.