Basics of differential geometry
(a.k.a. Mathematical methods of science)
Lector Yu.Burman
Syllabus
Items marked with * will be studied if time permits.
- Smooth manifolds.
Smooth manifolds, smooth maps, induced topology.
- Vector bundles.
Vector bundles. Operations over vector bundles (direct sum, tensor
product). Tangent bundles. Derivative of a smooth map.
- Vector fields.
Group of diffeomorphisms. Lie algebra of vector fields.
Lie derivative.
- *Transversality.
Paracompactness. Partition of unity. Thom's transversality theorem.
Existence of Morse functions.
- Differential forms.
Superalgebra of differential forms. Exterior derivative, d2=0.
Cartan's formula.
- *Frobenius' theorem.
Frobenius' theorem. Almost complex structure. Contact structure.
- Integration of differential forms.
Integral of a differential form. Stokes' theorem.
Lectures
Problem sets
Problem sets are loosely connected with lectures. If you solve these problems
(all or just some of them), please show the written solutions to the
lecturer. The problems given at the final exam will be minor variations of
the problems from the sets.
- Topological spaces and smooth
manifolds.
- Tangent bundle.
- Vector bundles.
- Paracompactness and
partition of unity.
- Vector fields.
- Differential forms.
- Integration and Stokes'
theorem.
- Lie derivative.
Recommended reading
- Raoul Bott, Loring W. Tu, Differential Forms
in Algebraic Topology.
- Raghavan Narasimhan, Lectures on Topics
in Analysis.
- Michael Spivak, Calculus on
Manifolds.