Basics of differential geometry

(a.k.a. Mathematical methods of science)

Lector Yu.Burman

Syllabus

Items marked with * will be studied if time permits.
  1. Smooth manifolds.
    Smooth manifolds, smooth maps, induced topology.
  2. Vector bundles.
    Vector bundles. Operations over vector bundles (direct sum, tensor product). Tangent bundles. Derivative of a smooth map.
  3. Vector fields.
    Group of diffeomorphisms. Lie algebra of vector fields. Lie derivative.
  4. *Transversality.
    Paracompactness. Partition of unity. Thom's transversality theorem. Existence of Morse functions.
  5. Differential forms.
    Superalgebra of differential forms. Exterior derivative, d2=0. Cartan's formula.
  6. *Frobenius' theorem.
    Frobenius' theorem. Almost complex structure. Contact structure.
  7. 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.
  1. Topological spaces and smooth manifolds.
  2. Tangent bundle.
  3. Vector bundles.
  4. Paracompactness and partition of unity.
  5. Vector fields.
  6. Differential forms.
  7. Integration and Stokes' theorem.
  8. Lie derivative.

Recommended reading

  1. Raoul Bott, Loring W. Tu, Differential Forms in Algebraic Topology.
  2. Raghavan Narasimhan, Lectures on Topics in Analysis.
  3. Michael Spivak, Calculus on Manifolds.

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