MA-352 "Differential Geometry and Mathematical Physics - II":

  1. Vector Bundles: Definitions, Examples; Module of Sections; Functorial constructions of new VB; VB with structure group G, Examples G=GLn+(R), SO, GLn(C); Jet-bundles Jk(M;N); Tangent, Cotangent bundles; Parallelizable Manifolds; Distributions, Curvature and Frobenius Theorem.
  2. Covariant differentiation: VB-valued differential forms; Linear connection: derivation , Parallel transport and Horizontal distribution, Equivalence of definitions; Torsion, Curvature; de Rham cohomology, Yang-Mills equation; Riemannian structures: Levi-Civita connection, geodesics, Riemannian curvature.
  3. Symplectic Geometry: Linear symplectic geometry, skew-orthogonality; group Sp(n), Lagrangian Grassmanian; Symplectic manifolds, Examples; Darboux theorem; Symplectomorphisms, Generating functions, Hamiltonian vector fields; Lagrange submanifolds.
  4. Contact Geometry: Definition via brackets and via forms; Examples of Contact manifolds; Darboux theorem; Contactomorphisms, Contact vector fields, Generating functions; Connection with Symplectic Geometry.