MA-352 "Differential Geometry and Mathematical Physics - II":
- 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
- Covariant differentiation: VB-valued differential forms; Linear
connection: derivation Ñ, Parallel transport and
Horizontal distribution, Equivalence of definitions; Torsion, Curvature; de
Rham cohomology, Yang-Mills equation; Riemannian structures: Levi-Civita
connection, geodesics, Riemannian curvature.
- 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.
- Contact Geometry: Definition via brackets and via forms; Examples
of Contact manifolds; Darboux theorem; Contactomorphisms, Contact vector
fields, Generating functions; Connection with Symplectic Geometry.