**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=GL_{n}^{+}(**R**), SO, GL_{n}(**C**); Jet-bundles J^{k}(M;N); Tangent, Cotangent bundles; Parallelizable Manifolds; Distributions, Curvature and Frobenius Theorem.__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.