Geometry of manifolds and bundles

Course synopsis

  1. Riemannian metrics. Curvature. Gauss--Bonnet formula.
  2. Covariant differentiation. Connections on a linear bundle --- various definitions and equivalence.
  3. Frobenius theorem.
  4. Euler class of a linear bundle.
  5. Levi--Civita connection.
  6. Symplectic structures and Poisson brackets. Symplectic leaves. Symplecticity of the geodesic flow.