Mathematics

The goal of the talk is to present a construction of the canonical connection and the curvature (generalizing Levi Civita connection and Riemannian sectional curvature) for a wide class of Hamiltonian systems and other vector fields on cotangent and tangent bundles. The construction is purely "dynamical": it involves only the flow generated by the vector field and the fiber bundle structure. The correspondent curvature enjoys properties similar to the Riemannian case, i.e. the case of the geodesic flow on the tangent bundle. In particular, it allows to state very general "Comparizon theorems". No preliminary knowledge in Riemannian geometry is required.