Contact projective structures and contact path geometries

A contact path geometry is a family of paths in a contact manifold each of which is everywhere tangent to the contact hyperplane and such that given a point and a one-dimensional subspace of the tangent space at that point there is a unique path of the family passing through the given point and tangent to the given subspace. A contact projective structure is a contact path geometry the paths of which are among the geodesics of some affine connection. In the manner of T. Y. Thomas there is associated to each contact projective structure an ambient affine connection on a symplectic manifold with one-dimensional fibers over the contact manifold and using this the local equivalence problem for contact projective geometries is solved by the construction of a canonical regular Cartan connection. The curvature function of this Cartan connection is co-closed if and only if an invariant contact torsion vanishes. The vanishing of this contact torsion implies the existence of canonical paths transverse to the contact structure. An analogue of the classical Beltrami theorem is proved for pseudo-hermitian manifolds with transverse symmetry. Prolongation methods of N. Tanaka, T. Morimoto, and A. Cap - H. Schichl are applied to solve the local equivalence problem for k-path geometries and for a subclass of contact path geometries for which the contact paths are generated by a vector field having a Hamiltonian character.