Connections on distributions and geodesic sprays

Almost contact metric structures defined by an $N$-prolonged connection

“…connection in the vector bundle (X, D)) splits into the direct sum of the vertical and horizontal distributions. In [2,3] it is shown that on the manifold D can be defined in a natural way an almost contact metric structure allowing e.g. to give an invariant character to the analytical description of the mechanics with constraints.…”

“…to give an invariant character to the analytical description of the mechanics with constraints. In [3], on the manifold D the geodesic pulverization of the connection over the distribution is defined; this is an analogue of the geodesic pulverization, defined on the space of the tangent bundle T X and having the following physical interpretation: the projections of the integral curves of the geodesic pulverization of the connection over the distribution coincide with the admissible geodesics (the trajectories of the mechanical system with constrains).…”