Lifted tensors and Hamilton-Jacobi separability

Starting from a bundle τ:E→R, the bundle π:J1τ∗→E, which is the dual of the first jet bundle J1τ and a sub-bundle of T∗E, is the appropriate manifold for the geometric description of time-dependent Hamiltonian systems. Based on previous work, we recall properties of the complete lifts of a type (1,1) tensor R on E to both T∗E and J1τ∗. We discuss how an interplay between both lifted tensors leads to the identification of related distributions on both manifolds. The integrability of these distributions, a coordinate free condition, is shown to produce exactly Forbat's conditions for separability of the time-dependent Hamilton-Jacobi equation in appropriate coordinates.