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Starting from a bundle τ : E → R , the bundle π : J 1 τ ∗ → E , which is the dual of the first jet bundle J 1 τ 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 J 1 τ ∗ . 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.

We study natural lifting operations from a bundle τ : E → R to the bundle π : J 1 τ ∗ → E which is the dual of the first-jet bundle J 1 τ . The main purpose is to define a complete lift of a type ( 1 , 1 ) tensor field on E and to understand all features of its construction. Various other lifting operations of tensorial objects on E are needed for that purpose. We prove that the complete lift of a type ( 1 , 1 ) tensor with vanishing Nijenhuis torsion gives rise to a Poisson–Nijenhuis structure on J 1 τ ∗ , and discuss in detail how the construction of associated Darboux–Nijenhuis coordinates can be carried out.

The modular vector field of a Poisson-Nijenhuis Lie algebroid $A$ is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian $A$-vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson-Nijenhuis structure., Comment: To appear in Letters in Mathematical Physics