Lifting geometric objects to the dual of the first jet bundle of a bundle fibred over RR

We study natural lifting operations from a bundle τ:E→Rτ:E→R to the bundle π:J1τ∗→Eπ:J1τ∗→E which is the dual of the first-jet bundle J1τJ1τ. The main purpose is to define a complete lift of a type (1,1)(1,1) tensor field on EE and to understand all features of its construction. Various other lifting operations of tensorial objects on EE are needed for that purpose. We prove that the complete lift of a type (1,1)(1,1) tensor with vanishing Nijenhuis torsion gives rise to a Poisson–Nijenhuis structure on J1τ∗J1τ∗, and discuss in detail how the construction of associated Darboux–Nijenhuis coordinates can be carried out.