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

TL;DR

This paper develops a comprehensive method for lifting tensor fields from a bundle over R to the dual of its first-jet bundle, establishing conditions under which these lifts form Poisson-Nijenhuis structures and exploring coordinate constructions.

Contribution

It introduces a complete lift of (1,1) tensor fields on bundles over R and characterizes when these lifts produce Poisson-Nijenhuis structures, including coordinate methods.

Findings

Complete lift of (1,1) tensors with vanishing Nijenhuis torsion yields Poisson-Nijenhuis structures.

Construction of Darboux-Nijenhuis coordinates is detailed.

Various tensor lifting operations are systematically analyzed.

Abstract

We study natural lifting operations from a bundle E over R to the dual bundle of its first-jet bundle. 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 the dual of the first-jet bundle, and discuss in detail how the construction of associated Darboux-Nijenhuis coordinates can be carried out.