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

Abstract

Abstract. 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.

Lifted tensors and Hamilton–Jacobi separability

“…One is the symmetry property referred to above (which equally holds for the projected Poisson structure on J 1 τ * ). A second one is the vanishing of the so-called Magri-Morosi concommitant; this was proved to be the case in detail on J 1 τ * in [25] and the proof for T * E can be found in [12]. The last condition is the vanishing of the Nijenhuis torsion, so that the statement then immediately follows from the preceding theorem.…”

“…the induced transformation of a time-dependent point transformation on E, which as such will not destroy the coordinate form of the Poisson tensor. We have proved in detail in [25] that this can be done for the Poisson-Nijenhuis structure (P, R) on J 1 τ * , and it is an easy matter to verify that the same point transformation on E will also induce a canonical transformation on T * E that does the job. Note further that in the course of proving the existence of an appropriate transformation (t, q) → (t, Q(t, q)) on E, we found that each eigenvalue λ i will in the new coordinates at most depend on the corresponding coordinate Q i .…”

“…A direct construction of a complete lift of R to J 1 τ * is not that straightforward. In [25], we developed a way to do it by an action on appropriately lifted vector fields. We will frequently refer to that paper for further properties and other lifting operations.…”

“…The property is well known for the lift to a cotangent bundle (see [26] or [11]). A proof for the lift to J 1 τ * was given in [25].…”

Dynamics of histories

“…1 Different authors use various appellations for Y, or for its associated bundle manifold: the covariant phase space bundle, the doubly extended phase space, the extended dual bundle [9], orthe extended multimomentum bundle [7] ...…”