Tulczyjew's Triplet with an Ehresmann connection I: Trivialization and Reduction

TL;DR

This paper explores the trivialization and reduction of Tulczyjew's triplet in the context of symmetries and Ehresmann connections, providing new insights into the structure of iterated tangent and cotangent bundles.

Contribution

It introduces methods for trivializing and reducing complex geometric structures associated with Tulczyjew's triplet using Ehresmann connections and symmetries.

Findings

Trivializations of $T^*TQ$, $TT^*Q$, and $T^*T^*Q$ are achieved.

Symplectomorphisms are properly trivialized and reduced.

Provides a framework for analyzing geometric structures with symmetries.

Abstract

We study the trivialization and the reduction of the Tulczyjew's triplet, in the presence of a symmetry and an Ehresmann connection associated to it. We thus obtain trivializations and reductions of iterated tangent and cotangent bundles $T^*TQ$, $TT^*Q$ and $T^*T^*Q$. Accordingly, the symplectomorphisms between these manifolds are properly trivialized and reduced.