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

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.