Bundles of acceleration on Banach manifolds

Abstract

We consider an infinite-dimensional manifold M modelled on a Banach space E and we construct smooth fiber bundle structures on the tangent bundle of order two T 2 M , which consists of all smooth curves of M that agree up to their acceleration, as well as on the corresponding second-order frame bundle L 2 M . These bundles prove to be associated with respect to the identity representation of the general linear group GL ( E ) that serves as the structure group of both of them. Moreover, a bijective correspondence between linear connections on T 2 M and connection forms of L 2 M is revealed.