Another Look at Connections

Abstract

In this note we make use of some properties of vector fields on a manifold to give an alternate proof to [F. Dumitrescu, Journal of Homotopy and Related Structures, vol 5, no 1 (2010)] for the equivalence between connections and parallel transport on vector bundles over manifolds. Out of the proof will emerge a new approach to connections on a bundle as a consistent way to lift the dynamics of the manifold to the bundle.