A new look at connections

TL;DR

This paper offers an alternative proof of the equivalence between connections and parallel transport on vector bundles, introducing a new perspective on lifting manifold dynamics to bundles.

Contribution

It presents a novel proof leveraging vector field properties and introduces a new approach to understanding connections as dynamic lifts.

Findings

Alternative proof of connection-parallel transport equivalence

New approach to connections via dynamic lifting

Insights into vector field properties on manifolds

Abstract

In this note we make use of some properties of vector fields on a manifold to give an alternate proof to [3] 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.