Covariant derivatives for Ehresmann connections

TL;DR

This paper develops a method for constructing covariant derivatives for general Ehresmann connections on fibre bundles, enabling separate treatment of vertical and horizontal components with applications to various bundle types.

Contribution

It introduces a vertical endomorphism to construct covariant derivatives on both vertical and horizontal distributions, then combines them for the total space, broadening the applicability of Ehresmann connections.

Findings

Constructed covariant derivatives using vertical endomorphism

Applied method to tangent, frame, and Hopf bundles

Demonstrated separate and combined treatment of distributions

Abstract

We deal with the construction of covariant derivatives for some quite general Ehresmann connections on fibre bundles. We show how the introduction of a vertical endomorphism allows construction of covariant derivatives separately on both the vertical and horizontal distributions of the connection which can then be glued together on the total space. We give applications across an important class of tangent bundle cases, frame bundles and the Hopf bundle.