Local connection forms revisited

TL;DR

This paper revisits local connection forms, emphasizing their role in simplifying the handling of principal bundle connections and extending their application to sheaf-theoretic and inverse limit frameworks.

Contribution

It characterizes connections related by bundle morphisms and introduces a sheaf-theoretic approach to principal connections, broadening their theoretical and practical scope.

Findings

Characterization of connections related by bundle morphisms

Application to connections on Banach and inverse limit bundles

Sheaf-theoretic globalization of local connection forms

Abstract

Local connection forms provide a very useful tool for handling connections on principal bundles, because they ignore any complexities of the total space and, essentially, involve only two fundamental features of the structure group, namely the adjoint representation and the left (logarithmic) differential. The main results of this note characterize connections related together by bundle morphisms, while applications (taken from various sources) refer to connections on (Banach) associated bundles, in particular vector bundles, and connections on inverse limit bundles (in the Fr\'echet framework). The role of local connection forms is further illustrated by their sheaf-theoretic globalization, resulting in a sheaf operator-like approach to principal connections. The latter point of view is naturally leading to a theory of connections on abstract principal sheaves.