dg.differential geometry - Gauge connections and Lie algebras?

First, I never liked working with principal bundles; vector bundles seem easier and more natural to me. Second, I never like thinking about abstract principal $G$-bundles. I prefer fixing a representation of $G$ and viewing the principal $G$ bundle as a reduced frame bundle associated with a vector bundle.

So let $E$ be a rank $k$ vector bundle and $F$ the bundle of arbitrary frames in $E$ (this is a principal $GL(k)$-bundle). Then $GL(k)$ acts on the right on $F$. Given a subgroup $G$ in $GL(k)$, let $F_G$ be a subbundle of $F$ such that if $f in F_G$, then so is $fcdot g$ for each $g in G$.

The primary example is $E = T_*M$ and $F_G$ is the bundle of orthonormal bases of the tangent space with respect to a Riemannian metric.

What is the critical property we want a $G$-connection to satisfy? Well, any connection allows you to parallel translate an arbitrary frame $f in F$ along a curve. We'd like the $G$-connection to be such that if $f in F_G$, then the parallel translation remains in $F_G$. This leads to the right definition of a $G$-connection.

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