  We extend authors' prior results on optimal regularity and Uhlenbeck
compactness for affine connections to general connections on vector bundles.
This is accomplished by deriving a vector bundle version of the RT-equations,
and establishing a new existence theory for these equations. These new
RT-equations, non-invariant elliptic equations, provide the gauge
transformations which transform the fibre component of a non-optimal connection
to optimal regularity, i.e., the connection is one derivative more regular than
its curvature in $L^p$. The existence theory handles curvature regularity all
the way down to, but not including, $L^1$. Taken together with the affine case,
our results extend optimal regularity of Kazden-DeTurck and the compactness
theorem of Uhlenbeck, applicable to Riemannian geometry and compact gauge
groups, to general connections on vector bundles over non-Riemannian manifolds,
allowing for both compact and non-compact gauge groups. In particular, this
extends optimal regularity and Uhlenbeck compactness to Yang-Mills connections
on vector bundles over Lorentzian manifolds as base space, the setting of
General Relativity.
