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In this paper we achieve the final step in our program to establish optimalregularity and Uhlenbeck compactness for connections on vector bundles overarbitrary base manifolds, including both Riemannian and Lorentzian manifolds,and allowing for both compact and non-compact Lie groups. Our proof is based onthe discovery of a non-linear system of elliptic equations, (the RT-equations),whose solutions provide the coordinate and gauge transformations that lift theregularity of a connection locally to one derivative above its $L^p$ curvature.The extra derivative gives a natural (local) extension of Uhlenbeck compactnessto non-Riemannian geometries. The step accomplished here is to extend authors'prior results for affine connections to general connections on vector bundlesby deriving a vector bundle version of the RT-equations, which augments theaffine RT-equations from our prior work, and proving existence of solutions.Taken together with the affine case, our results extend the optimal regularityresult of Kazdan-DeTurck and Uhlenbeck's compactness theorem from Riemanniangeometry, to connections on vector bundles over non-Riemannian manifolds,including Lorentzian manifolds of General Relativity and Yang-Millsconnections, allowing for both compact and non-compact gauge groups.

On the Optimal Regularity Implied by the Assumptions of Geometry II: Connections on Vector Bundles