  We study the weak continuity of two interrelated non-linear partial
differential equations, the Yang-Mills equations and the Gau{\ss}-Codazzi-Ricci
equations, involving $L^p$-integrable connections. Our key finding is that
underlying cancellations in the curvature form, especially the div-curl
structure inherent in both equations, are sufficient to pass to the limit in
the non-linear terms. We first establish the weak continuity of Yang-Mills
equations and prove that any weakly converging sequence of weak Yang-Mills
connections in $L^p$ converges to a weak Yang-Mills connection. We then prove
that, for a sequence of isometric immersions with uniformly bounded second
fundamental forms in $L^p$, the curvatures are weakly continuous, which leads
to the weak continuity of the Gau{\ss}-Codazzi-Ricci equations with respect to
sequences of isometric immersions with uniformly bounded second fundamental
forms in $L^p$. Our methods are independent of dimensions and do not rely on
gauge changes.
