  We establish the weak continuity of the Gauss-Coddazi-Ricci system for
isometric embedding with respect to the uniform $L^p$-bounded solution sequence
for $p>2$, which implies that the weak limit of the isometric embeddings of the
manifold is still an isometric embedding. More generally, we establish a
compensated compactness framework for the Gauss-Codazzi-Ricci system in
differential geometry. That is, given any sequence of approximate solutions to
this system which is uniformly bounded in $L^2$ and has reasonable bounds on
the errors made in the approximation (the errors are confined in a compact
subset of $H^{-1}_{\text{loc}}$), then the approximating sequence has a weakly
convergent subsequence whose limit is a solution of the Gauss-Codazzi-Ricci
system. Furthermore, a minimizing problem is proposed as a selection criterion.
For these, no restriction on the Riemann curvature tensor is made.
