Weak Continuity of the Gauss-codazzi-ricci System for Isometric Embedding

We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform L-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 and has reasonable bounds on the errors made in the approximation (the errors are confined in a compact subset of H 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.