Weak Continuity of the Cartan Structural System and Compensated Compactness on Semi-Riemannian Manifolds with Lower Regularity

We are concerned with the global weak continuity of Cartan structural system—or equivalently, Gauss–Codazzi–Ricci system—on semi-Riemannian manifolds lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over regularity (Theorem 3.2), extending classical quadratic compactness. then deduce $$L^p$$ system for $$p>2$$ : family $$\{{\mathcal {W}}_\varepsilon \}$$ connection 1-forms manifold (M, g), if is uniformly bounded in satisfies system, any limit also solution system. Moreover, it proved that isometric immersions into semi-Euclidean spaces can be constructed from solutions or 5.1), which leads to manifolds. As further applications, Einstein’s constraint equations, general immersed hypersurfaces, quasilinear wave equations established.