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