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We are concerned with the global weak continuity of the Cartan structuralsystem -- or equivalently, the Gauss--Codazzi--Ricci system -- onsemi-Riemannian manifolds with lower regularity. For this purpose, we firstformulate and prove a geometric compensated compactness theorem on vectorbundles over semi-Riemannian manifolds with lower regularity (Theorem 3.2),extending the classical quadratic theorem of compensated compactness. We thendeduce the $L^p$ weak continuity of the Cartan structural system for $p>2$: Fora family $\{\mathcal{W}_\varepsilon\}$ of connection $1$-forms on asemi-Riemannian manifold $(M,g)$, if $\{\mathcal{W}_\varepsilon\}$ is uniformlybounded 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 Cartanstructural system. Moreover, it is proved that isometric immersions ofsemi-Riemannian manifolds into semi-Euclidean spaces can be constructed fromthe weak solutions of the Cartan structural system or the Gauss--Codazzi--Riccisystem (Theorem 5.1), which leads to the $L^p$ weak continuity of theGauss--Codazzi--Ricci system on semi-Riemannian manifolds. As furtherapplications, the weak continuity of Einstein's constraint equations, generalimmersed hypersurfaces, and the quasilinear wave equations is also established.

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