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 are also established.

我们关注具有较低规则性的半黎曼流形上的Cartan结构系统（或等效地，Gauss-Codazzi-Ricci系统）的整体弱连续性。为此，我们首先在具有较低规则性的半黎曼流形上的矢量束上制定并证明了几何补偿紧致性定理（定理3.2），扩展了补偿紧致性的经典二次定理。然后，我们推导出\（L ^ P \）为嘉当结构系统的弱连续性\（P> 2 \） ：对于一个家庭\（\ {{\ mathcal {白}} _ \ varepsilon \} \）连接的如果\（\ {{\ mathcal {W}} _ \ varepsilon \} \）均匀地定界在半黎曼流形（M，  g）上的1个形式\（L ^ p \）并且满足Cartan结构系统，那么\（\ {{\ mathcal {W}} _ \ varepsilon \} \）的任何弱\（L ^ p \）限制也是解决方案Cartan结构系统。此外，已经证明，可以通过Cartan结构系统或Gauss-Codazzi-Ricci系统（定理5.1）的弱解来构造半黎曼流形等距浸入半欧空间中，从而导致\（L ^ p \）在半黎曼流形上的高斯-科达兹-里奇系统的弱连续性。作为进一步的应用，还建立了爱因斯坦约束方程，一般沉浸超曲面和拟线性波动方程的弱连续性。