  We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci
(GCR) equations on Riemannian manifolds and the corresponding isometric
immersions of Riemannian manifolds into the Euclidean spaces. We develop a
unified intrinsic approach to establish the global weak rigidity of both the
GCR equations and isometric immersions of the Riemannian manifolds, independent
of the local coordinates, and provide further insights of the previous local
results and arguments. The critical case has also been analyzed. To achieve
this, we first reformulate the GCR equations with div-curl structure
intrinsically on Riemannian manifolds and develop a global, intrinsic version
of the div-curl lemma and other nonlinear techniques to tackle the global weak
rigidity on manifolds. In particular, a general functional-analytic compensated
compactness theorem on Banach spaces has been established, which includes the
intrinsic div-curl lemma on Riemannian manifolds as a special case. The
equivalence of global isometric immersions, the Cartan formalism, and the GCR
equations on the Riemannian manifolds with lower regularity is established. We
also prove a new weak rigidity result along the way, pertaining to the Cartan
formalism, for Riemannian manifolds with lower regularity, and extend the weak
rigidity results for Riemannian manifolds with unfixed metrics.
