Compensated compactness in Banach spaces and weak rigidity of isometric immersions of manifolds

Abstract

We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div–curl lemma for tensor fields on Riemannian manifolds is obtained. Then we show how this intrinsic div–curl lemma can be employed to establish the global weak rigidity of the Gauss–Codazzi–Ricci equations, the Cartan formalism, and the corresponding isometric immersions of Riemannian submanifolds.