"In Riemannian geometry, the Gauss\u2013Codazzi\u2013Mainardi equations are fundamental equations in the theory of embedded hypersurfaces in a Euclidean space, and more generally submanifolds of Riemannian manifolds. They also have applications for embedded hypersurfaces of pseudo-Riemannian manifolds.In the classical differential geometry of surfaces, the Gauss\u2013Codazzi\u2013Mainardi equations consist of a pair of related equations."@en .