Noncommutative minimal embeddings and morphisms of pseudo-Riemannian calculi

In analogy with classical submanifold theory, we introduce morphisms of real metric calculi together noncommutative embeddings. We show that basic concepts, such as the second fundamental form and Weingarten map, translate into setting and, in particular, prove a analogue Gauss’ equations for curvature submanifold. Moreover, mean an embedding is readily introduced, giving natural definition minimal embedding, illustrate novel concepts by considering torus surface 3-sphere.