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