# Noncommutative minimal embeddings and morphisms of pseudo-Riemannian calculi

## TL;DR

This paper develops a noncommutative analogue of classical submanifold theory, introducing morphisms and embeddings of real metric calculi, and applies these concepts to define and analyze noncommutative minimal surfaces.

## Contribution

It introduces noncommutative embeddings and morphisms of real metric calculi, extending classical geometric concepts to the noncommutative setting, including curvature and minimality.

## Key findings

- Noncommutative Gauss equations derived for curvature.
- Definition of noncommutative mean curvature and minimal embeddings.
- Example of noncommutative torus as a minimal surface in noncommutative 3-sphere.

## Abstract

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.