Complex-valued extension of mean curvature for surfaces in Riemann-Cartan geometry

TL;DR

This paper generalizes mean curvature to complex-valued quantities for surfaces in Riemann-Cartan manifolds with torsion, linking it to classical surface theory and geometric concepts.

Contribution

It introduces a complex-valued mean curvature extension in Riemann-Cartan geometry, connecting torsion with surface curvature concepts.

Findings

Defines a 2-form associated with ambient torsion

Introduces a complex-valued mean curvature as a Hodge dual

Links the new curvature to Hopf differential and Gauss map

Abstract

We extend the framework of submanifolds in Riemannian geometry to Riemann-Cartan geometry, which addresses connections with torsion. This procedure naturally introduces a 2-form on submanifolds associated with the nontrivial ambient torsion, whose Hodge dual plays the role of an imaginary counterpart to mean curvature for surfaces in a Riemann-Cartan 3-manifold. We observe that this complex-valued geometric quantity interacts with a number of other geometric concepts including the Hopf differential and the Gauss map, which generalizes classical minimal surface theory.