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

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.