Complex-valued extension of mean curvature for surfaces in Riemann-Cartan geometry
Dongha Lee
TL;DR
This work develops a framework for submanifolds in Riemann-Cartan spaces with torsion by introducing a complex mean curvature $\boldsymbol{H}= H + \boldsymbol{i}\,{\star}\tau$, where $\tau$ is a torsion 2-form. The real and imaginary parts of $\boldsymbol{H}$ correspond to the divergence and curl of the Gauss map, and the authors extend Hopf-differential and third-fundamental-form machinery to RC and Weitzenböck geometries, establishing holomorphicity criteria and gauge-invariant properties of $|\boldsymbol{H}|$. They show minimal surfaces ($\boldsymbol{H}=0$) and conformality of Gauss maps extend naturally to torsionful ambient spaces, with analogues of classical results like the Hopf characterization and Gauss-Bonnet holding in this broader setting. The framework connects RC geometry to Weitzenböck geometry, providing a richer perspective on torsion-influenced submanifold theory and potential links to renormalized invariants in hyperbolic 3-manifolds.
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.