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arXiv · 2502.12647

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

Dongha Lee
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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.

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Dongha Lee. 2025-02-18. Complex-valued extension of mean curvature for surfaces in Riemann-Cartan geometry. https://doi.org/10.1016/j.geomphys.2025.105748

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