{
  "id": "2502.12647",
  "version": "v2",
  "published": "2025-02-18T08:53:08.000Z",
  "updated": "2025-05-08T09:15:37.000Z",
  "title": "Complex-valued extension of mean curvature for surfaces in Riemann-Cartan geometry",
  "authors": [
    "Dongha Lee"
  ],
  "comment": "26 pages, 1 figure. arXiv admin note: text overlap with arXiv:2310.04776",
  "categories": [
    "math.DG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-05-08T09:15:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "53C05"
    ],
    "keywords": [
      "riemann-cartan geometry",
      "mean curvature",
      "complex-valued extension",
      "generalizes classical minimal surface theory",
      "hodge dual plays"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}