{
  "id": "2306.04013",
  "version": "v1",
  "published": "2023-06-06T20:59:04.000Z",
  "updated": "2023-06-06T20:59:04.000Z",
  "title": "Catenaries in Riemannian Surfaces",
  "authors": [
    "Luiz C. B. da Silva",
    "Rafael López"
  ],
  "comment": "14 pages, 3 figures. Keywords: Catenary, Surface of revolution, Clairaut relation, Grusin plane",
  "categories": [
    "math.DG"
  ],
  "abstract": "The concept of catenary has been recently extended to the sphere and the hyperbolic plane by the second author [L\\'opez, arXiv:2208.13694]. In this work, we define catenaries on any Riemannian surface. A catenary on a surface is a critical point of the potential functional, where we calculate the potential with the intrinsic distance to a fixed reference geodesic. Adopting semi-geodesic coordinates around the reference geodesic, we characterize catenaries using their curvature. Finally, after revisiting the space-form catenaries, we consider surfaces of revolution (where a Clairaut relation is established), ruled surfaces, and the Gru\\v{s}in plane.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-06T20:59:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A04",
      "53B20",
      "49J05"
    ],
    "keywords": [
      "riemannian surface",
      "define catenaries",
      "clairaut relation",
      "hyperbolic plane",
      "intrinsic distance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}