{
  "id": "2307.13077",
  "version": "v1",
  "published": "2023-07-24T18:59:34.000Z",
  "updated": "2023-07-24T18:59:34.000Z",
  "title": "Ruled surfaces in $3$-dimensional Riemannian manifolds",
  "authors": [
    "Marco Castrillón",
    "M. Eugenia Rosado María",
    "Alberto Soria"
  ],
  "comment": "22 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "In this work ruled surfaces in $3$-dimensional Riemannian manifolds are studied. We determine the expression for the extrinsic and sectional curvature of a parametrized ruled surface, where the former one is shown to be non-positive. We also quantify the set of ruling vector fields along a given base curve which allow to define a relevant reference frame along it and that we refer to as \\emph{Sannia}. The fundamental Theorem of existence and equivalence of Sannia ruled surfaces in terms of a system of invariants is given. The second part of the article tackles the concept of striction curve, which is proven to be the set of points where the so-called \\emph{Jacobi evolution function} vanishes on a ruled surface. This provides independent proofs for their existence and uniqueness in space forms, and to disprove its existence or uniqueness in some other cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-24T18:59:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53B25",
      "53B20",
      "53A55"
    ],
    "keywords": [
      "dimensional riemannian manifolds",
      "relevant reference frame",
      "ruling vector fields",
      "base curve",
      "sectional curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}