{
  "id": "1911.04275",
  "version": "v1",
  "published": "2019-11-11T13:54:37.000Z",
  "updated": "2019-11-11T13:54:37.000Z",
  "title": "On Totally umbilical surfaces in the warped product $\\mathbb{M}(κ)_f\\times\\mathbb{R}$",
  "authors": [
    "Ady Cambraia Jr",
    "Abigail Folha",
    "Carlos Peñafiel"
  ],
  "comment": "15 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "In this article we classify the totally umbilical surfaces which are immersed into a wide class of Riemannian manifolds having a structure of warped product, more precisely, we show that a totally umbilical surface immersed into the warped product $\\mathbb{M}(\\kappa)_f\\times I$ (here, $\\mathbb{M}(\\kappa)$ denotes the 2-dimensional space form, having constant curvature $\\kappa$, $I$ an interval and $f$ the warping function) is invariant by an one-parameter group of isometries of the ambient space. We also find the first integral of the ordinary differential equation that the profile curve satisfies (we mean, the curve which generates a invariant totally umbilical surface). Moreover, we construct explicit examples of totally umbilical surfaces, invariant by one-parameter group of isometries of the ambient space, by considering certain non-trivial warping function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-11T13:54:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C42",
      "53C30"
    ],
    "keywords": [
      "warped product",
      "ambient space",
      "one-parameter group",
      "warping function",
      "ordinary differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}