{
  "id": "2210.14175",
  "version": "v1",
  "published": "2022-10-25T17:09:42.000Z",
  "updated": "2022-10-25T17:09:42.000Z",
  "title": "Line Congruences on singular surfaces",
  "authors": [
    "Débora Lopes",
    "Tito Alexandro Medina Tejeda",
    "Maria Aparecida Soares Ruas",
    "Igor Chagas Santos"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "This paper is a first step in order to extend Kummer's theory for line congruences to the case $\\lbrace x, \\xi \\rbrace $, where $x: U \\rightarrow \\mathbb{R}^3$ is a smooth map and $\\xi: U \\rightarrow \\mathbb{R}^3$ is a proper frontal. We show that if $\\lbrace x, \\xi \\rbrace$ is a normal congruence, the equation of the principal surfaces is a multiple of the equation of the developable surfaces, furthermore, the multiplicative factor is associated to the singular set of $\\xi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-25T17:09:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "line congruences",
      "singular surfaces",
      "extend kummers theory",
      "smooth map",
      "proper frontal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}