{
  "id": "1702.01328",
  "version": "v1",
  "published": "2017-02-04T19:25:19.000Z",
  "updated": "2017-02-04T19:25:19.000Z",
  "title": "Normal holonomy and rational properties of the shape operator",
  "authors": [
    "Carlos Olmos",
    "Richar Riaño-Riaño"
  ],
  "comment": "15 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $M$ be a most singular orbit of the isotropy representation of a simple symmetric space. Let $(\\nu _i, \\Phi _i)$ be an irreducible factor of the normal holonomy representation $(\\nu _pM, \\Phi (p))$. We prove that there exists a basis of a section $\\Sigma _i\\subset \\nu _i$ of $\\Phi _i$ such that the corresponding shape operators have rational eigenvalues (this is not in general true for other isotropy orbits). Conversely, this property, if referred to some non-transitive irreducible normal holonomy factor, characterizes the isotropy orbits. We also prove that the definition of a submanifold with constant principal curvatures can be given by using only the traceless shape operator, instead of the shape operator, restricted to a non-transitive (non necessarily irreducible) normal holonomy factor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-04T19:25:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C40",
      "53C42",
      "53C39"
    ],
    "keywords": [
      "shape operator",
      "rational properties",
      "isotropy orbits",
      "normal holonomy representation",
      "simple symmetric space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}