{
  "id": "2206.12566",
  "version": "v1",
  "published": "2022-06-25T05:55:57.000Z",
  "updated": "2022-06-25T05:55:57.000Z",
  "title": "Isoparametric submanifolds in Hilbert spaces and holonomy maps",
  "authors": [
    "Naoyuki Koike"
  ],
  "comment": "17pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $\\pi:P\\to B$ be a smooth $G$-bundle over a compact Riemannian manifold $B$ and $c$ a smooth loop in $B$ of constant seed, where $G$ is compact semi-simple Lie group. In this paper we prove that the holonomy map ${\\rm hol}_c:\\mathcal A_P^{H^s}\\to G$ is a homothetic submersion with minimal regularizable fibres, where $\\mathcal A_P^{H^s}$ is the Hilbert space of all $H^s$-connections of the bundle $P$ and $s$ is a positive number with $\\displaystyle{s>\\frac{1}{2}\\,{\\rm dim}\\,B\\,-\\,1}$. From this fact, we can derive that each component of the inverse image of any equifocal submanifold in $G$ by the holonomy map ${\\rm hol}_c:\\mathcal A_P^{H^s}\\to G$ is an isoparametric submanifold in the Hilbert space $\\mathcal A_P^{H^s}$. As the result, we obtain a new systematic construction of isoparametric submanifolds in a Hilbert space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-25T05:55:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hilbert space",
      "isoparametric submanifold",
      "holonomy map",
      "compact semi-simple lie group",
      "compact riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}