{
  "id": "2411.04231",
  "version": "v1",
  "published": "2024-11-06T19:56:12.000Z",
  "updated": "2024-11-06T19:56:12.000Z",
  "title": "On the Work of Cartan and Münzner on Isoparametric Hypersurfaces",
  "authors": [
    "Thomas E. Cecil"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "A hypersurface $M^n$ in a real space form ${\\bf R}^{n+1}$, $S^{n+1}$, or $H^{n+1}$ is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan and M\\\"{u}nzner on the theory of isoparametric hypersurfaces in real space forms, in particular, spheres. This work is contained in four papers of Cartan published during the period 1938--1940, and two papers of M\\\"{u}nzner that were published in preprint form in the early 1970's, and as journal articles in 1980--1981. These papers of Cartan and M\\\"{u}nzner have been the foundation of the extensive field of isoparametric hypersurfaces, and they have all been recently translated into English by the author. The paper concludes with a brief survey of the recently completed classification of isoparametric hypersurfaces in spheres.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-06T19:56:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C40",
      "53C42",
      "53B25"
    ],
    "keywords": [
      "isoparametric hypersurfaces",
      "real space form",
      "constant principal curvatures",
      "brief survey",
      "paper concludes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}