{
  "id": "1203.3790",
  "version": "v3",
  "published": "2012-03-16T19:33:07.000Z",
  "updated": "2012-07-13T18:25:35.000Z",
  "title": "Submanifolds of products of space forms",
  "authors": [
    "Bruno Mendonça",
    "Ruy Tojeiro"
  ],
  "comment": "31 pages. Final version. Some typos corrected, a few arguments added. To appear in Indiana Univ. Math. J",
  "categories": [
    "math.DG"
  ],
  "abstract": "We give a complete classification of submanifolds with parallel second fundamental form of a product of two space forms. We also reduce the classification of umbilical submanifolds with dimension $m\\geq 3$ of a product $\\Q_{k_1}^{n_1}\\times \\Q_{k_2}^{n_2}$ of two space forms whose curvatures satisfy $k_1+k_2\\neq 0$ to the classification of $m$-dimensional umbilical submanifolds of codimension two of $\\Sf^n\\times \\R$ and $\\Hy^n\\times \\R$. The case of $\\Sf^n\\times \\R$ was carried out in \\cite{mt}. As a main tool we derive reduction of codimension theorems of independent interest for submanifolds of products of two space forms.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-07-13T18:25:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53B25"
    ],
    "keywords": [
      "space forms",
      "parallel second fundamental form",
      "independent interest",
      "complete classification",
      "curvatures satisfy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.3790M"
    }
  }
}