{
  "id": "0707.1860",
  "version": "v1",
  "published": "2007-07-12T18:52:59.000Z",
  "updated": "2007-07-12T18:52:59.000Z",
  "title": "The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature",
  "authors": [
    "Eric L. Grinberg",
    "Li Haizhong"
  ],
  "comment": "10 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "In 1963, K.P.Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let M be an oriented closed surface in the Euclidean space R^3 with Euler characteristic \\chi(M), Gauss curvature G and unit normal vector field n. Grotemeyer's identity replaces the Gauss-Bonnet integrand G by the normal moment <a,n>^2G, where $a$ is a fixed unit vector. Grotemeyer showed that the total integral of this integrand is (2/3)pi times chi(M). We generalize Grotemeyer's result to oriented closed even-dimesional hypersurfaces of dimension n in an (n+1) ndimensional space form N^{n+1}(k).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-07-12T18:52:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C42",
      "53A10"
    ],
    "keywords": [
      "constant curvature",
      "gauss-bonnet-grotemeyer theorem",
      "unit normal vector field",
      "grotemeyers identity replaces",
      "pi times chi"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0707.1860G"
    }
  }
}