{
  "id": "0906.2111",
  "version": "v1",
  "published": "2009-06-11T14:05:05.000Z",
  "updated": "2009-06-11T14:05:05.000Z",
  "title": "On the scalar curvature of hypersurfaces in spaces with a Killing field",
  "authors": [
    "Alma L. Albujer",
    "Juan A. Aledo",
    "Luis J. Alias"
  ],
  "comment": "First version (April 2008). Final version (July 2008). To appear in Advances in Geometry",
  "categories": [
    "math.DG"
  ],
  "abstract": "We consider compact hypersurfaces in an $(n+1)$-dimensional either Riemannian or Lorentzian space $N^{n+1}$ endowed with a conformal Killing vector field. For such hypersurfaces, we establish an integral formula which, especially in the simpler case when $N=M^n\\times R$ is a product space, allows us to derive some interesting consequences in terms of the scalar curvature of the hypersurface. For instance, when $n=2$ and $M^2$ is either the sphere $\\mathbb{S}^2$ or the real projective plane $\\mathbb{RP}^2$, we characterize the slices of the trivial totally geodesic foliation $M^2\\times\\{t\\}$ as the only compact two-sided surfaces with constant Gaussian curvature in the Riemannian product $M^2\\times\\mathbb{R}$ such that its angle function does not change sign. When $n\\geq 3$ and $M^n$ is a compact Einstein Riemannian manifold with positive scalar curvature, we also characterize the slices as the only compact two-sided hypersurfaces with constant scalar curvature in the Riemannian product $M^n\\times\\mathbb{R}$ whose angle function does not change sign. Similar results are also established for spacelike hypersurfaces in a Lorentzian product $\\mathbb{M}\\times\\mathbb{R}_1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-06-11T14:05:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "53C42"
    ],
    "keywords": [
      "hypersurface",
      "killing field",
      "compact einstein riemannian manifold",
      "change sign",
      "angle function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.2111A"
    }
  }
}