{
  "id": "1801.00628",
  "version": "v1",
  "published": "2018-01-02T12:42:06.000Z",
  "updated": "2018-01-02T12:42:06.000Z",
  "title": "Deformation of the $σ_2$-curvature",
  "authors": [
    "Almir Silva Santos",
    "Maria Andrade"
  ],
  "comment": "19 pages. to appear in Annals of Global Analysis and Geometry",
  "categories": [
    "math.DG"
  ],
  "abstract": "Our main goal in this work is to deal with results concern to the $\\sigma_2$-curvature. First we find a symmetric 2-tensor canonically associated to the $\\sigma_2$-curvature and we present an Almost Schur Type Lemma. Using this tensor we introduce the notion of $\\sigma_2$-singular space and under a certain hypothesis we prove a rigidity result. Also we deal with the relations between flat metrics and $\\sigma_2$-curvature. With a suitable condition on the $\\sigma_2$-curvature we show that a metric has to be flat if it is close to a flat metric. We conclude this paper by proving that the 3-dimensional torus does not admit a metric with constant scalar curvature and non-negative $\\sigma_2$-curvature unless it is flat.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-02T12:42:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20",
      "53C21",
      "53C24"
    ],
    "keywords": [
      "deformation",
      "flat metric",
      "constant scalar curvature",
      "schur type lemma",
      "main goal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}