{
  "id": "1103.3877",
  "version": "v1",
  "published": "2011-03-20T19:03:20.000Z",
  "updated": "2011-03-20T19:03:20.000Z",
  "title": "Some Remarks on Nijenhuis Bracket, Formality, and Kähler Manifolds",
  "authors": [
    "Paolo de Bartolomeis",
    "Vladimir S. Matveev"
  ],
  "comment": "9 pages; no figures",
  "categories": [
    "math.DG",
    "math.CV",
    "math.SG"
  ],
  "abstract": "One (actually, almost the only effective) way to prove formality of a differentiable manifold is to be able to produce a suitable derivation $\\delta$ such that $d\\delta$-lemma holds. We first show that such derivation $\\delta$ generates a (1,1)-tensor field (we denote it by $R$). Then, we show that the supercommutation of $d$ and $\\delta$ (which is a natural, essentially necessary condition to get a $d\\delta$-lemma) is equivalent to vanishing of the Nijenhujis torsion of $R$. Then, we are looking for sufficient conditions that ensure the $d\\delta$-lemma holds: we consider the cases when $R$ is self adjoint with respect to a Riemannian metric or compatible with an almost symplectic structure. Finally, we show that if $R$ is scew-symmetric with respect to a Riemannian metric, has constant determinant, and if its Nijenhujis torsion vanishes, then the orthogonal component of $R$ in its polar decomposition is a complex structure compatible with the metric, which gives us a new characterization of K\\\"ahler structures",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-03-20T19:03:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "kähler manifolds",
      "nijenhuis bracket",
      "lemma holds",
      "riemannian metric",
      "nijenhujis torsion vanishes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.3877D"
    }
  }
}