{
  "id": "1204.2224",
  "version": "v5",
  "published": "2012-04-10T17:27:43.000Z",
  "updated": "2015-02-12T12:19:53.000Z",
  "title": "Dirac operators on foliations: the Lichnerowicz inequality",
  "authors": [
    "Weiping Zhang"
  ],
  "comment": "53 pages. Title, abstract and the main results changed. The vanishing consequence is not as strong as originally claimed. The originally claimed vanishing results will be dealt with in a separate paper",
  "categories": [
    "math.DG"
  ],
  "abstract": "We construct Dirac operators on foliations by applying the Bismut-Lebeau analytic localization technique to the Connes fibration over a foliation. The Laplacian of the resulting Dirac operators has better lower bound than that obtained by using the usual adiabatic limit arguments on the original foliation. As a consequence, we prove an extension of the Lichnerowicz-Hitchin vanishing theorem to the case of foliations.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-01-12T14:10:45.000Z",
      "title": "A Lichnerowicz-Hitchin vanishing theorem for foliations",
      "abstract": "We establish a generalization of the Lichnerowicz-Hitchin vanishing theorem to the case of foliations. As a consequence, we show that there is no foliation of positive leafwise scalar curvature on any torus. Our proof, which is inspired by the analytic localization techniques developed by Bismut and Lebeau, applies to give a purely geometric proof of the Connes vanishing theorem which also extends the Lichnerowicz vanishing theorem to the case of foliations.",
      "comment": "52 pages. The proof is refined by slightly changing the isometric embedding in Section 2.4",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2015-02-12T12:19:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lichnerowicz-hitchin vanishing theorem",
      "lichnerowicz vanishing theorem",
      "purely geometric proof",
      "positive leafwise scalar curvature",
      "analytic localization techniques"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.2224Z"
    }
  }
}