{
  "id": "0907.1222",
  "version": "v1",
  "published": "2009-07-07T14:09:53.000Z",
  "updated": "2009-07-07T14:09:53.000Z",
  "title": "Half-flat Structures and Special Holonomy",
  "authors": [
    "Vicente Cortés",
    "Thomas Leistner",
    "Lars Schäfer",
    "Fabian Schulte-Hengesbach"
  ],
  "comment": "40 pages",
  "journal": "Proc. Lond. Math. Soc. 2010",
  "doi": "10.1112/plms/pdq012",
  "categories": [
    "math.DG"
  ],
  "abstract": "It was proven by Hitchin that any solution of his evolution equations for a half-flat SU(3)-structure on a compact six-manifold M defines an extension of M to a seven-manifold with holonomy in G_2. We give a new proof, which does not require the compactness of M. More generally, we prove that the evolution of any half-flat G-structure on a six-manifold M defines an extension of M to a Ricci-flat seven-manifold N, for any real form G of SL(3,C). If G is noncompact, then the holonomy group of N is a subgroup of the noncompact form G_2^* of G_2^C. Similar results are obtained for the extension of nearly half-flat structures by nearly parallel G_2- or G_2^*-structures, as well as for the extension of cocalibrated G_2- and G_2^*-structures by parallel Spin(7)- and Spin(3,4)-structures, respectively. As an application, we obtain that any six-dimensional homogeneous manifold with an invariant half-flat structure admits a canonical extension to a seven-manifold with a parallel G_2- or G_2^*-structure. For the group H_3 \\times H_3, where H_3 is the three-dimensional Heisenberg group, we describe all left-invariant half-flat structures and develop a method to explicitly determine the resulting parallel G_2- or G_2^*-structure without integrating. In particular, we construct three eight-parameter families of metrics with holonomy equal to G_2 and G_2^*. Moreover, we obtain a strong rigidity result for the metrics induced by a half-flat structure (\\omega,\\rho) on H_3 \\times H_3 satisfying \\omega(Z,Z)=0 where Z denotes the centre. Finally, we describe the special geometry of the space of stable three-forms satisfying a reality condition. Considering all possible reality conditions, we find four different special K\\\"ahler manifolds and one special para-K\\\"ahler manifold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-07-07T14:09:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C10",
      "53C25",
      "53C29",
      "53C44",
      "53C50"
    ],
    "keywords": [
      "special holonomy",
      "reality condition",
      "invariant half-flat structure admits",
      "three-dimensional heisenberg group",
      "left-invariant half-flat structures"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1310336,
      "adsabs": "2009arXiv0907.1222C"
    }
  }
}