{
  "id": "math/0612392",
  "version": "v1",
  "published": "2006-12-14T15:07:43.000Z",
  "updated": "2006-12-14T15:07:43.000Z",
  "title": "Holonomy groups and special geometric structures of pseudo-Kählerian manifolds of index 2",
  "authors": [
    "Anton S. Galaev"
  ],
  "comment": "the corrected version of PhD thesis, Humbold University, Berlin",
  "categories": [
    "math.DG"
  ],
  "abstract": "The problem of classification of connected holonomy groups (equivalently of holonomy algebras) for pseudo-Riemannian manifolds is open. The classification of Riemannian holonomy algebras is a classical result. The classification of Lorentzian holonomy algebras was obtained recently. In the present paper weakly-irreducible not irreducible subalgebras of $\\su(1,n+1)$ ($n\\geq 0$) are classified. Weakly-irreducible not irreducible holonomy algebras of pseudo-K\\\"ahlerian and special pseudo-K\\\"ahlerian manifolds are classified. An example of metric for each possible holonomy algebra is given. This gives the classification of holonomy algebras for pseudo-K\\\"ahlerian manifolds of index 2. Finally we consider some examples and applications. We describe examples of 4-dimensional Lie groups with left-invariant pseudo-K\\\"ahlerian metrics and determine their holonomy algebras. We use our classification of holonomy algebras to give a new proof for the classification of simply connected pseudo-K\\\"ahlerian symmetric spaces of index 2 with weakly-irreducible not irreducible holonomy algebras. We consider time-like cones over Lorentzian Sasaki manifolds. These cones are also pseudo-K\\\"ahlerian manifolds of index 2. We describe the local DeRham-Wu decomposition of the cone in terms of the initial Lorentzian Sasaki manifold and we describe all possible weakly-irreducible not irreducible holonomy algebras of such cones.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-12-14T15:07:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C29",
      "53C50",
      "53B30"
    ],
    "keywords": [
      "special geometric structures",
      "holonomy groups",
      "pseudo-kählerian manifolds",
      "irreducible holonomy algebras",
      "classification"
    ],
    "tags": [
      "dissertation"
    ],
    "publication": {
      "journal": "Ph.D. Thesis",
      "year": 2006,
      "month": "Dec"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006PhDT.......193G"
    }
  }
}