{
  "id": "2109.15253",
  "version": "v2",
  "published": "2021-09-30T16:36:31.000Z",
  "updated": "2022-02-09T09:45:50.000Z",
  "title": "Differential geometry of $\\mathsf{SO}^\\ast(2n)$-type structures",
  "authors": [
    "Ioannis Chrysikos",
    "Jan Gregorovič",
    "Henrik Winther"
  ],
  "comment": "48 pages, title changed and minor corrections done. To appear in Annali di Matematica Pura ed Applicata (1923 -)",
  "categories": [
    "math.DG"
  ],
  "abstract": "We study $4n$-dimensional smooth manifolds admitting a $\\mathsf{SO}^*(2n)$- or a $\\mathsf{SO}^*(2n)\\mathsf{Sp}(1)$-structure, where $\\mathsf{SO}^*(2n)$ is the quaternionic real form of $\\mathsf{SO}(2n, \\mathbb{C})$. We show that such $G$-structures, called almost hypercomplex/quaternionic skew-Hermitian structures, form the symplectic analogue of the better known almost hypercomplex/quaternionic-Hermitian structures (hH/qH for short). We present several equivalent definitions of $\\mathsf{SO}^*(2n)$- and $\\mathsf{SO}^*(2n)\\mathsf{Sp}(1)$-structures in terms of almost symplectic forms compatible with an almost hypercomplex/quaternionic structure, a quaternionic skew-Hermitian form, or a symmetric 4-tensor, the latter establishing the counterpart of the fundamental 4-form in almost hH/qH geometries. The intrinsic torsion of such structures is presented in terms of Salamon's $\\mathsf{E}\\mathsf{H}$-formalism, and the algebraic types of the corresponding geometries are classified. We construct explicit adapted connections to our $G$-structures and specify certain normalization conditions, under which these connections become minimal. Finally, we present the classification of symmetric spaces $K/L$ with $K$ semisimple admitting an invariant torsion-free $\\mathsf{SO}^*(2n)\\mathsf{Sp}(1)$-structure. This paper is the first in a series aiming at the description of the differential geometry of $\\mathsf{SO}^*(2n)$- and $\\mathsf{SO}^*(2n)\\mathsf{Sp}(1)$-structures.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-02-09T09:45:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C10",
      "53C26",
      "53D15",
      "53B05",
      "53C35",
      "53A55"
    ],
    "keywords": [
      "differential geometry",
      "type structures",
      "hypercomplex/quaternionic skew-hermitian structures",
      "quaternionic real form",
      "quaternionic skew-hermitian form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}