{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $(M, g)$ be a connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $4n$.\nLet $Sp(n) \\cdot Sp(1) = (Sp(n) \\times Sp(1)) / \\set{\\pm 1}$, acting on $\\mathbb{H}^n = \\R^{4n}$ by $v \\mapsto a v \\bar{b}$ for $a \\in Sp(n)$ and unit quaternions $b \\in Sp(1)$.\nThe manifold $M$ is *quaternion-Kähler* if its [[holonomy|WHDZ3MK56B]] group is contained in $Sp(n) \\cdot Sp(1) \\subseteq SO(4n)$.\n\nEquivalently, there is a rank $3$ subbundle $Q \\subseteq \\End(TM)$ such that:\n\n1. $Q$ is preserved by the [[Levi-Civita connection|CAZYGE1T4E]];\n2. near each point, $Q$ has a local frame $I, J, K$ of [[almost complex structures|9SYWRGVTRS]] that are orthogonal for $g$ and satisfy $IJ = K$.\n\n## Remarks\n\nExample: the quaternionic projective space $\\mathbb{H}P^n$ with its standard symmetric metric, whose holonomy is all of $Sp(n) \\cdot Sp(1)$.\n\nNon-example: no manifold of dimension $6$ is quaternion-Kähler, since the dimension must be a multiple of $4$.\n\nFor $n \\ge 2$, a quaternion-Kähler manifold is [[Einstein|41S2EZ0KR3]].\n[[Hyperkähler manifolds|87AWY7NQS0]] satisfy the definition, with holonomy in $Sp(n) \\subseteq Sp(n) \\cdot Sp(1)$; many authors exclude them by asking for nonzero scalar curvature.\nIn dimension $4$ the holonomy condition is empty, since $Sp(1) \\cdot Sp(1) = SO(4)$, so one defines quaternion-Kähler $4$-manifolds instead as self-dual Einstein manifolds.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "NAJBH8GJMM",
    "kind": "definition",
    "msc": [
      "53C26",
      "53C29"
    ],
    "name": "Quaternion-Kähler manifold",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quaternion-K%C3%A4hler_manifold"
      }
    ],
    "tag": "W6J4MQ6D5N",
    "uses": [
      "7CBT4SM83Y",
      "WHDZ3MK56B",
      "9SYWRGVTRS",
      "CAZYGE1T4E",
      "87AWY7NQS0",
      "41S2EZ0KR3"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/W6J4MQ6D5N/",
  "links": [
    {
      "field": "defines",
      "target": "NAJBH8GJMM"
    },
    {
      "field": "uses",
      "target": "7CBT4SM83Y"
    },
    {
      "field": "uses",
      "target": "WHDZ3MK56B"
    },
    {
      "field": "uses",
      "target": "9SYWRGVTRS"
    },
    {
      "field": "uses",
      "target": "CAZYGE1T4E"
    },
    {
      "field": "uses",
      "target": "87AWY7NQS0"
    },
    {
      "field": "uses",
      "target": "41S2EZ0KR3"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/W6J4MQ6D5N.md",
  "msc": [
    "53C26",
    "53C29"
  ],
  "name": "Quaternion-Kähler manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/W6J/W6J4MQ6D5N.md",
  "status": null,
  "tag": "W6J4MQ6D5N",
  "uri": "m2://W6J4MQ6D5N",
  "url": "https://math2.tamnd.com/wiki/quaternion-kahler-manifold-definition/",
  "writing": {
    "tag": "W6J4MQ6D5N",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}