{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "AJ7P4G261J"
    }
  ],
  "body": "\n## Summary\n\nA Spin(7)-manifold is a Riemannian manifold of dimension $8$ whose holonomy group lies in the group $\\operatorname{Spin}(7)$, viewed inside $SO(8)$.\nThe structure is encoded in a parallel $4$-form, the Cayley form.\nSpin(7)-manifolds are Ricci-flat and form, with [[$G_2$ manifolds|P1TFW9QWAD]], the two exceptional cases of Berger's list.\n\n## Explanations\n\n### L1 human\n\nThe octonions form an eight-dimensional number system, and the group $\\operatorname{Spin}(7)$ preserves a special $4$-form on them.\nA Spin(7)-manifold is an eight-dimensional curved space in which such a form exists at every point and is unchanged by parallel transport.\nThe flat space $\\R^8$ is the simplest example.\nThese spaces are cousins of the seven-dimensional $G_2$ manifolds: the product of a $G_2$ manifold with a circle is an example.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "53C29",
      "53C27",
      "53C38"
    ],
    "name": "Spin(7)-manifold",
    "same_as": {
      "wikidata": "Q7577346",
      "wikipedia": "Spin(7)-manifold"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Spin(7)-manifold"
      }
    ],
    "specializes": [
      "7CBT4SM83Y"
    ],
    "tag": "1GMSWCHKHE"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/1GMSWCHKHE/",
  "links": [
    {
      "field": "specializes",
      "target": "7CBT4SM83Y"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/1GMSWCHKHE.md",
  "msc": [
    "53C29",
    "53C27",
    "53C38"
  ],
  "name": "Spin(7)-manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1GM/1GMSWCHKHE.md",
  "status": null,
  "tag": "1GMSWCHKHE",
  "uri": "m2://1GMSWCHKHE",
  "url": "https://math2.tamnd.com/wiki/spin-7-manifold/",
  "writing": {
    "tag": "1GMSWCHKHE",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}