{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "A8Q8RVFWWE"
    }
  ],
  "body": "\n## Summary\n\nLet $J$ be a field of endomorphisms of the tangent bundle.\nIts Nijenhuis tensor is a [[tensor field|WX302T93TW]] of type $(1,2)$ built from $J$ and the [[Lie bracket of vector fields|68FGQEBD6J]].\nIt is, up to a factor $2$, the [[Frölicher–Nijenhuis bracket|D46QECMC10]] of $J$ with itself.\nFor an [[almost complex structure|9SYWRGVTRS]], it measures the obstruction to integrability.\nBy the [[Newlander–Nirenberg theorem|VCT848WV1T]], an almost complex structure comes from a complex manifold structure exactly when its Nijenhuis tensor vanishes.\n\n## Explanations\n\n### L1 human\n\nAn almost complex structure lets one multiply tangent vectors by $i$ at each point.\nOne would like coordinates in which this multiplication is the usual one, as on $\\C^n$.\nThe Nijenhuis tensor is a test, computed from derivatives of $J$, that tells whether such coordinates exist.\nOn $\\C^n$ with its usual structure, the test gives zero; on the six-sphere with its standard almost complex structure from the octonions, it does not.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "53C15",
      "32Q60"
    ],
    "name": "Nijenhuis tensor",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Almost_complex_manifold"
      }
    ],
    "tag": "T46ZBWDPHQ"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/T46ZBWDPHQ/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/T46ZBWDPHQ.md",
  "msc": [
    "53C15",
    "32Q60"
  ],
  "name": "Nijenhuis tensor",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/T46/T46ZBWDPHQ.md",
  "status": null,
  "tag": "T46ZBWDPHQ",
  "uri": "m2://T46ZBWDPHQ",
  "url": "https://tamnd.github.io/math2/wiki/nijenhuis-tensor/",
  "writing": {
    "tag": "T46ZBWDPHQ",
    "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
      }
    ]
  }
}