{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "78JRW8KTH8"
    },
    {
      "field": "uses",
      "target": "AV8Q85BH70"
    },
    {
      "field": "uses",
      "target": "PCNPQ7YBDE"
    },
    {
      "field": "uses",
      "target": "TTXDZENFVB"
    },
    {
      "field": "defines",
      "target": "TW35GWE1XF"
    }
  ],
  "body": "\n## Summary\n\nThe torsion tensor of an [[affine connection|JX65KSVHSJ]] measures how far covariant derivatives are from being symmetric in the two vector fields.\nThe Lie bracket is subtracted to make this a tensor.\nIt is a tensor field of type $(1, 2)$, antisymmetric in its two arguments.\nConnections with zero torsion are called torsion-free or symmetric; the Levi-Civita connection is the unique torsion-free connection compatible with a given Riemannian metric.\nGeometrically, torsion describes how infinitesimal parallelograms built by parallel transport fail to close.\n\n## Explanations\n\n### L1 human\n\nStart at a point, move a little in direction $X$, then a little in direction $Y$, carrying each direction along with the connection.\nDo the same in the other order.\nFor a torsion-free connection the two endpoints agree to second order, so the small parallelogram closes.\nTorsion measures the small gap that remains.\nThe flat connection of the plane has no torsion, while a connection that twists directions as you move, like a screw, can have torsion.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "53B05"
    ],
    "name": "Torsion tensor",
    "same_as": {
      "wikidata": "Q13229797",
      "wikipedia": "Torsion tensor"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Torsion_tensor"
      }
    ],
    "specializes": [
      "WX302T93TW"
    ],
    "tag": "JZNQAN00ZJ"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/JZNQAN00ZJ/",
  "links": [
    {
      "field": "specializes",
      "target": "WX302T93TW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/JZNQAN00ZJ.md",
  "msc": [
    "53B05"
  ],
  "name": "Torsion tensor",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JZN/JZNQAN00ZJ.md",
  "status": null,
  "tag": "JZNQAN00ZJ",
  "uri": "m2://JZNQAN00ZJ",
  "url": "https://math2.tamnd.com/wiki/torsion-tensor/",
  "writing": {
    "tag": "JZNQAN00ZJ",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}