{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $M$ be a smooth manifold of dimension $n$, and let $p, q \\ge 0$ be integers with $p + q = n$.\nA *foliation of dimension $p$* (and *codimension $q$*) of $M$ is a maximal atlas of charts $\\varphi_i \\colon U_i \\to \\R^p \\times \\R^q$ with the following property.\nThe transition maps have the local form\n$$\\varphi_j \\circ \\varphi_i^{-1}(x, y) = \\bigl(g_{ij}(x, y), h_{ij}(y)\\bigr), \\qquad x \\in \\R^p, \\ y \\in \\R^q.$$\nIn such a chart, the sets $\\varphi_i^{-1}(\\R^p \\times \\set{y}) \\cap U_i$ are the *plaques*.\nCall two points of $M$ equivalent when they can be joined by a finite chain of plaques, each meeting the next.\nThe *leaves* of the foliation are the equivalence classes of this relation.\n\n## Remarks\n\nEach leaf is a connected injectively immersed submanifold of dimension $p$, and the leaves partition $M$.\nThe tangent spaces of the leaves form a subbundle of the [[tangent bundle|WY6K3P4YCG]] that is involutive; by the [[Frobenius theorem|5Q4BHBFXJY]], foliations of dimension $p$ correspond to involutive rank $p$ subbundles.\n\nExample: on the torus $\\R^2 / \\Z^2$, the lines of slope $\\alpha$ give a foliation of dimension $1$.\nThe charts are restrictions of the linear map $(s, t) \\mapsto (s + \\alpha t, t - \\alpha s)$ composed with local inverses of the quotient map, and transition maps are translations, which have the required form.\nIf $\\alpha$ is irrational, each leaf is dense in the torus.\n\nNon-example: on $\\R^3$ with coordinates $(x, y, z)$, the plane field spanned by $\\partial_x$ and $\\partial_y + x \\, \\partial_z$ is not involutive, because the bracket of these two fields is $\\partial_z$.\nSo it is not the tangent field of any foliation by surfaces.\n\n## Explanations\n\n### L1 human\n\nA foliation is a way to slice a manifold into thin sheets of one fixed dimension.\nIn small pieces the slicing looks like the horizontal slices of a stack.\nThe charts must respect the slices: when you change chart, slices go to slices.\nA leaf is what you get by following one slice as far as it goes.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "98HTB7MSPX",
    "kind": "definition",
    "msc": [
      "57R30",
      "53C12"
    ],
    "name": "Foliation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Foliation"
      }
    ],
    "tag": "S8TA91BAN8",
    "uses": [
      "F45VQ47GPV",
      "WY6K3P4YCG"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/S8TA91BAN8/",
  "links": [
    {
      "field": "defines",
      "target": "98HTB7MSPX"
    },
    {
      "field": "uses",
      "target": "F45VQ47GPV"
    },
    {
      "field": "uses",
      "target": "WY6K3P4YCG"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/S8TA91BAN8.md",
  "msc": [
    "57R30",
    "53C12"
  ],
  "name": "Foliation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/S8T/S8TA91BAN8.md",
  "status": null,
  "tag": "S8TA91BAN8",
  "uri": "m2://S8TA91BAN8",
  "url": "https://math2.tamnd.com/wiki/foliation-definition/",
  "writing": {
    "tag": "S8TA91BAN8",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}