{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $D = \\set{u \\in \\R^2 : \\abs{u} \u003c 1}$ be the open unit disc.\nOn $D \\times \\R$ consider, for each $c \\in \\R$, the surface\n$$L_c = \\set{(u, t) \\in D \\times \\R : t = c + e^{1/(1 - \\abs{u}^2)}}.$$\nThese surfaces are disjoint graphs over $D$, and they fill $D \\times \\R$.\nTogether with the boundary cylinder $\\partial D \\times \\R$, they form a codimension one [[foliation|98HTB7MSPX]] of the closed solid cylinder $\\bar D \\times \\R$.\nEach $L_c$ is a plane that bends upwards and approaches the boundary cylinder as $\\abs{u} \\to 1$, since $e^{1/(1 - \\abs{u}^2)} \\to \\infty$.\nThe translation $t \\mapsto t + 1$ maps $L_c$ to $L_{c+1}$, so the foliation passes to the quotient solid torus $\\bar D \\times (\\R / \\Z)$.\nThis foliated solid torus is a *Reeb component*.\nIts boundary torus is one leaf, and every interior leaf is a plane that spirals towards the boundary.\n\nThe $3$-sphere is the union of two solid tori glued along their boundary tori.\nThe gluing sends the meridian of each solid torus to the longitude of the other.\nPutting a Reeb component on each solid torus gives a smooth codimension one foliation of $S^3$, the *Reeb foliation*.\nThe leaves are one torus and infinitely many planes.\n\n## Remarks\n\nThe construction shows that $S^3$, which has Euler characteristic $0$, carries a codimension one foliation.\nNovikov proved that every codimension one foliation of class $C^2$ on $S^3$ has a compact leaf, a torus bounding a Reeb component.\nThe torus leaf has infinite fundamental group, so the [[Reeb stability theorem|Z93KMZ9R37]] does not force the other leaves to be compact, and indeed they are planes.\n\n## Explanations\n\n### L1 human\n\nImagine a stack of bowls inside a solid cylinder, each bowl a little higher than the last, with rims climbing ever higher along the wall.\nRoll the cylinder into a doughnut, and the bowls become sheets spiralling towards the doughnut's surface.\nTwo such doughnuts glued together make up the $3$-sphere, foliated by one torus and many planes.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "instance_of": [
      "98HTB7MSPX"
    ],
    "kind": "example",
    "msc": [
      "57R30",
      "53C12"
    ],
    "name": "Reeb foliation",
    "same_as": {
      "wikidata": "Q564842",
      "wikipedia": "Reeb foliation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Reeb_foliation"
      }
    ],
    "tag": "W58NYR1DGR"
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/W58NYR1DGR/",
  "links": [
    {
      "field": "instance_of",
      "target": "98HTB7MSPX"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/W58NYR1DGR.md",
  "msc": [
    "57R30",
    "53C12"
  ],
  "name": "Reeb foliation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/W58/W58NYR1DGR.md",
  "status": null,
  "tag": "W58NYR1DGR",
  "uri": "m2://W58NYR1DGR",
  "url": "https://math2.tamnd.com/wiki/reeb-foliation/",
  "writing": {
    "tag": "W58NYR1DGR",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}