{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be a compact Lie group with a smooth [[action|0YSE802YCQ]] on a [[smooth manifold|R2TQ82PCNB]] $M$.\nLet $x \\in M$, let $G x$ be its [[orbit|00F2JFA41J]], and let $H = G_x$ be its [[stabiliser|0RGGF8ZS6D]], a closed subgroup of $G$.\nLet $V = T_x M / T_x(G x)$ be the normal space of the orbit at $x$.\nThe group $H$ acts linearly on $V$ by the derivatives at $x$ of the maps $y \\mapsto h y$ for $h \\in H$; this is a [[representation|RK8RV2C998]] of $H$.\nLet $G \\times_H V$ be the quotient of $G \\times V$ by the action $h \\cdot (g, v) = (g h^{-1}, h v)$ of $H$, with $G$ acting by $g' [g, v] = [g' g, v]$.\nThen there are a $G$-invariant open neighbourhood $U$ of $G x$ in $M$ and a $G$-equivariant [[diffeomorphism|0Z1Y7JHGBR]] $\\Phi \\colon G \\times_H V \\to U$.\nMoreover $\\Phi([e, 0]) = x$, where $e$ is the identity of $G$.\n\n## Remarks\n\nThe idea of the proof is to average a Riemannian metric on $M$ over $G$, which is possible because $G$ is compact.\nThe new metric is $G$-invariant, so the exponential map on the normal bundle of the orbit $G x$ is $G$-equivariant.\nThe orbit is a compact submanifold diffeomorphic to $G / H$, and its normal bundle is $G \\times_H V$.\nThe [[tubular neighbourhood theorem|GCBY80CSC6]] gives $\\Phi$ from this exponential map on vectors of small length.\nThe image $S = \\Phi(\\set{[e, v] : v \\in V})$ is called a *slice* at $x$: it is $H$-invariant, and $g S \\cap S \\neq \\emptyset$ implies $g \\in H$.\nExample: let $G = SO(2)$ act on $\\R^2$ by rotations and let $x = (1, 0)$.\nThen $H$ is trivial, $V$ is the radial line $\\R$, and $\\Phi([g, t]) = g (e^t, 0)$ is an equivariant diffeomorphism from $SO(2) \\times \\R$ onto $U = \\R^2 \\setminus \\set{0}$.\nCompactness can be replaced by properness of the action, a result of Palais.\nWithout such a hypothesis the theorem fails.\nLet $\\R$ act on the torus $\\R^2 / \\Z^2$ by translations along a line of irrational slope.\nEvery orbit is dense, so no orbit has a neighbourhood of the form $G \\times_H V$ in which it is closed.\n\n## Explanations\n\n### L1 human\n\nWhen a compact group of symmetries moves a manifold around, the theorem describes how the manifold looks near one orbit.\nNear the orbit, the manifold is a thickened copy of the orbit.\nThe thickening is a small disk at each point of the orbit, and the symmetries that fix the point turn this disk by linear maps.\n\n### L2 human\n\nThe theorem reduces local questions about smooth actions of compact groups to linear algebra.\nIt shows that every orbit has a neighbourhood that retracts onto it equivariantly, and that the stabilisers of nearby points are conjugate to subgroups of $H$.\nFrom it one derives the principal orbit type theorem.\nIt also shows that the quotient of a manifold by a free action of a compact Lie group is again a manifold.\nWhen $G$ is finite and acts faithfully, it shows that $M / G$ is an [[orbifold|RPAVY0W0XS]].\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "0YSE802YCQ",
      "00F2JFA41J"
    ],
    "kind": "statement",
    "msc": [
      "57S15",
      "22E15"
    ],
    "name": "Slice theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q17093903",
      "wikipedia": "Slice theorem (differential geometry)"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Slice_theorem_(differential_geometry)"
      }
    ],
    "tag": "VQB2N60ARE",
    "uses": [
      "R2TQ82PCNB",
      "0YSE802YCQ",
      "0RGGF8ZS6D",
      "00F2JFA41J",
      "RK8RV2C998",
      "0Z1Y7JHGBR",
      "GCBY80CSC6"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/VQB2N60ARE/",
  "links": [
    {
      "field": "about",
      "target": "0YSE802YCQ"
    },
    {
      "field": "about",
      "target": "00F2JFA41J"
    },
    {
      "field": "uses",
      "target": "R2TQ82PCNB"
    },
    {
      "field": "uses",
      "target": "0YSE802YCQ"
    },
    {
      "field": "uses",
      "target": "0RGGF8ZS6D"
    },
    {
      "field": "uses",
      "target": "00F2JFA41J"
    },
    {
      "field": "uses",
      "target": "RK8RV2C998"
    },
    {
      "field": "uses",
      "target": "0Z1Y7JHGBR"
    },
    {
      "field": "uses",
      "target": "GCBY80CSC6"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/VQB2N60ARE.md",
  "msc": [
    "57S15",
    "22E15"
  ],
  "name": "Slice theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VQB/VQB2N60ARE.md",
  "status": null,
  "tag": "VQB2N60ARE",
  "uri": "m2://VQB2N60ARE",
  "url": "https://math2.tamnd.com/wiki/slice-theorem/",
  "writing": {
    "tag": "VQB2N60ARE",
    "kind": "statement",
    "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
      }
    ]
  }
}