{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $M$ be a [[topological manifold|QQQ9B1DK4P]] of dimension $n$, so $M$ is Hausdorff, second countable and locally homeomorphic to $\\R^n$.\nA *chart* on $M$ is a pair $(U, \\varphi)$ where $U \\subset M$ is open and $\\varphi \\colon U \\to \\varphi(U)$ is a [[homeomorphism|0QT00PB5G2]] onto an open subset of $\\R^n$.\nTwo charts $(U, \\varphi)$ and $(V, \\psi)$ are *smoothly compatible* if $U \\cap V = \\emptyset$ or the transition map\n$$\\psi \\circ \\varphi^{-1} \\colon \\varphi(U \\cap V) \\to \\psi(U \\cap V)$$\nis a [[smooth|0V33BNKZ9S]] ($C^\\infty$) map with smooth inverse.\nA *smooth atlas* on $M$ is a family of pairwise smoothly compatible charts whose domains cover $M$.\nA *smooth structure* is a smooth atlas that is maximal: it contains every chart that is smoothly compatible with all of its charts.\nA *smooth manifold* of dimension $n$ is a topological manifold of dimension $n$ together with a smooth structure.\n\n## Remarks\n\nThe unit circle $S^1 \\subset \\R^2$ is a smooth manifold.\nLet $N = (0, 1)$ and $S = (0, -1)$, and let $\\sigma_N \\colon S^1 \\setminus \\set{N} \\to \\R$ and $\\sigma_S \\colon S^1 \\setminus \\set{S} \\to \\R$ be the stereographic projections, $\\sigma_N(x, y) = x/(1 - y)$ and $\\sigma_S(x, y) = x/(1 + y)$.\nOn the overlap, $\\sigma_N \\sigma_S = x^2/(1 - y^2) = 1$, so the transition map is $t \\mapsto 1/t$ on $\\R \\setminus \\set{0}$, which is smooth with smooth inverse.\n\nThe charts $(\\R, \\id)$ and $(\\R, t \\mapsto t^3)$ on $\\R$ are not smoothly compatible.\nThe transition map $t \\mapsto t^{1/3}$ is not differentiable at $0$.\nEach of them alone defines a smooth structure on $\\R$, and the two structures are different, although the resulting manifolds are diffeomorphic.\n\nEvery smooth atlas is contained in a unique maximal one, so in practice a smooth manifold is given by any smooth atlas.\nReplacing $C^\\infty$ by $C^k$ gives $C^k$ manifolds.\n\n## Explanations\n\n### L1 human\n\nA smooth manifold is a space covered by coordinate charts in which every change of coordinates is a smooth map.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "F45VQ47GPV",
    "kind": "definition",
    "msc": [
      "58A05"
    ],
    "name": "Smooth manifold",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Differentiable_manifold"
      }
    ],
    "tag": "WBZHXGKNSJ",
    "uses": [
      "QQQ9B1DK4P",
      "0QT00PB5G2",
      "0V33BNKZ9S"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/WBZHXGKNSJ/",
  "links": [
    {
      "field": "defines",
      "target": "F45VQ47GPV"
    },
    {
      "field": "uses",
      "target": "QQQ9B1DK4P"
    },
    {
      "field": "uses",
      "target": "0QT00PB5G2"
    },
    {
      "field": "uses",
      "target": "0V33BNKZ9S"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/WBZHXGKNSJ.md",
  "msc": [
    "58A05"
  ],
  "name": "Smooth manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WBZ/WBZHXGKNSJ.md",
  "status": null,
  "tag": "WBZHXGKNSJ",
  "uri": "m2://WBZHXGKNSJ",
  "url": "https://math2.tamnd.com/wiki/smooth-manifold-definition/",
  "writing": {
    "tag": "WBZHXGKNSJ",
    "kind": "definition",
    "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
      }
    ]
  }
}