{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(M, g)$ and $(N, h)$ be [[Riemannian manifolds|7CBT4SM83Y]].\nLet $\\varphi \\colon M \\to N$ be a [[diffeomorphism|0KZZ70ZHXP]], and let $d\\varphi_p$ be its [[differential|6P9PXQDSEN]] at $p \\in M$.\nThe map $\\varphi$ is an *isometry* from $(M, g)$ to $(N, h)$ if, for all $p \\in M$ and all $v, w \\in T_pM$,\n$$h_{\\varphi(p)}\\bigl(d\\varphi_p(v), d\\varphi_p(w)\\bigr) = g_p(v, w).$$\nEquivalently, $\\varphi^* h = g$.\n\n## Remarks\n\nExample: let $A \\in O(n + 1)$ be an orthogonal matrix.\nThe map $x \\mapsto Ax$ restricts to a diffeomorphism of the unit sphere $S^n$ with differential $v \\mapsto Av$.\nSince $\\ip{Av}{Aw} = \\ip{v}{w}$, it is an isometry of the round sphere.\n\nNon-example: the map $x \\mapsto 2x$ of $\\R^n$ is a diffeomorphism, but $\\ip{2v}{2w} = 4 \\ip{v}{w}$, so it does not preserve the metric.\n\nLet $M$ and $N$ have the same dimension.\nA smooth map $\\varphi$ with $\\varphi^* h = g$, not required to be a diffeomorphism, is a local isometry.\nThe covering map $\\R \\to S^1$, $t \\mapsto (\\cos t, \\sin t)$, is an example.\nAn isometry preserves the [[Riemannian distance|B4QR306D8G]]; conversely, by the [[Myers–Steenrod theorem|Z3VRMW8ETQ]], a distance-preserving bijection between connected Riemannian manifolds is an isometry in this sense.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "R9YWQS15AA",
    "kind": "definition",
    "msc": [
      "53C20",
      "53B20"
    ],
    "name": "Isometry of Riemannian manifolds",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Isometry"
      }
    ],
    "tag": "99RQHAGNW1",
    "uses": [
      "7CBT4SM83Y",
      "0KZZ70ZHXP",
      "6P9PXQDSEN"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/99RQHAGNW1/",
  "links": [
    {
      "field": "defines",
      "target": "R9YWQS15AA"
    },
    {
      "field": "uses",
      "target": "7CBT4SM83Y"
    },
    {
      "field": "uses",
      "target": "0KZZ70ZHXP"
    },
    {
      "field": "uses",
      "target": "6P9PXQDSEN"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/99RQHAGNW1.md",
  "msc": [
    "53C20",
    "53B20"
  ],
  "name": "Isometry of Riemannian manifolds",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/99R/99RQHAGNW1.md",
  "status": null,
  "tag": "99RQHAGNW1",
  "uri": "m2://99RQHAGNW1",
  "url": "https://math2.tamnd.com/wiki/isometry-of-riemannian-manifolds-definition/",
  "writing": {
    "tag": "99RQHAGNW1",
    "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
      }
    ]
  }
}