{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and $p \\in M$.\nFor $r \u003e 0$, let $B_r(0) = \\set{v \\in T_pM : g_p(v, v) \u003c r^2}$.\nLet $A_p$ be the set of $r \u003e 0$ such that:\n\n1. $B_r(0)$ lies in the domain of the [[exponential map|Y7Z0AGY1A7]] $\\exp_p$;\n2. $\\exp_p$ restricted to $B_r(0)$ is a [[diffeomorphism|0KZZ70ZHXP]] onto an open subset of $M$.\n\nThe *injectivity radius* of $M$ at $p$ is $\\operatorname{inj}(p) = \\sup A_p$.\nThe injectivity radius of $M$ is the [[infimum|0QF1MZ9FRG]] $\\operatorname{inj}(M) = \\inf_{p \\in M} \\operatorname{inj}(p)$.\n\n## Remarks\n\nExample: on the unit sphere $S^n$, for any $p$, $\\exp_p$ is a diffeomorphism from $B_\\pi(0)$ onto $S^n$ minus the antipode $-p$.\nIt maps the whole sphere $\\set{\\abs{v} = \\pi}$ to $-p$, so no larger ball works, and $\\operatorname{inj}(p) = \\pi$.\n\nNon-example: for the same $p$ on $S^n$, the radius $r = 2\\pi$ does not belong to $A_p$, because $\\exp_p$ is not injective on $B_{2\\pi}(0)$.\n\nFor $\\R^n$ with the Euclidean metric, $\\operatorname{inj}(p) = \\infty$ for every $p$.\nThe injectivity radius is always positive at each point; if $M$ is compact, then $\\operatorname{inj}(M) \u003e 0$.\nIf $M$ is complete, $\\operatorname{inj}(p)$ is the distance from $p$ to its [[cut locus|X5WKGC5Q6T]].\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "Y6ZSKVAE3V",
    "kind": "definition",
    "msc": [
      "53C22",
      "53C20"
    ],
    "name": "Injectivity radius",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Glossary_of_Riemannian_and_metric_geometry"
      }
    ],
    "tag": "E8X03PNHM0",
    "uses": [
      "7CBT4SM83Y",
      "Y7Z0AGY1A7",
      "0KZZ70ZHXP",
      "0QF1MZ9FRG"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/E8X03PNHM0/",
  "links": [
    {
      "field": "defines",
      "target": "Y6ZSKVAE3V"
    },
    {
      "field": "uses",
      "target": "7CBT4SM83Y"
    },
    {
      "field": "uses",
      "target": "Y7Z0AGY1A7"
    },
    {
      "field": "uses",
      "target": "0KZZ70ZHXP"
    },
    {
      "field": "uses",
      "target": "0QF1MZ9FRG"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/E8X03PNHM0.md",
  "msc": [
    "53C22",
    "53C20"
  ],
  "name": "Injectivity radius",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/E8X/E8X03PNHM0.md",
  "status": null,
  "tag": "E8X03PNHM0",
  "uri": "m2://E8X03PNHM0",
  "url": "https://math2.tamnd.com/wiki/injectivity-radius-definition/",
  "writing": {
    "tag": "E8X03PNHM0",
    "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
      }
    ]
  }
}