{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a connected [[complex manifold|JSGWAN2RJV]] with [[Kobayashi pseudodistance|DY8QQ5GTTX]] $k_X$.\nThe manifold $X$ is *Kobayashi hyperbolic* if $k_X(p, q) \u003e 0$ for all $p, q \\in X$ with $p \\ne q$, that is, if $k_X$ is a [[distance|0VKD1TP7F3]] on $X$.\n\n## Remarks\n\nWhen $X$ is hyperbolic, $k_X$ induces the manifold topology of $X$, by a theorem of Barth.\nExample: a bounded domain $\\Omega \\subseteq \\C^n$ is hyperbolic.\nCheck: if $\\abs{z} \u003c R$ on $\\Omega$ and $p \\ne q$, choose a complex linear function $\\ell$ with $\\ell(p) \\ne \\ell(q)$ and $\\abs{\\ell(z)} \u003c 1$ on $\\Omega$; since $\\ell \\colon \\Omega \\to \\mathbb{D}$ decreases Kobayashi distances, $k_\\Omega(p, q) \\ge k_{\\mathbb{D}}(\\ell(p), \\ell(q)) \u003e 0$.\nNon-example: $\\C^n$ is not hyperbolic, since $k_{\\C^n} \\equiv 0$.\nMore generally, if there is a nonconstant holomorphic map $f \\colon \\C \\to X$, then $k_X(f(a), f(b)) \\le k_\\C(a, b) = 0$ with $f(a) \\ne f(b)$ for some $a, b$, so $X$ is not hyperbolic.\nHence complex tori and complex projective spaces are not hyperbolic.\nA Riemann surface is hyperbolic exactly when its universal covering is the unit disc.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "HRPAX8V7HA",
    "kind": "definition",
    "msc": [
      "32Q45",
      "32F45"
    ],
    "name": "Kobayashi hyperbolic manifold",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Kobayashi_metric"
      }
    ],
    "tag": "6V6WDRHRTF",
    "uses": [
      "JSGWAN2RJV",
      "DY8QQ5GTTX",
      "0VKD1TP7F3"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/6V6WDRHRTF/",
  "links": [
    {
      "field": "defines",
      "target": "HRPAX8V7HA"
    },
    {
      "field": "uses",
      "target": "JSGWAN2RJV"
    },
    {
      "field": "uses",
      "target": "DY8QQ5GTTX"
    },
    {
      "field": "uses",
      "target": "0VKD1TP7F3"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/6V6WDRHRTF.md",
  "msc": [
    "32Q45",
    "32F45"
  ],
  "name": "Kobayashi hyperbolic manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6V6/6V6WDRHRTF.md",
  "status": null,
  "tag": "6V6WDRHRTF",
  "uri": "m2://6V6WDRHRTF",
  "url": "https://tamnd.github.io/math2/wiki/kobayashi-hyperbolic-manifold-definition/",
  "writing": {
    "tag": "6V6WDRHRTF",
    "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
      }
    ]
  }
}