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  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]].\nAn *entire curve* in $X$ is a nonconstant [[holomorphic map|3MQTWKBBBT]] $f \\colon \\C \\to X$.\nThe manifold $X$ is *Brody hyperbolic* if there is no entire curve in $X$, that is, if every holomorphic map $\\C \\to X$ is constant.\n\n## Remarks\n\nExample: the unit disc $\\mathbb{D}$ is Brody hyperbolic, since a holomorphic map $\\C \\to \\mathbb{D}$ is a bounded entire function, hence constant by [[Liouville's theorem|0EHKVRF4J6]].\nExample: a compact Riemann surface of genus $g \\ge 2$ is Brody hyperbolic, because an entire curve would lift to a holomorphic map from $\\C$ to its universal covering, the unit disc.\nNon-example: $\\C^n$ and the complex projective space $\\mathbb{P}^n$ contain lines $t \\mapsto (t, 0, \\dots, 0)$ and $t \\mapsto [1 : t : 0 : \\cdots : 0]$, so they are not Brody hyperbolic.\nEvery [[Kobayashi hyperbolic manifold|HRPAX8V7HA]] is Brody hyperbolic, since an entire curve $f$ gives $k_X(f(a), f(b)) \\le k_\\C(a, b) = 0$.\nFor compact $X$ the two notions agree by [[Brody's lemma|VEV5E2077C]]; for noncompact $X$ they can differ.\n",
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