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  "body": "\n## Definition\n\nLet $\\mathbb{D} = \\set{\\zeta \\in \\C : \\abs{\\zeta} \u003c 1}$ and let $\\rho(a, b) = \\tanh^{-1} \\abs{(a - b)/(1 - \\bar a b)}$ be the Poincaré distance on $\\mathbb{D}$.\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] and $p, q \\in X$.\nThe *Carathéodory pseudodistance* of $X$ is\n$$c_X(p, q) = \\sup\\set{\\rho(f(p), f(q)) : f \\colon X \\to \\mathbb{D} \\text{ holomorphic}} .$$\nThe *Carathéodory–Reiffen pseudometric* is $\\gamma_X(p, v) = \\sup\\set{\\abs{df_p(v)} : f \\colon X \\to \\mathbb{D} \\text{ holomorphic},\\ f(p) = 0}$ for $v$ tangent to $X$ at $p$.\n\n## Remarks\n\n$c_X$ is a [[pseudodistance|0A3HJX3KTF]], as a supremum of pseudodistances $(p, q) \\mapsto \\rho(f(p), f(q))$.\nIf $g \\colon X \\to Y$ is a [[holomorphic map|3MQTWKBBBT]], then $c_Y(g(p), g(q)) \\le c_X(p, q)$, since $f \\circ g$ is admissible for $X$ whenever $f$ is admissible for $Y$.\nExample: $c_{\\mathbb{D}} = \\rho$; the identity gives $c_{\\mathbb{D}} \\ge \\rho$, and the [[Schwarz–Pick theorem|V1QYA9RW73]] gives $\\rho(f(a), f(b)) \\le \\rho(a, b)$ for every $f$.\nExample: if $X$ is a bounded domain in $\\C^n$, then $c_X$ is a distance, since a suitable linear function, rescaled, maps $X$ into $\\mathbb{D}$ and separates any two given points.\nNon-example: $c_{\\C^n} \\equiv 0$, since every bounded holomorphic function on $\\C^n$ is constant.\nFor every connected $X$, $c_X \\le k_X$, where $k_X$ is the [[Kobayashi pseudodistance|DY8QQ5GTTX]].\n",
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