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  "body": "\n## Definition\n\nLet $\\mathbb{D} = \\set{\\zeta \\in \\C : \\abs{\\zeta} \u003c 1}$ and let $\\rho$ be the Poincaré distance on $\\mathbb{D}$,\n$$\\rho(a, b) = \\tanh^{-1} \\abs{\\frac{a - b}{1 - \\bar a b}} .$$\nLet $X$ be a connected [[complex manifold|JSGWAN2RJV]] and $p, q \\in X$.\nA *holomorphic chain* from $p$ to $q$ consists of points $p = p_0, p_1, \\dots, p_k = q$ of $X$, points $a_1, b_1, \\dots, a_k, b_k$ of $\\mathbb{D}$, and [[holomorphic maps|3MQTWKBBBT]] $f_1, \\dots, f_k \\colon \\mathbb{D} \\to X$ with $f_j(a_j) = p_{j-1}$ and $f_j(b_j) = p_j$.\nThe *Kobayashi pseudodistance* of $X$ is\n$$k_X(p, q) = \\inf \\sum_{j=1}^k \\rho(a_j, b_j),$$\nthe infimum over all holomorphic chains from $p$ to $q$.\n\n## Remarks\n\nHolomorphic chains exist because $X$ is connected.\nThe function $k_X$ is a [[pseudodistance|0A3HJX3KTF]]: it is symmetric, satisfies the triangle inequality and vanishes on the diagonal, but it may vanish for distinct points.\nSome authors use $2\\rho$, the distance of curvature $-1$, which doubles $k_X$.\nThe infinitesimal form is the Kobayashi–Royden pseudometric $F_X(p, v) = \\inf\\set{1/R : \\text{there is } f \\colon \\mathbb{D} \\to X \\text{ holomorphic with } f(0) = p,\\ f'(0) = R v}$ for $v$ tangent at $p$; by a theorem of Royden, $k_X$ is the integrated form of $F_X$.\nDistance-decreasing property: if $g \\colon X \\to Y$ is holomorphic, then $k_Y(g(p), g(q)) \\le k_X(p, q)$, because $g$ maps holomorphic chains to holomorphic chains.\nExample: $k_{\\mathbb{D}} = \\rho$; the inequality $k_{\\mathbb{D}} \\le \\rho$ comes from the chain with $f = \\id$, and $k_{\\mathbb{D}} \\ge \\rho$ comes from the [[Schwarz–Pick theorem|V1QYA9RW73]].\nNon-example of a distance: $k_\\C \\equiv 0$; for $p \\ne q$ in $\\C$ and $R \u003e 1$, the map $f(\\zeta) = p + R \\zeta (q - p)$ sends $0$ to $p$ and $1/R$ to $q$, so $k_\\C(p, q) \\le \\rho(0, 1/R) \\to 0$ as $R \\to \\infty$.\n",
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