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  "body": "\n## Summary\n\nThe Kobayashi metric, or Kobayashi pseudodistance, of a complex manifold $X$ is a pseudodistance on $X$ built from holomorphic discs.\nIt is the largest pseudodistance for which every [[holomorphic map|3MQTWKBBBT]] from the unit disc, with its Poincaré distance, is distance-decreasing.\nIt is built from chains of holomorphic discs, and Shoshichi Kobayashi introduced it in 1967.\nEvery holomorphic map between complex manifolds is distance-decreasing for the Kobayashi pseudodistances, so it is a biholomorphic invariant.\nIt vanishes identically on $\\C^n$ and is a true distance on bounded domains.\nThe manifolds on which it is a distance are the [[Kobayashi hyperbolic manifolds|HRPAX8V7HA]].\n\n## Explanations\n\n### L1 human\n\nThe unit disc carries a natural distance, the hyperbolic distance of the [[Poincaré disk model|85MM2EA0TK]], which holomorphic self-maps of the disc never increase.\nTo measure the distance between two points of a complex manifold, connect them by a chain of holomorphic discs and add up the hyperbolic lengths of the pieces.\nThe Kobayashi distance is the smallest possible total.\nIn $\\C$ one can join two points by a huge disc, in which they are very close in the hyperbolic sense, so the Kobayashi distance of $\\C$ is $0$.\nIn the unit disc itself, the Kobayashi distance is the hyperbolic distance.\n",
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