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  "body": "\n## Summary\n\nA complex geodesic of a complex manifold $X$ is a holomorphic map from the unit disc into $X$ that preserves distances.\nThe disc carries its Poincaré distance, and $X$ carries its [[Kobayashi pseudodistance|DY8QQ5GTTX]].\nIts image is a holomorphic disc along which the invariant geometry of $X$ is exactly that of the hyperbolic disc.\nEdoardo Vesentini introduced the notion, first with the [[Carathéodory distance|SYNST4W3FR]].\nBy [[Lempert's theorem|2PANNSQ3G6]], in a bounded convex domain any two distinct points lie on a complex geodesic.\n\n## Explanations\n\n### L1 human\n\nIn the unit disc, holomorphic self-maps never increase hyperbolic distances.\nA holomorphic disc in a larger manifold usually makes points look farther apart in the disc than they are in the manifold.\nA complex geodesic is a disc for which no distance is lost: the distance of two points of the disc in the manifold equals their hyperbolic distance in the disc.\nIn the unit ball of $\\C^n$, every slice of the ball by a complex line through the centre gives a complex geodesic.\n",
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