Complex geodesic

Summary

A complex geodesic of a complex manifold $X$ is a holomorphic map from the unit disc into $X$ that preserves distances. The disc carries its Poincaré distance, and $X$ carries its Kobayashi pseudodistance. Its image is a holomorphic disc along which the invariant geometry of $X$ is exactly that of the hyperbolic disc. Edoardo Vesentini introduced the notion, first with the Carathéodory distance. By Lempert's theorem, in a bounded convex domain any two distinct points lie on a complex geodesic.