---
tag: 275YT975A8
kind: concept
name: Complex geodesic
msc: [32F45, 32H02]
sources:
  - url: https://en.wikipedia.org/wiki/Kobayashi_metric
---

## 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|DY8QQ5GTTX]].
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|SYNST4W3FR]].
By [[Lempert's theorem|2PANNSQ3G6]], in a bounded convex domain any two distinct points lie on a complex geodesic.

## Explanations

### L1 human

In the unit disc, holomorphic self-maps never increase hyperbolic distances.
A holomorphic disc in a larger manifold usually makes points look farther apart in the disc than they are in the manifold.
A 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.
In the unit ball of $\C^n$, every slice of the ball by a complex line through the centre gives a complex geodesic.