---
tag: SYNST4W3FR
kind: concept
name: Carathéodory metric
msc: [32F45]
sources:
  - url: https://en.wikipedia.org/wiki/Carath%C3%A9odory_metric
same_as:
  wikidata: Q5037756
  wikipedia: Carathéodory metric
---

## Summary

The Carathéodory metric, or Carathéodory pseudodistance, of a complex manifold $X$ uses holomorphic maps from $X$ to the unit disc.
The distance of two points is the largest Poincaré distance between their images under such maps.
It is the smallest pseudodistance for which all holomorphic maps from $X$ to the disc are distance-decreasing, and it is a biholomorphic invariant.
It is always at most the [[Kobayashi metric|DY8QQ5GTTX]], and on bounded convex domains the two agree by [[Lempert's theorem|2PANNSQ3G6]].
Constantin Carathéodory introduced it in the 1920s for domains in $\C^n$.

## Explanations

### L1 human

Bounded holomorphic functions separate the points of a bounded domain.
The Carathéodory distance uses them as rulers: map the domain into the unit disc by a holomorphic function and measure the hyperbolic distance of the two image points.
The largest value over all such maps is the Carathéodory distance.
On $\C$ every bounded holomorphic function is constant by Liouville's theorem, so the Carathéodory distance of $\C$ is $0$.
On the unit disc it is the hyperbolic distance.