---
tag: DY8QQ5GTTX
kind: concept
name: Kobayashi metric
msc: [32F45, 32Q45]
sources:
  - url: https://en.wikipedia.org/wiki/Kobayashi_metric
same_as:
  wikidata: Q6424284
  wikipedia: Kobayashi metric
---

## Summary

The Kobayashi metric, or Kobayashi pseudodistance, of a complex manifold $X$ is a pseudodistance on $X$ built from holomorphic discs.
It is the largest pseudodistance for which every [[holomorphic map|3MQTWKBBBT]] from the unit disc, with its Poincaré distance, is distance-decreasing.
It is built from chains of holomorphic discs, and Shoshichi Kobayashi introduced it in 1967.
Every holomorphic map between complex manifolds is distance-decreasing for the Kobayashi pseudodistances, so it is a biholomorphic invariant.
It vanishes identically on $\C^n$ and is a true distance on bounded domains.
The manifolds on which it is a distance are the [[Kobayashi hyperbolic manifolds|HRPAX8V7HA]].

## Explanations

### L1 human

The unit disc carries a natural distance, the hyperbolic distance of the [[Poincaré disk model|85MM2EA0TK]], which holomorphic self-maps of the disc never increase.
To 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.
The Kobayashi distance is the smallest possible total.
In $\C$ 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 $0$.
In the unit disc itself, the Kobayashi distance is the hyperbolic distance.