---
tag: HRPAX8V7HA
kind: concept
name: Kobayashi hyperbolic manifold
msc: [32Q45, 32F45]
sources:
  - url: https://en.wikipedia.org/wiki/Kobayashi_metric
---

## Summary

A complex manifold is Kobayashi hyperbolic when its [[Kobayashi pseudodistance|DY8QQ5GTTX]] is a true distance, that is, it separates points.
Bounded domains in $\C^n$ and compact Riemann surfaces of genus at least $2$ are hyperbolic.
The spaces $\C^n$, the projective spaces and the complex tori are not.
A hyperbolic manifold admits no nonconstant holomorphic map from $\C$, and for compact manifolds this property characterises hyperbolicity by [[Brody's lemma|VEV5E2077C]].
Hyperbolicity is linked with the geometry of the canonical bundle and with the arithmetic of varieties, through the [[Kobayashi conjecture|1X3EV2TZNN]] and the [[Green–Griffiths–Lang conjecture|AD9GD9EM9R]].

## Explanations

### L1 human

A complex manifold is hyperbolic when holomorphic discs inside it cannot be too large.
Then the Kobayashi construction, which measures distances with chains of discs, sees every pair of distinct points at a positive distance.
In $\C$ there are discs of every radius, so all Kobayashi distances are $0$, and $\C$ is not hyperbolic.
The unit disc is hyperbolic, and its Kobayashi distance is the usual hyperbolic distance.
A domain such as $\C$ minus two points is also hyperbolic, which is one way to see the little Picard theorem.