---
tag: NXEAT9DMWD
kind: concept
name: Oka manifold
msc: [32Q56]
specializes: [JSGWAN2RJV]
sources:
  - url: https://en.wikipedia.org/wiki/Stein_manifold
---

## Summary

An Oka manifold is a complex manifold $Y$ that receives many holomorphic maps from Euclidean spaces.
Holomorphic maps from neighbourhoods of compact convex sets in $\C^n$ to $Y$ can be approximated by entire maps $\C^n \to Y$.
For such targets the Oka principle holds: every continuous map from a Stein manifold to $Y$ can be deformed to a holomorphic map.
Oka manifolds are the opposite of Kobayashi hyperbolic manifolds, which receive no nonconstant entire maps from $\C$.

## Explanations

### L1 human

Some complex manifolds are "flexible": they contain many holomorphic images of $\C$ and of $\C^n$, as $\C^n$ itself does.
Others are "rigid": every holomorphic map from $\C$ into them is constant, as for the unit disc, by Liouville's theorem.
Oka manifolds are the flexible ones, made precise by an approximation property.
Examples are $\C^n$, the complex projective spaces, and complex Lie groups such as $\C^* = \C \setminus \set{0}$.