Kobayashi hyperbolic manifold

Summary

A complex manifold is Kobayashi hyperbolic when its Kobayashi pseudodistance 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. Hyperbolicity is linked with the geometry of the canonical bundle and with the arithmetic of varieties, through the Kobayashi conjecture and the Green–Griffiths–Lang conjecture.