Brody hyperbolicity

Summary

A complex manifold is Brody hyperbolic if every holomorphic map from $\C$ into it is constant. Nonconstant holomorphic maps $\C \to X$ are called entire curves, so Brody hyperbolicity means that $X$ contains no entire curve. Every Kobayashi hyperbolic manifold is Brody hyperbolic, and for compact manifolds the converse holds by Brody's lemma. The notion is named after Robert Brody, who proved the converse in 1978.