Brody hyperbolicity

Remarks

Example: the unit disc $\mathbb{D}$ is Brody hyperbolic, since a holomorphic map $\C \to \mathbb{D}$ is a bounded entire function, hence constant by Liouville's theorem. Example: a compact Riemann surface of genus $g \ge 2$ is Brody hyperbolic, because an entire curve would lift to a holomorphic map from $\C$ to its universal covering, the unit disc. Non-example: $\C^n$ and the complex projective space $\mathbb{P}^n$ contain lines $t \mapsto (t, 0, \dots, 0)$ and $t \mapsto [1 : t : 0 : \cdots : 0]$, so they are not Brody hyperbolic. Every Kobayashi hyperbolic manifold is Brody hyperbolic, since an entire curve $f$ gives $k_X(f(a), f(b)) \le k_\C(a, b) = 0$. For compact $X$ the two notions agree by Brody's lemma; for noncompact $X$ they can differ.