Kobayashi hyperbolic manifold

Remarks

When $X$ is hyperbolic, $k_X$ induces the manifold topology of $X$, by a theorem of Barth. Example: a bounded domain $\Omega \subseteq \C^n$ is hyperbolic. Check: if $\abs{z} < R$ on $\Omega$ and $p \ne q$, choose a complex linear function $\ell$ with $\ell(p) \ne \ell(q)$ and $\abs{\ell(z)} < 1$ on $\Omega$; since $\ell \colon \Omega \to \mathbb{D}$ decreases Kobayashi distances, $k_\Omega(p, q) \ge k_{\mathbb{D}}(\ell(p), \ell(q)) > 0$. Non-example: $\C^n$ is not hyperbolic, since $k_{\C^n} \equiv 0$. More generally, if there is a nonconstant holomorphic map $f \colon \C \to X$, then $k_X(f(a), f(b)) \le k_\C(a, b) = 0$ with $f(a) \ne f(b)$ for some $a, b$, so $X$ is not hyperbolic. Hence complex tori and complex projective spaces are not hyperbolic. A Riemann surface is hyperbolic exactly when its universal covering is the unit disc.