Complex geodesic

Remarks

Since holomorphic maps decrease Kobayashi distances, the inequality $\le$ always holds; the condition is the reverse inequality. Some authors use the Carathéodory distance $c_X$ in place of $k_X$; on bounded convex domains the two notions agree, since $c_X = k_X$ there. Example: in the unit ball $B \subset \C^n$, $\varphi(\zeta) = (\zeta, 0, \dots, 0)$ is a complex geodesic. Check: the projection $\pi(z) = z_1$ maps $B$ into $\mathbb{D}$ and $\pi \circ \varphi = \id$, so $\rho(a, b) = \rho(\pi\varphi(a), \pi\varphi(b)) \le k_B(\varphi(a), \varphi(b)) \le \rho(a, b)$. Non-example: $\psi(\zeta) = (\zeta / 2, 0, \dots, 0)$ is holomorphic into $B$, but $k_B(\psi(a), \psi(b)) = \rho(a/2, b/2) < \rho(a, b)$ for $a \ne b$, by the same argument and the Schwarz–Pick lemma for $\zeta \mapsto \zeta/2$. A complex geodesic is injective, since $\rho(a, b) > 0$ for $a \ne b$.