Kobayashi metric

Remarks

Holomorphic chains exist because $X$ is connected. The function $k_X$ is a pseudodistance: it is symmetric, satisfies the triangle inequality and vanishes on the diagonal, but it may vanish for distinct points. Some authors use $2\rho$, the distance of curvature $-1$, which doubles $k_X$. The infinitesimal form is the Kobayashi–Royden pseudometric $F_X(p, v) = \inf\set{1/R : \text{there is } f \colon \mathbb{D} \to X \text{ holomorphic with } f(0) = p,\ f'(0) = R v}$ for $v$ tangent at $p$; by a theorem of Royden, $k_X$ is the integrated form of $F_X$. Distance-decreasing property: if $g \colon X \to Y$ is holomorphic, then $k_Y(g(p), g(q)) \le k_X(p, q)$, because $g$ maps holomorphic chains to holomorphic chains. Example: $k_{\mathbb{D}} = \rho$; the inequality $k_{\mathbb{D}} \le \rho$ comes from the chain with $f = \id$, and $k_{\mathbb{D}} \ge \rho$ comes from the Schwarz–Pick theorem. Non-example of a distance: $k_\C \equiv 0$; for $p \ne q$ in $\C$ and $R > 1$, the map $f(\zeta) = p + R \zeta (q - p)$ sends $0$ to $p$ and $1/R$ to $q$, so $k_\C(p, q) \le \rho(0, 1/R) \to 0$ as $R \to \infty$.