spherical metric

Suppose that $\widehat{ℂ}:=ℂ\cup \{\mathrm{\infty }\}$ is the extended complex plane (the Riemann sphere).

Definition.

Suppose $\gamma :[0,1]\to \widehat{ℂ}$ is a path in $\widehat{ℂ}$.The spherical length of $\gamma$ is defined as

$$\mathrm{\ell }(\gamma ):=2{\int }_{\gamma }\frac{|dz|}{1+{|z|}^2}=2{\int }_0^1\frac{|{\gamma }^{\prime }(t)|}{1+{|\gamma (t)|}^2}𝑑t.$$

Definition.

Let $z_1,z_2\in \widehat{ℂ}$, and let $\mathrm{\Gamma }$ be the set of all pathsin $\widehat{ℂ}$ from $z_1$ to $z_2$, then the distance from$z_1$ to $z_2$ in the spherical metric is defined as

$$\sigma (z_1,z_2):=\underset{\gamma \in \mathrm{\Gamma }}{inf}\mathrm{\ell }(\gamma ).$$

More intuitivelly this is the shortest distance to travel from $z_1$ to$z_2$ if we think of these points as being on the Riemann sphere, and we canonly travel on the Riemann sphere itself (we cannot “drill” a straight linefrom $z_1$ to $z_2$).