# spherical metric

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

###### Definition.

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

 $\ell(\gamma):=2\int_{\gamma}\frac{\lvert dz\rvert}{1+\lvert z\rvert^{2}}=2\int% _{0}^{1}\frac{\lvert\gamma^{\prime}(t)\rvert}{1+\lvert\gamma(t)\rvert^{2}}dt.$
###### Definition.

Let $z_{1},z_{2}\in\hat{\mathbb{C}}$, and let $\Gamma$ be the set of all paths in $\hat{\mathbb{C}}$ from $z_{1}$ to $z_{2}$, then the distance from $z_{1}$ to $z_{2}$ in the spherical metric is defined as

 $\sigma(z_{1},z_{2}):=\inf_{\gamma\in\Gamma}\ell(\gamma).$

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 can only travel on the Riemann sphere itself (we cannot “drill” a straight line from $z_{1}$ to $z_{2}$).

## References

• 1 Theodore B. Gamelin. . Springer-Verlag, New York, New York, 2001.
Title spherical metric SphericalMetric 2013-03-22 14:18:41 2013-03-22 14:18:41 jirka (4157) jirka (4157) 6 jirka (4157) Definition msc 54-00 msc 30A99 spherical length