spherical metric
SphericalMetric
2004-04-16 21:10:17
2005-03-07 20:20:29
Definition
spherical length
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Suppose that $\hat{\mathbb{C}} := {\mathbb{C}} \cup \{ \infty \}$ is the extended complex plane (the Riemann sphere).
\begin{defn}
Suppose $\gamma \colon [0,1] \to \hat{\mathbb{C}}$ is a path in $\hat{\mathbb{C}}$.
The {\em spherical length} of $\gamma$ is defined as
\begin{equation*}
\ell (\gamma) :=
2 \int_\gamma \frac{\lvert dz \rvert}{1+\lvert z \rvert^2}
=
2 \int_0^1 \frac{\lvert \gamma'(t) \rvert}{1+\lvert \gamma(t) \rvert^2} dt.
\end{equation*}
\end{defn}
\begin{defn}
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 {\em spherical metric} is defined as
\begin{equation*}
\sigma(z_1,z_2) := \inf_{\gamma \in \Gamma} \ell(\gamma) .
\end{equation*}
\end{defn}
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 $z_2$).
\begin{thebibliography}{9}
\bibitem{Gamelin:complex}
Theodore~B.\@ Gamelin.
{\em \PMlinkescapetext{Complex Analysis}}.
Springer-Verlag, New York, New York, 2001.
\end{thebibliography}