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'spherical metric'
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| Title of object: |
spherical metric |
| Canonical Name: |
SphericalMetric |
| Type: |
Definition |
| Created on: |
2004-04-16 21:10:17 |
| Modified on: |
2004-04-20 12:08:05 |
| Classification: |
msc:30A99, msc:54-00 |
| Defines: |
spherical length |
Revision comment (for changes between this and next version):
| moved thm stuff to preamble |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
\theoremstyle{definition}
\newtheorem*{defn}{Definition}
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} |
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