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spherical metric
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(Definition)
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Suppose that $\hat{\mathbb{C}} := {\mathbb{C}} \cup \{ \infty \}$ is the extended complex plane (the Riemann sphere).
Definition 1 Suppose $\gamma \colon [0,1] \to \hat{\mathbb{C}}$ is a path in $\hat{\mathbb{C}}$ The 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*}
Definition 2 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 \begin{equation*} \sigma(z_1,z_2) := \inf_{\gamma \in \Gamma} \ell(\gamma) . \end{equation*}
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 $z_2$ .
- 1
- Theodore B. Gamelin. Complex Analysis. Springer-Verlag, New York, New York, 2001.
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"spherical metric" is owned by jirka.
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spherical length |
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Cross-references: line, straight, points, distance, path, Riemann sphere, extended complex plane
There are 3 references to this entry.
This is version 3 of spherical metric, born on 2004-04-16, modified 2005-03-07.
Object id is 5775, canonical name is SphericalMetric.
Accessed 5226 times total.
Classification:
| AMS MSC: | 30A99 (Functions of a complex variable :: General properties :: Miscellaneous) | | | 54-00 (General topology :: General reference works ) |
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Pending Errata and Addenda
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