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spherical metric
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(Definition)
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Suppose that
 is the extended complex plane (the Riemann sphere).
Definition 1 Suppose
![$ \gamma \colon [0,1] \to \hat{\mathbb{C}}$](http://images.planetmath.org:8080/cache/objects/5775/l2h/img2.png "$ \gamma \colon [0,1] \to \hat{\mathbb{C}}$") is a path in
 . The spherical length of  is defined as
Definition 2 Let
 , and let  be the set of all paths in
 from  to  , then the distance from  to  in the spherical metric is defined as
More intuitivelly this is the shortest distance to travel from  to  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  to  ).
- 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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(view preamble)
| Also defines: |
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 4225 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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