spherical metric

Suppose that ^:={} is the extended complex planeMathworldPlanetmath (the Riemann sphere).


Suppose γ:[0,1]^ is a path in ^. The spherical length of γ is defined as


Let z1,z2^, and let Γ be the set of all paths in ^ from z1 to z2, then the distanceMathworldPlanetmath from z1 to z2 in the spherical metric is defined as


More intuitivelly this is the shortest distance to travel from z1 to z2 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 z1 to z2).


  • 1 Theodore B. Gamelin. . Springer-Verlag, New York, New York, 2001.
Title spherical metric
Canonical name SphericalMetric
Date of creation 2013-03-22 14:18:41
Last modified on 2013-03-22 14:18:41
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 6
Author jirka (4157)
Entry type Definition
Classification msc 54-00
Classification msc 30A99
Defines spherical length