Riemann sphere [SEP] named after
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point "∞" is near to very large numbers, just as the point "0" is near to very small numbers.
The extended complex numbers are useful in complex analysis because they allow for
Fuchsian model
In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface "R" as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation. The concept is named after Lazarus Fuchs.
A more precise definition.
By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. More precisely this theorem states that a Riemann surface formula_1 which is not isomorphic to either the Riemann sphere (the elliptic case) or