Definition:Riemann Sphere

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In particular: closed line interval, Riemann map
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Definition

Let $f_1: \C \to \R^2$ be defined as:

$\forall z \in \C: \map {f_1} z = \tuple {\map \Re z, \map \Im z}$

Let $f_2: \R^2 \to \R^3$ be the inclusion map:

$\forall \tuple {a, b} \in \C^2: \map {f_2} {a, b} = \tuple {a, b, 0}$

Let $f = f_2 \circ f_1$.

Let $F: \C \to \map \PP {\R^3}$ be defined as the mapping which takes $z$ to the closed line interval from $\tuple {0, 0, 1}$ to $\map f z$ for all $z \in \C$.


Let $G = \set {x, y, z: x^2 + y^2 + z^2 = 1}$.

Then the Riemann map $R: \C \to \mathbb S^2$ is defined as:

$\map R x = \map F z \cap G$

The set $R \sqbrk \C \cup \set {\tuple {0, 0, 1} } $ is called the Riemann sphere, with the understanding that $\map f \infty = \tuple {0, 0, 1}$.


This article is complete as far as it goes, but it could do with expansion.
In particular: According to the definition in Clapham & Nicholson, include the definition as the extended complex plane under a stereographic projection
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Also see

  • Results about Riemann spheres can be found here.


Source of Name

This entry was named for Georg Friedrich Bernhard Riemann.


Sources

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