Definition:Riemann Sphere

From ProofWiki
Jump to: navigation, search

Definition

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

$\forall z \in \C: f_1 \left({z}\right) = \left({\Re \left({z}\right), \Im \left({z}\right)}\right)$

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

$\forall \left({a, b}\right) \in \C^2: f_2 \left({a, b}\right) = \left({a, b, 0}\right)$

Let $f = f_2 \circ f_1$.

Let $F: \C \to \mathcal P \left({\R^3}\right)$ be defined as the mapping which takes $z$ to the closed line interval from $\left({0, 0, 1}\right)$ to $f \left({z}\right)$ for all $z \in \C$.


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

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

$R \left({z}\right) = F \left({z}\right) \cap G$

The set $R \left[{\C}\right] \cup \left\{{\left({0, 0, 1}\right)}\right\} $ is called the Riemann sphere, with the understanding that $f \left({\infty}\right) = \left({0, 0, 1}\right)$.


Also see


Source of Name

This entry was named for Georg Friedrich Bernhard Riemann.


Sources