# Definition:Riemann Sphere

Jump to navigation
Jump to search

This article needs to be linked to other articles.In particular: closed line interval, Riemann mapYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{MissingLinks}}` from the code. |

## 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 projectionYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Also see

## Source of Name

This entry was named for Georg Friedrich Bernhard Riemann.

## Sources

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Spherical Representation of Complex Numbers. Stereographic Projection - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Riemann sphere**