Riemann sphere

English

Etymology

Named after German mathematician Template:W.

Noun

1. Template:Lb The complex numbers extended with the number ∞; the complex plane (representation of the complex numbers as a Euclidean plane) extended with a single idealised point at infinity and consequently homeomorphic to a sphere in 3-dimensional Euclidean space.
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2. Template:Lb The 2-sphere embedded in Euclidean three-dimensional space and often represented as a unit sphere, regarded as a homeomorphic representation of the extended complex plane and thus the extended complex numbers.
   • 1967 [Prentice-Hall], Richard A. Silverman, Introductory Complex Analysis, Dover, 1972, page 22,

     Every circle $\gamma$ on the Riemann sphere $\mathrm{\Sigma }$ which does not go through a given point $P^{*}\in \mathrm{\Sigma }$ divides $\mathrm{\Sigma }$ into two parts, such that one part contains $P^{*}$ and the other does not.

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Usage notes

• A suitable (and often cited) homeomorphism is the one represented geometrically as a stereographic projection. In graphic representations of the projection, the Riemann sphere is an object in Euclidean space, while the projective plane (i.e., the complex plane) is itself a Euclidean representation of the complex numbers.

Synonyms

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Translations

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See also

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Further reading

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