Riemann sphere

== English == === Etymology === Named after German mathematician Bernhard Riemann. === Noun === Riemann sphere (plural Riemann spheres) (topology, complex analysis) 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. (topology, complex analysis) 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 γ {\displaystyle \gamma } on the Riemann sphere Σ {\displaystyle \Sigma } which does not go through a given point P ∗ ∈ Σ {\displaystyle P^{*}\in \Sigma } divides Σ {\displaystyle \Sigma } into two parts, such that one part contains P ∗ {\displaystyle P^{*}} and the other does not. ==== 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 ==== (complex numbers with infinity): extended complex numbers (complex plane with point at infinity): closed complex plane, extended complex plane ==== Translations ==== === See also === Argand plane complex manifold complex plane complex projective line (differently constructed set homeomorphic to the Riemann sphere) Riemann surface === Further reading === Complex plane on Wikipedia.Wikipedia Riemann surface on Wikipedia.Wikipedia