Riemann sphere

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Riemann sphere

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Etymology

Named after German mathematician Bernhard Riemann.

Noun

Riemann sphere (plural Riemann spheres)

1. (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.
   • 2002, Wenhua Zhao, “Some Generalizations of Genus Zero Two-Dimensional Conformal Field Theory”, in Stephen Berman, Paul Fendley, Yi-Zhi Huang, Kailash Misra, Brian Parshall, editors, Recent Developments in Infinite-dimensional Lie Algebras and Conformal Field Theory, American Mathematical Society, page 306:

     We use ${\displaystyle {\hat {\mathbb {C} }}^{\times }}$ (resp. ${\displaystyle \mathbb {C} }$) to denote the subset ${\displaystyle \{z\in {\hat {\mathbb {C} }}:z\neq 0\}}$ (resp. ${\displaystyle \{z\in {\hat {\mathbb {C} }}:z\neq \infty \}}$) in the Riemann sphere ${\displaystyle {\hat {\mathbb {C} }}}$.

2. (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 , Richard A. Silverman, Introductory Complex Analysis, Dover, 1972, page 22,

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

   • 2002, Yue Kuen Kwok, Applied Complex Variables for Scientists and Engineers, Cambridge University Press, page 27:

     In order to visualize the point at infinity, we consider the Riemann sphere that has radius ${\displaystyle 1/2}$ and is tangent to the complex plane at the origin (see Figure 1.8). We call the point of contact the south pole (denoted by ${\displaystyle S}$) and the point diametrically opposite ${\displaystyle S}$ the north pole (denoted by ${\displaystyle N}$). Let ${\displaystyle z}$ be an arbitrary point in the complex plane, represented by the point ${\displaystyle P}$. We draw the line ${\displaystyle PN}$ which intersects the Riemann sphere at the unique point ${\displaystyle P'}$, distinct from ${\displaystyle N}$. Conversely, to each point ${\displaystyle P'}$ on the sphere, other than the north pole ${\displaystyle N}$, we draw the line ${\displaystyle P'N}$ which cuts uniquely one point ${\displaystyle P}$ in the complex plane. Clearly, there exists a one-to-one correspondence between points on the Riemann sphere, except ${\displaystyle N}$, and all the finite points on the complex plane. We assign the north pole ${\displaystyle N}$ as the point at infinity. With such an assignment, we then establish a one-to-one correspondence between all the points on the Riemann sphere and all the points in the extended complex plane. This correspondence is known as the stereographic projection.

   • 2003, Philip L. Bowers, Monica K. Hurdal, “Planar Conformal Mappings of Piecewise Flat Surfaces”, in Hans-Christian Hege, Konrad Polthier, editors, Visualization and Mathematics III, Springer, page 6:

     We find that many mathematicians, even those who specialize in complex analysis and conformal geometry, are not familiar with the inversive distance between pairs of circles in the Riemann sphere.

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