Riemann sphere

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Riemann sphere

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Named after German mathematician Bernhard Riemann.

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.

• 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.

• (complex numbers with infinity): extended complex numbers
• (complex plane with point at infinity): closed complex plane, extended complex plane

• Argand plane
• complex manifold
• complex plane
• complex projective line (differently constructed set homeomorphic to the Riemann sphere)
• Riemann surface

• Complex plane on Wikipedia.Wikipedia
• Riemann surface on Wikipedia.Wikipedia