Meaning of Riemann sphere | Babel Free

Definitions

1. 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.
2. 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.

Equivalents

Examples

“We use #92;hat#123;#92;Complex#125;#92;times (resp. #92;Complex) to denote the subset #92;#123;z#92;in#92;hat#92;Complex#58;z#92;ne 0#92;#125; (resp. #92;#123;z#92;in#92;hat#92;Complex#58;z#92;ne#92;infty#92;#125;) in the Riemann sphere #92;hat#92;Complex.”

“1967 [Prentice-Hall], Richard A. Silverman, Introductory Complex Analysis, Dover, 1972, page 22, Every circle γ on the Riemann sphere Σ which does not go through a given point P^*∈Σ divides Σ into two parts, such that one part contains P^* and the other does not.”

“In order to visualize the point at infinity, we consider the Riemann sphere that has radius 1#47;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 S) and the point diametrically opposite S the north pole (denoted by N). Let z be an arbitrary point in the complex plane, represented by the point P. We draw the line PN which intersects the Riemann sphere at the unique point P', distinct from N. Conversely, to each point P' on the sphere, other than the north pole N, we draw the line P'N which cuts uniquely one point P in the complex plane. Clearly, there exists a one-to-one correspondence between points on the Riemann sphere, except N, and all the finite points on the complex plane. We assign the north pole 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.”

“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.”

CEFR level