# Riemann sphere

In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. It consists of the complex plane plus the point at infinity

[itex]\hat{\mathbb{C}} = \mathbb{C}\cup\{\infty\}.[itex]

This is just the one-point compactification of the complex plane, also known as the extended complex plane. Topologically, it is just a sphere, S2. The Riemann sphere is named after the geometer Bernhard Riemann.

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## Complex structure

The complex manifold structure on the Riemann sphere is specified by an atlas with two charts as follows

[itex]f:\hat{\mathbb{C}}\setminus\{\infty\} \to \mathbb{C},\ f(z)=z[itex]
[itex]g:\hat{\mathbb{C}}\setminus\{0\} \to \mathbb{C},\ g(z)=\frac{1}{z}\mbox{ and }g(\infty) = 0.[itex]

The overlap of these two charts is all points except 0 and ∞. On this overlap the transition function is given by z → 1/z, which is clearly holomorphic and so defines a complex structure.

The Riemann sphere has the same topology as S2, that is, the sphere of radius 1 centered at the origin in the Euclidean space R3. A homeomorphism between them is given by the stereographic projection tangent to the South Pole onto the complex plane. Labeling the points in S2 by (x1, x2, x3) where [itex]x_1^2 + x_2^2 + x_3^2 = 1[itex], the homeomorphism is

[itex](x_1, x_2, x_3)\to \frac{x_1-i x_2}{1-x_3}.[itex]

This maps the South Pole to the origin of the complex plane and the North Pole to ∞.

In terms of standard spherical coordinates (θ, φ), this map can be given as

[itex](\theta, \phi)\to e^{-i\phi}\cot\frac{\theta}{2}.[itex]

One can also use the stereographic projection tangent to the North Pole, which will map the North Pole to the origin and the South Pole to ∞. The formula is

[itex](x_1, x_2, x_3) \to \frac{x_1+i x_2}{1+x_3}[itex]

or, in spherical coordinates

[itex](\theta, \phi)\to e^{i\phi}\tan\frac{\theta}{2}.[itex]

## The complex projective line

The Riemann sphere can also be realized as the complex projective line, CP1. Explicitly, the isomorphism is given by

[itex][z_1, z_2]\leftrightarrow z_1/z_2[itex]

where [z1,z2] are homogeneous coordinates on CP1.

## Properties

In the category of Riemann surfaces, the automorphism group of the Riemann sphere is the group of Möbius transformations. These are just the projective linear transformations PGL2 C on CP1. When the sphere is given the round metric the isometry group is the subgroup PSU2 C (which is isomorphic to rotation group SO(3)).

The Riemann sphere is one of three simply-connected Riemann surfaces. The other two being the complex plane and the hyperbolic plane. This statement, known as the uniformization theorem, is important to the classification of Riemann surfaces.

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