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

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.

The complex manifold structure on the Riemann sphere is specified by an atlas with two charts and coordinates z and w

The transition function between the two patches is w = 1/z, which is clearly holomorphic and so defines a complex structure. To see that these charts give the topology of the sphere note that we can give an atlas on S² by stereographic projection onto the complex planes tangent to the north and south poles respectively. Labeling points in S² by (x₁, x₂, x₃) where , we have

which satisfies w = 1/z. In terms of standard spherical coordinates (θ, φ)

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

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

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.