Riemann sphere
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In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. It consists of the complex plane plus the point at infinity
- <math>\hat{\mathbb{C}} = \mathbb{C}\cup\{\infty\}.<math>
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
- <math>f:\hat{\mathbb{C}}\setminus\{\infty\} \to \mathbb{C},\ f(z)=z<math>
- <math>g:\hat{\mathbb{C}}\setminus\{0\} \to \mathbb{C},\ g(z)=\frac{1}{z}\mbox{ and }g(\infty) = 0.<math>
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 <math>x_1^2 + x_2^2 + x_3^2 = 1<math>, the homeomorphism is
- <math>(x_1, x_2, x_3)\to \frac{x_1-i x_2}{1-x_3}.<math>
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
- <math>(\theta, \phi)\to e^{-i\phi}\cot\frac{\theta}{2}.<math>
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
- <math>(x_1, x_2, x_3) \to \frac{x_1+i x_2}{1+x_3}<math>
or, in spherical coordinates
- <math>(\theta, \phi)\to e^{i\phi}\tan\frac{\theta}{2}.<math>
The complex projective line
The Riemann sphere can also be realized as the complex projective line, CP1. Explicitly, the isomorphism is given by
- <math>[z_1, z_2]\leftrightarrow z_1/z_2<math>
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.