Riemann sphere
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fr:Sphère de Riemann
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 and coordinates z and w
- <math>z:\hat{\mathbb{C}}\setminus\{\infty\} \to \mathbb{C}\,<math>
- <math>w:\hat{\mathbb{C}}\setminus\{0\} \to \mathbb{C}\,<math>
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 <math>x_1^2 + x_2^2 + x_3^2 = 1<math>, we have
- <math>z = \frac{x_1+i x_2}{1+x_3}<math>
- <math>w = \frac{x_1-i x_2}{1-x_3}<math>
which satisfies w = 1/z. In terms of standard spherical coordinates (θ, φ)
- <math>z = e^{i\phi}\tan\frac{\theta}{2}<math>
- <math>w = e^{-i\phi}\cot\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.
