nLab Riemann sphere

Idea

The 2-sphere with its canonical structure of a complex manifold is called the Riemann sphere.

As such this is complex projective space $\mathbb{C}P^1$.

Properties

The biholomorphisms, i.e. the bijective conformal transformations from the Riemann sphere to itself are the Möbius transformations.

Related concepts

• geometric quantization of the 2-sphere.

• celestial sphere

• branched cover of Riemann sphere

• stereographic projection

• complex projective space

  • complex projective plane

  • complex projective 3-space

References

See also

• Wikipedia, Riemann sphere

On the homotopy type of the space of rational functions from the Riemann sphere to itself (related to the moduli space of monopoles in $\mathbb{R}^3$ and to the configuration space of points in $\mathbb{R}^2$):

• Graeme Segal, The topology of spaces of rational functions, Acta Math. Volume 143 (1979), 39-72 (euclid:1485890033)