nLab Riemann sphere

Idea

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

As such this is:

• the complex 1-dimensional complex projective space $\mathbb{C}P^1$, hence the complex projective line,

• the complex Grassmannian of 1-dimensional complex linear subspaces of the complex plane, hence the double coset space of the unitary group $\mathrm{U}(2)$ by U(1), hence the quotient space by its maximal torus $\mathrm{U}(1) \times \mathrm{U}(1)$:

  $$\mathbb{C}P^1 \,\simeq\, Gr_1(\mathbb{C}^2) \,\simeq\, \mathrm{U}(2)/\big(\mathrm{U}(1) \times \mathrm{U}(1)\big) \,.$$

  (NB: There is also the famous coset space realization of the 2-sphere as $S^2 \,\simeq\, SU(2)/\mathrm{U}(1)$, witnessing the complex Hopf fibration, but this quotient presentation does not determine the complex structure that makes the 2-sphere into the Riemann sphere.)

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)