# nLab Riemann sphere

Contents

### Context

#### Spheres

n-sphere

low dimensional n-spheres

complex geometry

# Contents

## 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.

## References

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 $\mathbb{R}^2$):