The Riemann Sphere,

**P**^{1} is the one point

compactification of the

complex plane **C**. The extra point is called

infinity. You can imagine the Riemann Sphere as a unit sphere in 3-d space centred at the origin, and the complex plane inhabiting the x-y plane of 3-space; the two intersecting on the unit circle. Map between the two as follows:

- If x on the sphere is not at (0,0,1) (the north pole), draw a line between the north pole and x, and the image of x in
**C** is the point where the line intersects the plane (this is called *stereographic projection*).
- If x is the north pole, define its image in
**P**^{1} to be the point at infinity.

This gives a bijection (indeed, homeomorphism) between

**P**^{1} and the sphere.

What is nice about this is that the rotations of the sphere correspond with (projective special unitary)

MÃ¶bius Transformations of the complex plane via this map. You can use this to show then that SO

_{3}(

**R**) is isomorphic to PSU

_{2}(

**C**).

The Riemann sphere is the simplest interesting example of a Riemann surface.