Riemann Sphere (thing)

The Riemann Sphere, P¹ 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 P¹ to be the point at infinity.

This gives a bijection (indeed, homeomorphism) between P¹ and the sphere.
What is nice about this is that the rotations of the sphere correspond with (projective special unitary) Möbius Transformations of the complex plane via this map. You can use this to show then that SO₃(R) is isomorphic to PSU₂(C).

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