The Riemann Sphere, P1
is the one point compact
ification 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 Transformation
s of the complex plane via this map. You can use this to show then that SO3
) is isomorphic to PSU2
The Riemann sphere is the simplest interesting example of a Riemann surface.