Uniformization theorem

In mathematics, Uniformization theorem for surfaces says that any surface admits a metric of constant curvature in thermal coordinates. In other words, any surface has a complex structure and a metric of constant curvature - either 0, -1, or +1.

From this, a classification of surfaces follows. Surface is a quotient of either a complex plane (curvature 0), Riemanian sphere (curvature +1) or unit disc (curvature -1 - hyperbolic plane) by a discrete group.

The first case is just a cylinder, torus or a complex plane.

The second casewe can have only Riemanian sphere itself.

The last case is the most important, and almost all surfaces are hyperbolic.