From this, a classification of surfaces follows. A surface is a quotient of one of: the complex plane (curvature 0), the Riemann sphere (curvature +1) or the unit disc (curvature -1 - hyperbolic plane) by a discrete group.

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

In the second case we can have only the Riemann sphere itself.

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