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Uniformization theorem

In mathematics, the 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. 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.