Uniformization theorem

In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gauss curvature. In fact, one can find a metric with constant Gauss curvature in any given conformal class.

From this, a classification of surfaces follows. A surface is a quotient of one of the following by a free action of a discrete subgroup of an isometry group:

1. the Euclidean plane (curvature 0),
2. the sphere (curvature +1), or
3. the hyperbolic plane (curvature -1)

The first case include all surfaces with zero Euler characteristic: a cylinder, torus, Möbius strip, Klein bottle or Euclidean plane. In the second case we have all surfaces with positive Euler characteristic: only the sphere and projective plane. The last case we have all surfaces with negative Euler characteristic; almost all surfaces are hyperbolic.

See also: Uniformization theorem, Almost all, Conformal map, Curvature, Cylinder, Discrete group, Euclidean plane, Euler characteristic, Group action, Hyperbolic plane