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:
- the Euclidean plane (curvature 0),
- the sphere (curvature +1), or
- 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.
