Uniformization theorem

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

The first case include all surfaces with positive Euler characteristic: the sphere and the real projective plane. The second includes all surfaces with vanishing Euler characteristic: the Euclidean plane, cylinder, Möbius strip, torus, and Klein bottle. The third case covers all surfaces with negative Euler characteristic: almost all surfaces are hyperbolic.