Editing Uniformization theorem (section)

{{short description|Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere}}

In mathematics, the '''uniformization theorem''' states that every [[simply connected]] [[Riemann surface]] is [[Conformal equivalence|conformally equivalent]] to one of three Riemann surfaces: the open [[unit disk]], the [[complex plane]], or the [[Riemann sphere]]. The theorem is a generalization of the [[Riemann mapping theorem]] from simply connected [[open set|open]] [[subset]]s of the plane to arbitrary simply connected Riemann surfaces.

Since every Riemann surface has a [[Universal Cover|universal cover]] which is a simply connected Riemann surface, the uniformization theorem leads to a classification of Riemann surfaces into three types: those that have the Riemann sphere as universal cover ("elliptic"), those with the plane as universal cover ("parabolic") and those with the unit disk as universal cover ("hyperbolic"). It further follows that every Riemann surface admits a [[Riemannian metric]] of [[constant curvature]], where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case. 

The uniformization theorem also yields a similar classification of closed [[Orientable manifold|orientable]] [[Riemannian manifold|Riemannian 2-manifolds]] into elliptic/parabolic/hyperbolic cases. Each such manifold has a conformally equivalent Riemannian metric with constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case.