Editing Uniformization theorem (section)

==Classification of connected Riemann surfaces==

Every [[Riemann surface]] is the quotient of the free, proper and holomorphic action of a [[discrete group]] on its universal covering and this universal covering, being a simply connected Riemann surface, is holomorphically isomorphic (one also says: "conformally equivalent" or "biholomorphic") to one of the following:

#the [[Riemann sphere]]
#the complex plane
#the unit disk in the complex plane.
For compact Riemann surfaces, those with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1, all with non-abelian fundamental group; those with universal cover the complex plane are the Riemann surfaces of genus 1, namely the complex tori or [[elliptic curve]]s with fundamental group {{math|'''Z'''<sup>2</sup>}}; and those with universal cover the Riemann sphere are those of genus zero, namely the Riemann sphere itself, with trivial fundamental group.