Editing Riemann surface (section)

== Classification of Riemann surfaces ==
The set of all Riemann surfaces can be divided into three subsets: hyperbolic, parabolic and elliptic Riemann surfaces. Geometrically, these correspond to surfaces with negative, vanishing or positive constant [[sectional curvature]]. That is, every connected Riemann surface ''X'' admits a unique [[completeness (topology)|complete]] 2-dimensional real [[Riemannian manifold|Riemann metric]] with constant curvature equal to −1, 0 or 1 that belongs to the conformal class of Riemannian metrics determined by its structure as a Riemann surface. This can be seen as a consequence of the existence of [[isothermal coordinates]]. 

In complex analytic terms, the Poincaré–Koebe [[uniformization theorem]] (a generalization of the [[Riemann mapping theorem]]) states that every [[simply connected]] Riemann surface is conformally equivalent to one of the following:
* The Riemann sphere {{nowrap|1={{overset|lh=0.5|^|'''C'''}} := '''C''' ∪ {{mset|∞}}}}, which is isomorphic to [[complex projective line|'''P'''<sup>1</sup>('''C''')]]; 
* The complex plane '''C''';
* The [[open disk]] {{nowrap|1='''D''' := {{mset|''z'' ∈ '''C''' : {{abs|z}} < 1}}}}, which is isomorphic to the [[upper half-plane]] {{nowrap|1='''H''' := {{mset|''z'' ∈ '''C''' : Im(''z'') > 0}}}}. 
A Riemann surface is elliptic, parabolic or hyperbolic according to whether its [[universal cover]] is isomorphic to '''P'''<sup>1</sup>('''C'''), '''C''' or '''D'''. The elements in each class admit a more precise description. 

=== Elliptic Riemann surfaces ===
The Riemann sphere '''P'''<sup>1</sup>('''C''') is the only example, as there is no [[Group (mathematics)|group]] [[Group action (mathematics)|acting]] on it by biholomorphic transformations [[Group_action_(mathematics)#Types_of_actions|freely]] and [[Group_action_(mathematics)#Types_of_actions|properly discontinuously]] and so any Riemann surface whose universal cover is isomorphic to '''P'''<sup>1</sup>('''C''') must itself be isomorphic to it.

=== Parabolic Riemann surfaces ===
If ''X'' is a Riemann surface whose universal cover is isomorphic to the complex plane '''C''' then it is isomorphic to one of the following surfaces: 
* '''C''' itself; 
* The quotient {{nowrap|'''C''' / '''Z'''}}; 
* A quotient {{nowrap|'''C''' / ('''Z''' + ''τ'''''Z''')}}, where {{nowrap|''τ'' ∈ '''C'''}} with {{nowrap|Im(''τ'') > 0}}. 
Topologically there are only three types: the plane, the cylinder and the [[torus]]. But while in the two former case the (parabolic) Riemann surface structure is unique, varying the parameter ''τ'' in the third case gives non-isomorphic Riemann surfaces. The description by the parameter ''τ'' gives the [[Teichmüller space]] of "marked" Riemann surfaces (in addition to the Riemann surface structure one adds the topological data of a "marking", which can be seen as a fixed homeomorphism to the torus). To obtain the analytic [[moduli space]] (forgetting the marking) one takes the quotient of Teichmüller space by the [[mapping class group of a surface|mapping class group]]. In this case it is the [[modular curve]].

=== Hyperbolic Riemann surfaces ===
In the remaining cases, ''X'' is a hyperbolic Riemann surface, that is isomorphic to a quotient of the upper half-plane by a [[Fuchsian group]] (this is sometimes called a [[Fuchsian model]] for the surface). The topological type of ''X'' can be any orientable surface save the [[torus]] and [[sphere]]. 

A case of particular interest is when ''X'' is compact. Then its topological type is described by its [[genus (mathematics)|genus]] {{nowrap|''g'' ≥ 2}}. Its Teichmüller space and moduli space are {{nowrap|(6''g'' − 6)}}-dimensional. A similar classification of Riemann surfaces of finite type (that is homeomorphic to a closed surface minus a finite number of points) can be given. However, in general the moduli space of Riemann surfaces of infinite topological type is too large to admit such a description.