Editing Uniformization theorem (section)

==Classification of closed oriented Riemannian 2-manifolds==
On an oriented 2-manifold, a [[Riemannian metric]] induces a complex structure using the passage to [[isothermal coordinates]]. If the Riemannian metric is given locally as

:<math> ds^2 = E \, dx^2 + 2F \, dx \, dy + G \, dy^2,</math>

then in the complex coordinate ''z'' = ''x'' + i''y'', it takes the form

:<math> ds^2 = \lambda|dz +\mu \, d\overline{z}|^2,</math>

where

:<math>\lambda = \frac14 \left( E + G + 2\sqrt{EG - F^2} \right),\ \ 
\mu = \frac1{4\lambda} (E - G + 2iF),</math>

so that ''λ'' and ''μ'' are smooth with ''λ'' > 0 and |''μ''| < 1. In isothermal coordinates (''u'', ''v'') the metric should take the form

:<math> ds^2 = \rho (du^2 + dv^2)</math>

with ''ρ'' > 0 smooth. The complex coordinate ''w'' = ''u'' + i ''v'' satisfies

:<math>\rho \, |dw|^2 = \rho |w_z|^2 \left| dz + {w_{\overline{z}}\over w_z} \, d\overline{z}\right|^2,</math>

so that the coordinates (''u'', ''v'') will be isothermal locally provided the [[Beltrami equation]]

:<math> {\partial w\over \partial \overline{z}} = \mu {\partial w\over \partial z}</math>

has a locally diffeomorphic solution, i.e. a solution with non-vanishing Jacobian.

These conditions can be phrased equivalently in terms of the [[exterior derivative]] and the [[Hodge star operator]] {{math|∗}}.<ref>{{harvnb|DeTurck|Kazdan|1981}}; {{harvnb|Taylor|1996a|pp=377&ndash;378}}</ref>
{{math|''u''}} and {{math|''v''}} will be isothermal coordinates if {{math|1=∗''du'' = ''dv''}}, where {{math|∗}} is defined
on differentials by {{math|1=∗(''p'' ''dx'' + ''q'' ''dy'') = −''q'' ''dx'' + ''p'' ''dy''}}.
Let {{math|1=∆ = ∗''d''∗''d''}} be the [[Laplace–Beltrami operator]]. By standard elliptic theory, {{math|''u''}} can be chosen to be [[harmonic]] near a given point, i.e. {{math|1=Δ ''u'' = 0}},  with {{math|''du''}} non-vanishing. By the [[Poincaré lemma]] {{math|1= ''dv'' = ∗''du''}} has a local solution {{math|''v''}} exactly when {{math|1=''d''(∗''du'') = 0}}. This condition is equivalent to {{math|1=Δ ''u'' = 0}}, so can always be solved locally. Since {{math|''du''}} is non-zero and the square of the Hodge star operator is &minus;1 on 1-forms, {{math|''du''}} and {{math|''dv''}} must be linearly independent, so that {{math|''u''}} and {{math|''v''}} give local isothermal coordinates.

The existence of isothermal coordinates can be proved by other methods, for example using the [[Beltrami equation#Solution in L2 for smooth Beltrami coefficients|general theory of the Beltrami equation]], as in {{harvtxt|Ahlfors|2006}}, or by direct elementary methods, as in {{harvtxt|Chern|1955}} and {{harvtxt|Jost|2006}}.

From this correspondence with compact Riemann surfaces, a classification of closed orientable Riemannian 2-manifolds follows. Each such is conformally equivalent to a unique closed 2-manifold of [[constant curvature]], so a [[Quotient space (topology)|quotient]] of one of the following by a [[Group action (mathematics)|free action]] of a [[discrete group|discrete subgroup]] of an [[isometry group]]:

#the [[sphere]] (curvature +1)
#the [[Euclidean plane]] (curvature 0)
#the [[Hyperbolic space|hyperbolic plane]] (curvature &minus;1).
<gallery>
File:Orange Sphere.png|genus 0
File:Orange Torus.png|genus 1
File:Orange Genus 2 Surface.png|genus 2
File:Orange Genus 3 Surface.png|genus 3
</gallery>
The first case gives the 2-sphere, the unique 2-manifold with constant positive curvature and hence positive [[Euler characteristic]] (equal to 2). The second gives all flat 2-manifolds, i.e. the [[torus|tori]], which have Euler characteristic 0. The third case covers all 2-manifolds of constant negative curvature, i.e. the ''hyperbolic'' 2-manifolds all of which have negative Euler characteristic. The classification is consistent with the [[Gauss–Bonnet theorem]], which implies that for a closed surface with constant curvature, the sign of that curvature must match the sign of the Euler characteristic. The Euler characteristic is equal to 2 – 2''g'', where ''g'' is the genus of the 2-manifold, i.e. the number of "holes".