# The Uniformization theorem The Uniformization Theorem: any [[Simply-connected topological space|simply connected]] [[Riemann surface]] there is a holomorphic bijection map to one and only one of the spaces $\widehat{\mathbb C}, \mathbb C, \mathbb D$ . This means that there are only 3 simply-connected Riemann surfaces (up to biholomorphism). $\widehat{\mathbb C}, \mathbb C, \mathbb D$ can be given unique canonical metrics with constant curvature: +1, 0, -1 respectively. These metrics are called spherical, Euclidean and hyperbolic. A well known theorem in Topology is the "classification theorem of closed surfaces" that state that 2D-manifold that is compact, closed, orientable and without boundary can be classified according to its genus. If its genus $g$ is 0 then the 2D-manifold is homeomorphic to the Sphere, if $g=1$ it is homeomorphic to a Torus and for $g>1$ it is homeomorphic to $g$ copies of Torus which are glued together (see image below). Note that this is a topological results that does not require the complex structure of a Riemann surface. ![[Pasted image 20260603174442.png|282]] A 2D-manifolds can be given a complex structure (not trivial fact) and so can be treated as Riemann surface. This Riemann surface (not necessarily simply-connected) has a covering space which is simply connected Riemann surface and so must be one of $\widehat{\mathbb C}, \mathbb C, \mathbb D$. There is a local-isometry from the covering space and the Riemann manifold (not a trivial fact) and so a canonical constant-curvature metric on the covering space is induced locally to the Riemann manifold. So each Riemann manifold has unique constant-curvature metric (canonical metric). The Gauss-Bonnet theorem state that for a closed, compact, orientable 2D-manifold $M$ without boundary we have $\int_M KdA=4\pi(1-g)$ where $g$ is the genus, $K$ is Gauss curvature , $dA$ area element. It is interesting theorem that connect two branches of mathematics: geometry on the left and topology on the right. A conclusion from this theorem is that $K=0\rightarrow g=1$ , $K=1\rightarrow g=0$ and for $K=-1\rightarrow g>1$. This is a remarkable result regarding the possible canonical geometries on 2D-manifolds that are compact, closed, orientable without boundary : Each has exactly one canonical geometry. The sphere has spherical geometry, the torus has Euclidean geometry and all the rest have hyperbolic geometry. ## Created 2026-05-29 13:40