Uniformization theorem

Every simply connected Riemann surface is conformally equivalent to the Riemann sphere, the complex plane, or the unit disk. This classifies all Riemann surfaces through their universal covering surfaces.

Riemann surface: A one-dimensional complex manifold whose charts have holomorphic transition maps. Uniformization classifies these surfaces by identifying the conformal type of each simply connected one.
Fuchsian group: A discrete subgroup of the group of orientation-preserving isometries of the hyperbolic plane. Many hyperbolic Riemann surfaces arise by quotienting the disk by a Fuchsian group.
Bernhard Riemann: A German mathematician whose work founded major parts of modern analysis, geometry, and number theory. Riemann introduced the surfaces and mapping problems that led to uniformization.
Conformal map: A differentiable map between surfaces that preserves angles and orientation at each point. Conformal equivalence is the relation asserted between a simply connected surface and its model.
Universal covering space: A simply connected covering space that maps onto a given connected, locally well-behaved space. Each Riemann surface is represented as a quotient of its simply connected universal cover.
Kleinian group: A discrete subgroup of the group of Möbius transformations of the Riemann sphere. Kleinian groups use conformal actions on the sphere to study hyperbolic three-dimensional geometry.
Felix Klein: A German mathematician known for work in geometry, group theory, and the Erlangen program. Klein developed a major approach to uniformization through automorphic functions and transformation groups.
Simply connected space: A path-connected space in which every loop can be continuously contracted to a point. Simple connectivity is the hypothesis that restricts the theorem to three models.
Deck transformation: An automorphism of a covering space that leaves the projection to the base space unchanged. Deck transformations describe how a surface is assembled as a quotient of its uniformizing model.
Moduli space of Riemann surfaces: A space whose points represent isomorphism classes of Riemann surfaces of a specified topological type. Uniformization supplies canonical geometric representatives for surfaces studied in moduli space.

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