Riemann Surface - Classification of Riemann Surfaces

Classification of Riemann Surfaces

The realm of Riemann surfaces can be divided into three regimes: hyperbolic, parabolic and elliptic Riemann surfaces, with the distinction given by the uniformization theorem. Geometrically, these correspond to negative curvature, zero curvature/flat, and positive curvature: stating the uniformization theorem in terms of conformal geometry, every connected Riemann surface X admits a unique complete 2-dimensional real Riemann metric with constant curvature −1, 0 or 1 inducing the same conformal structure – every metric is conformally equivalent to a constant curvature metric. The surface X is called hyperbolic, parabolic, and elliptic, respectively.

For simply connected Riemann surfaces, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of the following:

elliptic

the Riemann sphere C ∪ {∞}, also denoted P1C

parabolic

the complex plane C, or

hyperbolic

the open disk D := {z ∈ C : |z| < 1} or equivalently the upper half-plane H := {z ∈ C : Im(z) > 0}.

The existence of these three types parallels the several non-Euclidean geometries.

The general technique of associating to a manifold X its universal cover Y, and expressing the original X as the quotient of Y by the group of deck transformations gives a first overview over Riemann surfaces.