Definition:Riemann Surface

Definition
A Riemann Surface is a complex manifold of dimension 1.

Properties
Every Riemann surface is path-connected, since it is connected and locally path-connected.

Every Riemann surface is second countable (that is, its topology has a countable base) by Radò's Theorem.

The universal cover of any Riemann Surface is conformally isomorphic to either the Riemann Sphere, the complex plane or the unit disk by the Riemann Uniformization Theorem.

It follows that every Riemann Surface admits a metric of constant curvature. In particular, every Riemann Surface is metrizable.

Elliptic, Parabolic and Hyperbolic Riemann Surface
A Riemann Surface $X$ is called
 * elliptic if its universal cover is the Riemann Sphere;
 * parabolic if its universal cover is the complex plane;
 * hyperbolic if its universal cover is the unit disk.

Equivalently, a surface is elliptic, parabolic or hyperbolic depending on whether it admits a metric of constant positive, zero or negative curvature, respectively.

The Riemann Sphere is the only elliptic Riemann Surface (up to conformal isomorphism).

A parabolic Riemann Surface is conformally isomorphic to either the complex plane, the punctured plane $\C \setminus \left\{{0}\right\}$, or a torus. Hence most Riemann Surfaces are hyperbolic.