Definition:Riemann Surface

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

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.

Also see

 * Properties of Riemann Surface


 * Riemann Surface is Path-Connected
 * Riemann Surface is Second Countable
 * Conformal Isomorphism of Universal Cover of Riemann Surface
 * Riemann Surface is Metrizable
 * Riemann Surface admits Metric of Constant Curvature