Definition:Riemann Surface

Definition
A Riemann surface is a complex manifold of dimension $1$.

Also defined as
Some authors do not require a Riemann surface to be connected or second-countable.

Note that by Radó's Theorem, a connected Riemann surface is automatically second-countable.

Elliptic, Parabolic and Hyperbolic Riemann Surface
Riemann surfaces can be categorised according to their shape:

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


 * Riemann Sphere is only Elliptic Riemann Surface
 * Parabolic Riemann Surface is Plane, Punctured Plane or Torus

Hence most Riemann surfaces are hyperbolic.