Definition:Riemann Surface

Definition

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

Also defined as

Some authors do not require a Ri|emann 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:

Elliptic

A Riemann surface is elliptic if and only if its universal cover is the Riemann sphere.

Parabolic

A Riemann surface is parabolic if and only if its universal cover is the complex plane.

Hyperbolic

A Riemann surface is hyperbolic if and only if its universal cover is the unit disk.

Also see

Hence most Riemann surfaces are hyperbolic.

• Results about Riemann surfaces can be found here.

Source of Name

This entry was named for Georg Friedrich Bernhard Riemann.

Historical Note

The Riemann surface was a device invented by Bernhard Riemann as a device for clarifying the nature of multiple-valued functions in the complex plane.

This invention led directly to the Riemann Mapping Theorem.