# Definition:Riemann Surface

## Contents

## 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

- 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.

- 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.