Definition:Riemann Surface

Definition

A Riemann surface is a connected 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:

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

• 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.

• 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 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.

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Riemann surface