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 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.

Sources

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