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:
A Riemann surface is elliptic if and only if its universal cover is the Riemann sphere.
A Riemann surface is parabolic if and only if its universal cover is the complex plane.
A Riemann surface is hyperbolic if and only if its universal cover is the unit disk.
- 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.
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.
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Riemann surface