They are one foundation for the theory of Riemann surfaces.

It natural to ask which Riemann surfaces arise in this way.

Considered as a Riemann surface, the open unit disk is therefore different from the complex plane.

Riemann surface theory shows that some restriction on M will be required.

In the following, note that all Riemann surfaces are orientable.

The complex plane C is the most basic Riemann surface.

Again a Riemann surface can be constructed, but this time the "hole" is horizontal.

There are several equivalent definitions of a Riemann surface.

It is the analogue of a Riemann surface in indefinite signature.

Suppose is a 1-form on a Riemann surface.

The map carrying the structure of the complex plane to the Riemann surface is called a chart.

One way of depicting holomorphic functions is with a Riemann surface.

This is a hyperbolic surface, in fact, a Riemann surface.

Thereby the notion of a meromorphic function can be defined for every Riemann surface.

The definition of a residue can be generalized to arbitrary Riemann surfaces.

The main point of Riemann surfaces is that holomorphic functions may be defined between them.

Cole later showed that this result cannot be extended to all open Riemann surfaces .

He also proved that a Riemann surface is topologically equivalent to a box with holes in it.

This is an example of a branched covering of Riemann surfaces.

These rotational transforms are connected to the theory of Riemann surfaces.

Therefore, the Riemann surface, or more simply its genus is a birational invariant.

Therefore, the genus is an important topological invariant of a Riemann surface.

In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important.

This yields the same Riemann surface "R" and function log as before.

Two conformally equivalent Riemann surfaces are for all practical purposes identical.