Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a special shape that helps us understand complex numbers better. These surfaces were first studied by the mathematician Bernhard Riemann and are named after him.

Imagine taking the complex plane, which is like a flat piece of paper where every point has a complex number, and then bending or stretching it in interesting ways. Even though each small part of a Riemann surface looks like the complex plane, the whole shape can be very different — it might look like a sphere, a torus, or even several sheets glued together.

Riemann surfaces are useful because they help us handle functions that have more than one value. For example, the square root of a number isn’t just one value; it has two possible answers. Riemann surfaces let us keep track of all these values in a neat way.

Every Riemann surface is a two-dimensional shape, but it also has extra information called a complex structure. This makes it different from ordinary surfaces. Only certain shapes can become Riemann surfaces — they need to be orientable and metrizable. Shapes like a sphere or a torus can be Riemann surfaces, but others like the Möbius strip or the Klein bottle cannot. These surfaces are important in many areas of math and physics because they connect geometry and complex numbers in powerful ways.

Definitions

Further information: Complex manifold and Conformal geometry

A Riemann surface is a special space that looks like the complex plane up close but can be stretched or twisted in interesting ways. Think of covering a ball with maps — each map shows a small part clearly, and where maps overlap, they match up smoothly. This helps mathematicians study shapes using rules from complex numbers.

It is also like a smooth surface, such as a sphere or a torus, that lets us measure angles in a consistent way. This makes Riemann surfaces useful for understanding how complex functions work on curved spaces.

Examples

Riemann surfaces are made from special math rules, or functions, that help us understand complex numbers better. Some of these functions include arcsin z, log z, and the square root of z.

Pictures of these surfaces can show how the rules twist and turn in interesting ways, like the cube root of z or the fourth root of z. These images help us see how Riemann surfaces look different from the regular complex plane.

Further definitions and properties

A function between two Riemann surfaces is called holomorphic if it follows special smooth rules. When two Riemann surfaces can be matched perfectly with such a function, they are called biholomorphic.

Every Riemann surface can be given a direction, making it orientable. Some Riemann surfaces allow for changing holomorphic functions, while others only allow constant functions. These ideas help mathematicians understand the shapes and properties of complex numbers and functions.

Main article: Riemann mapping theorem

Maps between Riemann surfaces

In mathematics, there are special rules for how different Riemann surfaces can be connected. Moving from some types of Riemann surfaces to others is easy, but going the other way is very limited.

For example, maps from one type to another often do not change and stay the same, or "constant."

When we look at a Riemann sphere with missing points, or "punctures," we see different behaviors. A sphere with no punctures acts in one way, while adding punctures changes its properties. Some maps between these punctured spheres are possible, but many end up being constant. These ideas help us understand how different Riemann surfaces relate to each other.

Isometries of Riemann surfaces

The isometry group of a Riemann surface shows its shape. For example, the sphere's isometry group is the Möbius group.

For surfaces with genus 1, like a torus, the isometry group includes moving and turning.

Main article: Hurwitz's automorphisms theorem

Function-theoretic classification

Mathematicians classify Riemann surfaces in two main ways. Geometers use the geometric classification, which looks at the shape and curves of the surface. Complex analysts use the function-theoretic classification.

This method calls a Riemann surface parabolic if there are no special functions called negative subharmonic functions on it. If such functions exist, the surface is hyperbolic.

The hyperbolic group can be divided more based on how certain functions act. This helps mathematicians understand and compare different Riemann surfaces better.