Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a special kind of 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, like the square root of a complex number or the logarithm. 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 kind of space that looks like the complex plane up close but can be stretched or twisted in interesting ways overall. Imagine covering a ball with maps — each map shows a small part clearly, and where maps overlap, they match up smoothly. This idea helps mathematicians study shapes using rules from complex numbers.

Another way to think about it is as a smooth surface, like a sphere or a torus, that also carries extra information allowing angles to be measured consistently. This makes Riemann surfaces important for understanding how complex functions behave on curved spaces.

Examples

Riemann surfaces can be created 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, meaning they are the same in practical use.

Every Riemann surface can be given a direction, making it orientable. Non-compact Riemann surfaces allow for changing holomorphic functions, while compact ones only allow constant functions, though they can still have special functions called meromorphic functions. All compact Riemann surfaces can also be represented as algebraic curves in higher spaces.

Each type of Riemann surface has special properties. For example, the Riemann sphere is perfectly round, while other surfaces can be stretched or twisted in various ways. 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, special rules govern how different Riemann surfaces can be connected through maps. Moving from certain 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 end up being unchanging, or "constant."

When we think of 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 the relationships between different Riemann surfaces.

Isometries of Riemann surfaces

The isometry group of a Riemann surface shows its geometry. For example, the sphere's isometry group is the Möbius group, while the plane's isometry group fixes infinity.

For surfaces with genus 1, like a torus, the isometry group includes translations and a 180° rotation. For higher genus surfaces (genus g ≥ 2), the isometry group is finite and has a maximum order of 84(g − 1). Special surfaces like the Bolza surface, Klein quartic, Bring's surface, and Macbeath surface maximize this order for their respective genera.

Main article: Hurwitz's automorphisms theorem

Function-theoretic classification

In studying Riemann surfaces, mathematicians use two main ways to classify them. Geometers often use what is called the geometric classification, which looks at the shape and curvature of the surface. However, complex analysts use a different method known as the function-theoretic classification. This method classifies a Riemann surface as parabolic if there are no special types of functions called negative subharmonic functions on it. If such functions exist, the surface is called hyperbolic.

The hyperbolic group can be further divided based on how certain types of functions behave. For example, some Riemann surfaces have special properties where only very simple functions can exist. This helps mathematicians understand and compare different Riemann surfaces better.