Riemann Surfaces

For a quick definition of many of the terms used here, you may refer to the Glossary.

Contents

• Introduction
• The formal definition of a Riemann surface
• Symmetries of Riemann surfaces
• Algebraic functions

Introduction

In simplest terms, Riemann surfaces were invented because the square root operation is not a single-valued function. That is, the equation y^2 = x does not define a single valued function (the dependent variable y) of the independent variable x. For almost all values of x (a complex number), there are exactly two possible values of y.

A multi-valued function isn't really a function at all in the mathematical sense, and can't be dealt with directly except in the most cumbersome ways. (Especially with respect to anything more complicated than a square root.) Yet multi-value functions arise quite often, as inverse functions for instance. The square root is the inverse of x->x^2. Another example is the logarithm, which is the inverse of x->e^x.

The resolution of this problem is obvious and ingenious at the same time. Since the problem is that there can't be two different values y corresponding to the same x, we simply increase the number of x's so that there are enough for each possible y. We need to do this consistenly, so that there is a concept of when two points are near to each other, i. e. a topology. Intuitively, x is a point in the complex numbers C, so what we do is work with a suitable numbers of copies of C.

How many copies of C we need depends on the function in question. For y^2 = x we need two copies. For any given x, we pick one of two possible values of y to be sqrt(x) and let that be the value of the function at x on one of the copies of C. We let -sqrt(x) be the value of the function at x on the other copy of C. Since we are dealing with a continuous function, it is well-defined what the function value should be in an entire neighborhood of x on each copy of C.

This gives us a perfectly well-defined single-valued function on the new topological space consisting of two copies of C, where each copy has the same topology as C itself. This, really, is all there is to Riemann surfaces.

There is one technical difficulty, which is what to do when there is some collapsing of multi-valuedness. With y^2 = x, the point x=0 has only the corresponding value y=0. Since there is only one value here, we don't have any clue how to define which of the two possible value to pick for the function at points in the neighborhood of 0.

The problem is that in any neighborhood of 0 it is not clear how to perform the operation of "analytic continuation". A point like this is called a point of ramification, and it has to be dealt with specially when defining the topology on the Riemann surface.

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