From what I’ve heard, I go to a school that somewhat downplays complex analysis: we have the obligatory third-year course, but nobody really talks much, if at all, about the freakier things like analytic continuation and Riemann surfaces. I took analytic number theory a year ago, and it reared its head here and there, for example in continuing the Riemann zeta function to all of and brandishing the scimitar of residue calculus to slay a few belligerent contour integrals, but it’s not as if I was swinging around Rouché’s theorem every 5 minutes or anything.
That being said, at some point down the sleep-deprived line I call my life, I chugged peach juice ad nauseum, and then decided I wanted to learn about Riemann surfaces. Before I begin studying them formally, I thought it would be fitting to have a likely-useless discussion at the outset of which I claim I’m going to give you a very colloquial overview of the subject, but in reality just end up throwing around a bunch of buzzwords that, if you’re lucky, might leave some flavour in your mouth, which you can then decide is either wonderful or disgusting.
I initially planned this post to be a video, but I realized it would consist of me standing in front of a blackboard for 20 minutes ranting about stuff without writing anything down. If you’re reading this, hopefully you’ve got complex analysis behind you, or at least know what a holomorphic/complex differentiable/analytic function is. If not, I guess you can just take things on faith and use your imagination!
A Riemann surface, very roughly, is nothing more than a space that locally “looks like” the complex plane, where “looks like” means something like “carries the analytic structure of”: thus, as I will eventually elaborate, their geometry is locally rather simple, but globally can be highly nontrivial.
The study of Riemann surfaces is a branch of the enormous field known as complex geometry, which is concerned with complex-analytic manifolds. These are just like real manifolds except they locally look like instead of , and the “transition functions” between the charts have to be analytic in the sense of complex analysis (if you know what that means). Indeed, Riemann surfaces are simply complex manifolds of dimension one (although we sometimes assume them to be connected and stuff). It’s a special case where things tend to work out particularly elegantly and beautifully.
Although the geometry of these things can be quite difficult, complex analysis is also extremely rigid. We have a pretty huge repertoire with which to attack these creatures: the Cauchy integral formula, the residue theorem, Liouville’s theorem, the maximum modulus principle, the argument principle, Rouché’s theorem, and so on.
Two immediate examples of Riemann surfaces are the complex plane itself, and the “extended complex plane” (or Riemann sphere) , which you can get by stereographic projection. To be more explicit, imagine a sphere sitting on the complex plane, and call its north pole . Given any point in the plane, you draw a line from to , which will intersect the sphere in exactly one other point. In this sense, every point on the sphere except is mapped to its own point in the plane. To make this into a bijection we just throw in another point to , call it , and map it to . Of course, the resulting surface is now compact! This can be pretty convenient.
The familiar concepts from complex analysis, like the notions of holomorphic and meromorphic functions, poles, essential singularities, and so on, all carry over to the setting of Riemann surfaces, and we find ourselves in a position to formulate and prove theorems like “every holomorphic function on a compact Riemann surface is constant” (which may remind you of Liouville’s theorem), or “every meromorphic function on the Riemann sphere is rational“. We usually concentrate on meromorphic functions (whose singularities are limited to being poles), because a theorem of Picard says that all hell breaks loose near an essential singularity, so screw those things.
In fact, there are these things called “divisors“: basically they’re just formal sums over all points of your Riemann surface , with all integers, and only finitely many nonzero. By measuring the orders of zeroes and poles of a given function at each point , we naturally obtain a divisor (how?). Divisors obtainable in this fashion are called principal divisors, and one can ask the nontrivial question “which divisors are principal?”. What’s more, on a compact Riemann surface, the meromorphic functions are determined up to multiplication by a constant by their divisor! Again we observe the rigidity so characteristic of complex analysis.
When any survivor of a course in complex analysis hears the words “complex logarithm“, they will no doubt break out in a cold sweat, shuddering as the memories of their own horrendous struggles to define the logarithm on a subset of the complex plane flash through their minds. What’s more, you had to go against your most fundamental morals, perpetrating inhumane cruelties like branch cuts — and at the end of the day your hands were covered in the blood of multi-valued functions.
You may not know it, but what you did was wrong. In fact, the logarithm is not happy to be defined on a subset of , and polar coordinates (specifically, the argument of a complex number ) lie at the heart of its contention. Rather, its natural domain of definition is an instance of a Riemann surface!
This was precisely Riemann’s idea: to a given (sufficiently nice) complex function, one can systematically associate a Riemann surface, on which that function must be studied. Furthermore, the geometry of this resulting surface reflects profound characteristics of the function. For example, all Riemann surfaces are orientable, so if they’re compact, we can associate to them a very important topological invariant, their genus. Another crowning achievement is the so-called Riemann existence theorem which shows that, in some sense, we do not really lose anything by studying those Riemann surfaces arising in this way.
Okay, buzzwords! Think, think! Um… you may have heard of elliptic curves. They seem algebraically simple, but they admit an awe-inspiringly intricate arithmetic. A lot of number theorists, geometers and cryptographers practically rubberneck at their very mention, and their understanding played a turnkey role in Wiles’ celebrated proof of Fermat’s Last Theorem. Hanc marginis exiguitas non caparet indeed. Anyway, check it: one can actually show that elliptic curves are Riemann surfaces — more specifically, they’re complex tori! One can ask: what are the meromorphic functions on a complex torus? This question leads directly to the so-called “theta functions” of number-theoretic ubiquity.
There are also many interesting questions you can ask about, say, covering spaces of Riemann surfaces. It turns out that you can get a Galois-like correspondence: loosely speaking, the -sheeted coverings of certain Riemann surfaces correspond exactly to the index subgroups of an associated fundamental group.
What else? Well, just as you have things like the Haar integral in the setting of Lie groups, you can also talk about differential forms, and integration on Riemann surfaces. Also, since Riemann surfaces carry a “conformal geometry”, that means that any notions invariant under conformal transformations, like the Laplace operator, make sense. So it turns out you can attack certain problems with the machinery of elliptic partial differential equations, and even some physical intuition. Finally, there are a lot of other methods like Cech cohomology; despite knowing a decent amount of category theory, I’m admittedly pretty hopeless homologically, which is something I hope to change soon.
Of course, at the end of the day, all geometry is basically just the study of spaces equipped with some particular kind of sheaf…