# Abstract Nonsense

## Riemann Surfaces (Pt. I)

Point of Post: In this post we formally introduce the notion of Riemann surfaces and discuss some important examples.

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Motivation

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We now begin what is, in my very humble and uninformed opinion, one of the most beautiful subjects in the entirety of  basic graduate mathematics–Riemann surfaces. Such a bold statement begs two immediate questions: what are Riemann surfaces, and why are they so pretty?

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The first question is one which has a simple, albeit somewhat esoteric, response–a Riemann surface is merely a one-dimensional connected complex manifold. What is such an object? Well, anyone who is likely to get a lot out of these posts probably is familiar with the concept of a smooth manifold, which is merely topological spaces with a well-defined notion of how to ‘do calculus’ on them. From this, it’s not hard to guess what a complex manifold is, it’s a topological space that has well-defined way of doing complex analysis on them. So, Riemann Surfaces are nothing more than (connected!) topological spaces which locally look like open subsets of $\mathbb{C}$, and that this local notion pieces together nicely enough to give a global notion of what a holomorphic mapping between two such Riemann surfaces looks like.

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Now that we have a very rough idea of what a Riemann surface should be, we can at least try to explain why the theory of Riemann surfaces is so beautiful. Everyone who has taken complex analysis at an advanced undegraduate/graduate level is aware of the fact that complex analysis is much more intimately (or at least more immediately!) related to algebra and topology than real analysis is. For example, for a domain $D\subseteq\mathbb{C}$ one has that the simple connectedness of $D$ is equivalent to every harmonic function $u:D\to\mathbb{R}$ admitting a holomorphic function $f:D\to\mathbb{C}$ such that $\text{Re}(f)=u$, which is equivalent to every holomorphic function admitting a primitive on $D$ (these allow us to define cohomology via complex analysis!).

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This pervasive feeling of deep algebraic and geometric connections will continue when we discuss Riemann surfaces. We shall prove some truly deep, and truly beautiful theorems in this vein. For example, we shall prove that, in a very precise sense, doing work with compact Riemann surfaces is the same thing as working with projective plane curves–in particular, we shall see that every algebraic function field (algebraic extension of $\mathbb{C}(z)$) is just the meromorphic functions on some compact Riemann surface. While I could go on and on about how interesting and amazing this subject is, I think that it would be better that I attempt to inject my paltry insight as we go along, and let you see for yourself why this subject makes me so excited.

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Riemann Surfaces and Examples in $\mathbb{C}$

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Let $X$ be a $2$-dimensional topological manifold where we shall identify $\mathbb{R}^2$ with $\mathbb{C}$. We call two charts $(U,\varphi)$ and $(V,\psi)$ on $X$ holomorphically compatible, or just compatible for short, if either $U\cap V=\varnothing$ or that $\psi\circ\varphi^{-1}:\varphi(U\cap V)\to\psi(U\cap V)$ is a biholmorphism. We call a collection $\{(U_\alpha,\varphi_\alpha)\}$ of compatible charts a holomorphic atlas (or just atlas for short) if $\{U_\alpha\}$ covers $X$. Just as in the case of smooth manifolds (the intuition for why we would want to consider maximal atlases is also there!) every atlas of $X$ is contained in a unique maximal atlas $\mathfrak{A}$, also called a complex structure on $X$. A Riemann surface is a connected $2$-dimensional topological manifold $X$ along with a maximal (holomorphic) atlas $\mathfrak{A}$. Just as the case of smooth manifolds, we shall often merely specify an atlas for a manifold with the intention of giving the manifold the Riemann structure of the maximal atlas that the given atlas is contained in.

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Note that if $X$ is any Riemann surface and $p\in X$, then there exists a chart $(U,\varphi)$ at $p$ such that $\varphi(U)$ is a disc centered at $p$. Indeed, let $(V,\psi)$ be any chart at $p$. Since $\psi(V)$ is open we know that there exists a disc $D_r(p)\subseteq \psi(V)$. We see then that $U=\psi^{-1}(D_r(p))$ is a neighborhood of $p$ and clearly that $(U,\varphi)$, with $\varphi=\psi\mid_U$, is a chart at $p$ such that $\varphi(U)=D_p(r)$ is a disc centered at $p$. We shall call such a chart a coordinate disc centered at $p$

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This is a good point out to put out an interesting difference between Riemann surfaces and smooth surfaces. The above proof, of course, goes through in the case of smooth manifolds and tells us that every point $p$ of a smooth manifold $M$ has a chart $(U,\varphi)$ with $\varphi(U)$ a disc $D_r(p)$ centered at $p$. Now, from this we see that every point of $M$ has a chart whose image is $\mathbb{R}^2$. Indeed, this follows immediately from the fact that $D_r(p)$ is diffeomorphic to $\mathbb{R}^2$. Of course, this does not work for Riemann surfaces, because $D_r(p)$ is not biholmorphic to $\mathbb{C}$ (else we’d have a non-constant entire map $f:\mathbb{C}\to D_r(p)$ which contradicts Liouville’s theorem).

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Remark: It turns out that assuming that $X$ is second countable is unnecessary, but this further level of generality isn’t of particular interest to us.

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Let’s give some examples of Riemann surfaces:

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The most obvious example of a Riemann surface is obviously the complex numbers $\mathbb{C}$ along with the atlas $\{\text{id}_\mathbb{C}\}$. We shall call this the standard structure on $\mathbb{C}$, and shall always (unless specified otherwise) give $\mathbb{C}$ this complex structure.

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If $X$ is a Riemann surface with atlas $\{(U_\alpha,\varphi_\alpha)\}$ and $W\subseteq X$ is a domain, it’s trivial that $\{(U_\alpha\cap W,\varphi_\alpha\mid_{W\cap U_\alpha})\}$ is an atlas on $W$. We call the pursuant complex structure the induced structure on $W$.

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The above tells us that for any domain $D\subseteq\mathbb{C}$ one has that $D$ carries the structure induced by the standard structure on $\mathbb{C}$–this shall be called the standard structure on $D$. A particularly important example of this is when $D=\mathbb{D}$, where $\mathbb{D}$ is the unit disc.

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Now, being the simple minded students of mathematics that we are, we continue looking for new Riemann surfaces inside of $\mathbb{C}$ (obviously meaning domains with the induced structure), and wanting to keep things topologically simple, we decide that we want simply connected Riemann surfaces. We look for a while, and we can’t find (up to equivalence!) any new ones. This is not surprising. In fact, the famous Riemann Mapping Theorem says that we’ve already found them all–the disc and $\mathbb{C}$. To be more explicit, the Riemann Mapping Theorem tells us that if $D$ is a simply connected domain $D\subset \mathbb{C}$ then $D$ is biholomorphic to $\mathbb{D}$, and is thus the ‘same’ Riemann surface.

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References;

[1] Varolin, Dror. Riemann Surfaces by Way of Complex Analytic Geometry. Providence, RI: American Mathematical Society, 2011. Print.

[2] Miranda, Rick. Algebraic Curves and Riemann Surfaces. Providence, RI: American Mathematical Society, 1995. Print.

[3] Forster, Otto. Lectures on Riemann Surfaces. New York: Springer-Verlag, 1981. Print.

[4] Conway, John B. Functions of One Complex Variable. New York: Springer-Verlag, 1978. Print.

[5] Gong, Sheng. Concise Complex Analysis. Singapore: World Scientific, 2001. Print

[6] Markley, Nelson Groh. Topological Groups: An Introduction. Hoboken, NJ: Wiley, 2010. Print.