Lecture 10: Wednesday, September 11, 2013

In the late 1700’s and early 1800’s, the French mathematician Augustin-Louis Cauchy (1789 – 1857) and German mathematician Bernhard Riemann (1826 – 1866) worked independently to create a theory of differentiating complex-valued functions $f: \mathbb C \to \mathbb C$ . In order to define such functions, they realized they would need a way to define their domains of definition. In this lecture, we review some of the ideas surrounding Riemann Surfaces.

Riemann Surfaces

A compact, connected Riemann Surface is a triple $\left( X, \, \{ U_\alpha \}, \, \{ \mu_\alpha \} \right)$ satisfying the following five properties:

Atlas: For each $\alpha$ from a finite indexing set $I$ , the pair $(U_\alpha, \, \mu_\alpha)$ is a chart. That is, $X = \bigcup_{\alpha \in I} U_\alpha$ is a “cover” of $X$ , and $\mu_\alpha: \ U_\alpha \hookrightarrow \mathbb C$ is an injective map.

I like to think of an atlas $\bigl \{ (U_\alpha, \, \mu_\alpha) \, \bigl| \, \alpha \in I \bigr \}$ as a book, where the $\alpha \in I$ are the page numbers, the table contents is the collection $\bigl \{ \mu_\alpha \, \bigl| \, \alpha \in I \}$ of injective maps, and each page is a set $U_\alpha$ . Indeed, if you have an atlas of the Earth, you can lay out various photos of the globe as flat sheets of paper, but when you combine all of the photos in the atlas you recover a spherical planet.

Locally Euclidean: Each $\mu_\alpha(U_\alpha)$ is a connected, open subset of $\mathbb C$ . Moreover, the transition maps defined as the compositions $\omega_{\beta \alpha} = \mu_\beta \circ {\mu_\alpha}^{-1}$ are analytic functions, that is, complex-valued functions where the Cauchy-Riemann equations hold: $\dfrac {\partial u}{\partial x} = \dfrac {\partial v}{\partial y}$ and $\dfrac {\partial u}{\partial y} = - \dfrac {\partial v}{\partial x}$ where $\tau = x + i \, y$ while $\omega_{\beta \alpha} = u + i \, v$ .

The following diagram may be helpful to keep track of the transition maps:

$\begin{matrix} & \mathbb C & & X & & \mathbb C \\ & \uparrow & & \uparrow & & \uparrow \\ \omega_{\beta \alpha}: & \mu_\alpha \left( U_\alpha \cap U_\beta \right) & \overset{{\mu_\alpha}^{-1}}{\longrightarrow} & U_\alpha \cap U_\beta & \overset{\mu_\beta}{\longrightarrow} & \mu_\beta \left( U_\alpha \cap U_\beta \right) \end{matrix}$

Hausdorff: Points in $X$ can be separated by neighborhoods. That is, for distinct $z \in U_\alpha$ and $w \in U_\beta$ there exist open subsets ${\mathcal U}_\alpha, \, {\mathcal U}_\beta \subseteq \mathbb C$ such that
$\mu_\alpha(z) \in {\mathcal U}_\alpha \subseteq \mu_\alpha(U_\alpha)$ and $\mu_\beta(w) \in {\mathcal U}_\beta \subseteq \mu_\beta(U_\beta)$ in $\mathbb C$ while ${\mu_\alpha}^{-1}({\mathcal U}_\alpha) \cap {\mu_\beta}^{-1}({\mathcal U}_\beta) = \emptyset$ in $X$ .

Compact: Say that $\{ V_\gamma \}$ is an open cover of $X$ , that is, $X = \bigcup_{\gamma \in J} V_k$ while $\mu_\alpha(V_\gamma) \subseteq \mathbb C$ are open sets. Then we require that it has a finite subcover $X = V_{\gamma_1} \cup V_{\gamma_2} \cup \cdots \cup V_{\gamma_n}$ .

Connected: The only continuous functions $\omega: X \to \{ 0, \, 1 \}$ are the constant functions.

Here, we endow $\{ 0, \, 1 \}$ with the discrete topology (that is, every subset $Y \subseteq \{ 0, \, 1 \}$ is considered open), and say that a map $\phi: X \to \{ 0, \, 1 \}$ is continuous if the inverse image $\mu_\alpha \bigl( \phi^{-1}(Y) \bigr) \subseteq \mathbb C$ is open for every subset $Y \subseteq \{ 0, \, 1 \}$ and $\alpha \in I$ .

We often abuse notation and say “ $X$ is a Riemann surface” instead of “ $\left( X, \, \{ U_\alpha \}, \, \{ \mu_\alpha \} \right)$ is a compact, connected Riemann surface.” We say that a function $\phi: X \to \mathbb C$ is analytic on $X$ if each composition $\omega_\alpha = \phi \circ {\mu_\alpha}^{-1}$ is analytic as a function on the subset $\mu_\alpha(U_\alpha) \subseteq \mathbb C$ for each index $\alpha \in I$ .

Instead of defining $\left( X, \, \{ U_\alpha \}, \, \{ \mu_\alpha \} \right)$ as a Riemann surface, we can define it as an $n$ -dimensional real manifold if we replace $\mathbb C = \mathbb A^1(\mathbb C)$ with $\mathbb A^n(\mathbb R)$ and “analytic function” with “smooth function.” For this reason, we may think of a Riemann surface as a 1-dimensional complex manifold, that is, a complex curve.

The Riemann Sphere

Recall that the sphere of radius $r$ is the 2-dimensional surface $S^2(\mathbb R) = \left \{ (u,v,w) \in \mathbb A^3(\mathbb R) \, \biggl| \, u^2 + v^2 + w^2 = r^2 \right \}$ . We explain why $X = S^2(\mathbb R)$ is a Riemann surface.

We will use the following charts:

$\begin{aligned} U_1 & = \left \{ (u,v,w) \in S^2(\mathbb R) \ \biggl| \ (u,v,w) \neq (0,0,+r) \right \}, & & & \qquad \mu_1(u,v,w) & = \dfrac {u + i \, v}{+r - w}; \\[5pt] U_2 & = \left \{ (u,v,w) \in S^2(\mathbb R) \ \biggl| \ (u,v,w) \neq (0,0,-r) \right \}, & & & \mu_2(u,v,w) & = \dfrac {u - i \, v}{-r - w}. \\[5pt] \end{aligned}$

Since $U_1$ is just the sphere minus the “north pole” while $U_2$ is the sphere minus the “south pole”, we have the union $X = U_1 \cup U_2$ . It is easy to see that $\mu_1: U_1 \to \mathbb C$ and $\mu_2: U_2 \to \mathbb C$ are well-defined, injective maps because $\mu_1$ , for example, has inverse

$\sigma(x + i \, y) = \left( \dfrac {2 \, x}{x^2 + y^2 + 1} \, r, \ \dfrac {2 \, y}{x^2 + y^2 + 1} \, r, \ \dfrac {x^2 + y^2 - 1}{x^2 + y^2 + 1} \, r \right)$ .

This map is usually called stereographic projection; you should compare this formula with those given in Lecture 2. You’ll note that $U_1$ and $U_2$ are open sets regardless of whether you use the Euclidean topology on $S^2(\mathbb R) \simeq \mathbb P^1(\mathbb C)$ or the Zariski topology on $\text{Spec} \, \mathbb R[u,v,w]/(u^2 + v^2 + w^2 - r^2)$ .

Finally, it is easy to check that we have the transition maps on $\mu_\alpha(U_1 \cap U_2) = \mathbb C^\times = \mathbb C - \{ 0 \}$ :

$\bigl( \mu_\beta \circ {\mu_\alpha}^{-1} \bigr) (\tau) = \begin{cases} \tau & \text{if} \ \alpha = \beta, \\[5pt] -1/\tau & \text{if} \ \alpha \neq \beta. \end{cases}$
Clearly $\omega_{11}(\tau) = \tau$ and $\omega_{12}(\tau) = - 1/\tau$ are both examples of analytic functions because they are rational functions. We leave it as an exercise to show that points $(u,v,w)$ on $S^2(\mathbb R)$ can be separated by neighborhoods, that $S^2(\mathbb R)$ is compact, and that it is connected.

Endomorphisms of the Sphere

We end today’s lecture with a discussion about how rational functions are related to rotations of the sphere.

Let me begin with an observation and a definition which will be useful later. Fix two relatively prime polynomials $p(z) = a_n \, z^n + \cdots + a_1 \, z + a_0$ and $q(z) = b_m \, z^m + \cdots + b_1 \, z + b_0$ , that is, polynomials such that the $(n+m) \times (n+m)$ determinant

$\text{res}(p,q) = \det \left[ \begin{matrix} a_n & a_{n-1} & \cdots & a_0 & 0 & \cdots & 0 & 0 \\ 0 & a_n & \cdots & a_1 & a_0 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & a_n & a_{n-1} & \cdots & a_0 & 0 \\ 0 & 0 & \cdots & 0 & a_n & \cdots & a_1 & a_0 \\ b_m & b_{m-1} & \cdots & b_0 &0 & \cdots & 0 & 0 \\ 0 & b_m & \cdots & b_1 & b_0 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & b_m & b_{m-1} & \cdots & b_0 & 0 \\ 0 & 0 & \cdots & 0 & b_m & \cdots & b_1 & b_0 \\ \end{matrix} \right]$

of the Sylvester Matrix is nonzero. The rational function $f(z) = p(z) / q(z)$ defines a mapping $f$ which sends $\mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ , so we can use the ideas above to extend to a mapping $f^\ast= \sigma \circ f \circ \sigma^{-1}$ which sends $S^2(\mathbb R) \to S^2(\mathbb R)$ . I’ll explain.

For any radius $r$ , we define a map using stereographic projection: $f^\ast(u,v,w) = \left( \dfrac {2 \, \text{Re} \, \bigl[ p(z) \, \overline {q(z)} \bigr]}{|p(z)|^2 + |q(z)|^2} \, r, \ \dfrac {2 \, \text{Im} \, \bigl[ p(z) \, \overline {q(z)} \bigr]}{|p(z)|^2 + |q(z)|^2} \, r, \ \dfrac {|p(z)|^2 - |q(z)|^2}{|p(z)|^2 + |q(z)|^2} \, r \right)$ in terms of the complex number $z = \dfrac {u + i \, v}{r-w}$ . (Recall that $\overline{\tau} = x - i \, y$ is the complex conjugate of $\tau = x + i \, y$ .) Amazingly, even there are issues defining $f: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ when the denominator $q(z) = 0$ , the function $f^\ast: S^2(\mathbb R) \to S^2(\mathbb R)$ is always well-defined. We may consider this to be an endomorphism of the sphere because it maps the sphere to itself.

Let me give a simple example. Say that $f(z) = a \, z$ in terms of a complex number with $|a| = 1$ . Since $a = e^{i \theta}$ for some angle $\theta$ , we see that $f^\ast(u,v,w) = \bigl( u \, \cos \theta - v \, \sin \theta, \ u \, \sin \theta + v \, \cos \theta, \ w \bigr)$ is a rotation through angle $\theta$ .

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