Finally, it’s time to get to the definition of a Riemann surface. A Riemann surface is a connected one-dimensional complex manifold. In other words, it’s a Hausdorff space ${M}$ which is locally homeomorphic to ${\mathbb{C}}$ via charts (i.e., homeomorphisms) ${\phi_i:U_i \rightarrow V_i}$ for ${U_i \subset M, V_i \subset \mathbb{C}}$ open and such that ${\phi_j \circ \phi_i^{-1}: V_i \cap V_j \rightarrow V_i \cap V_j}$ is holomorphic.

Examples

Evidently an open subset of a Riemann surface is a Riemann surface. In particular, an open subset of ${\mathbb{C}}$ is a Riemann surface in a natural manner.

The Riemann sphere ${P^1(\mathbb{C}) := \mathbb{C} \cup \{ \infty \}}$ or ${S^2}$ is a Riemann sphere with the open sets ${U_1 = \mathbb{C}, U_2 = \mathbb{C} - \{0\} \cup \{\infty\}}$ and the charts

$\displaystyle \phi_1 =z, \ \phi_2 = \frac{1}{z}.$

The transition map is ${\frac{1}{z}}$ and thus holomorphic on ${U_1 \cap U_2 = \mathbb{C}^*}$.

An important example comes from analytic continuation, which I will briefly sketch below. A function element is a pair ${(f,V)}$ where ${f: V \rightarrow \mathbb{C}}$ is holomorphic and ${V \subset \mathbb{C}}$ is an open disk. Two function elements ${(f,V), (g,W)}$ are said to be direct analytic continuations of each other if ${V \cap W \neq \emptyset}$ and ${f \equiv g }$ on ${V \cap W}$. By piecing together direct analytic continuations on a curve, we can talk about the analytic continuation of a function element along a curve (which may or may not exist, but if it does, it is unique).

Starting with a given function element ${\gamma = (f,V)}$, we can consider the totality ${X}$ of all equivalence classes of function elements that can be obtained by continuing ${\gamma}$ along curves in ${\mathbb{C}}$. Then ${X}$ is actually a Riemann surface.

Indeed, we must first put a topology on ${X}$. If ${(g,W) \in X}$ with ${W=D_r(w_0)}$ centered at ${w_0}$, then let a neighborhood of ${g}$ be given by all function elements ${(g_w, W')}$ for ${w \in W, W' \subset W}$; these form a basis for a suitable topology on ${X}$. Then the coordinate projections ${(g,W) \rightarrow w_0}$ form appropriate local coordinates. In fact, there is a globally defined map ${X \rightarrow \mathbb{C}}$, whose image in general will be a proper subset of ${\mathbb{C}}$.

Basic facts

Since we have local coordinates, we can define a map ${f: X \rightarrow Y}$ of Riemann surfaces to be holomorphic or regular if it is so in local coordinates. In particular, we can define a holomorphic complex function as a holomorphic map ${f: X \rightarrow \mathbb{C}}$; for meromorphicity, this becomes ${f: X \rightarrow S^2}$.

Many of the usual theorems of elementary complex analysis (that is to say, the local ones) transfer immediately to the case of Riemann surfaces. For instance, we can locally get a Laurent expansion, etc.

Theorem 1 Let ${f: X \rightarrow Y}$ be a regular map. If ${X}$ is compact and ${f}$ is nonconstant, then ${f}$ is surjective and ${Y}$ compact.

To see this, note that ${f(X)}$ is compact, and by the open mapping theorem, open, so the result follows by connectedness of ${Y}$.

Complexified differentials

Since a Riemann surface ${X}$ is a smooth 2-dimensional manifold, it is possible to do the usual exterior calculus. We could consider a 1-form to be a section of the (usual) cotangent bundle ${T^*(X)}$, but it is more natural to take the complexified cotangent bundle ${\mathbb{C} \otimes_{\mathbb{R}} T^*(X)}$, which I will in the future just abbreviate ${T^*(X)}$; this will not be confusing since I will only do this when I talk about complex manifolds. Sections of this bundle will be called (complex-valued) 1-forms. Similarly, we do the same for 2-forms.

I think the natural setting of all this leads to the theory of almost complex manifolds, which sometime soon I will try to learn and/or blog about.

If ${z = x+iy}$ is a local coordinate on ${X}$, defined say on ${U \subset X}$, define the (complex) differentials

$\displaystyle dz = dx + i dy , \ d\bar{z} = dx - idy.$

These form a basis for the complexified cotangent space at each point of ${U}$. There is also a dual basis

$\displaystyle \frac{\partial}{\partial z } := \frac{1}{2}\left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y}\right), \ \frac{\partial}{\partial \bar{z} } := \frac{1}{2}\left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)$

for the complexified tangent space.

I now claim that we can split the tangent space ${T(X) = T^{1,0}(X) + T^{0,1}(X)}$, where the former consists of multiples of ${\frac{\partial}{\partial z}}$ and the latter of multiples of ${\frac{\partial}{\partial \bar{z}}}$; clearly a similar thing is possible for the cotangent space. This is always possible locally, and a holomorphic map preserves the decomposition. One way to see the last claim quickly is that given ${g: U \rightarrow \mathbb{C}}$ for ${U \subset \mathbb{C}}$ open and ${0 \in U}$ (just for convenience), we can write

$\displaystyle g(z) = g(0) + Az + A' \bar{z} + o(|z|)$

where ${A = \frac{\partial g }{\partial z }(0), A' = \frac{\partial g }{\partial \bar{z} }(0)}$, which we will often abbreviate as ${g_z(0), g_{\bar{z}}(0)}$. If ${\psi: U' \rightarrow U}$ is holomorphic and conformal sending ${z_0 \in U' \rightarrow 0 \in U}$, we have

$\displaystyle g(\phi(\zeta)) = g(\phi(0)) + A \phi'(z_0)(\zeta-z_0) + A' \overline{ \phi'(z_0)(\zeta-z_0)} + o(|z|);$

in particular, ${\phi}$ preserves the decomposition of ${T_0(\mathbb{C})}$.

Given ${f: X \rightarrow \mathbb{C}}$ smooth, we can consider the projections of the 1-form ${df}$ onto ${T^{1,0}(X)}$ and ${T^{0,1}(X)}$, respectively; these will be called ${\partial f, \overline{\partial} f}$. Similarly, we define the corresponding operators on 1-forms: to define ${\partial \omega}$, first project onto ${T^{0,1}(M)}$ (the reversal is intentional!) and then apply ${d}$, and vice versa for ${\overline{\partial} \omega}$.

In particular, if we write in local coordinates ${\omega = u dz + v d\bar{z}}$, then

$\displaystyle \partial \omega = d( v d \bar{z}) = v_z dz \wedge d\bar{z},$

and

$\displaystyle \overline{\partial} \omega = d( u dz) = u_{\bar{z}} d\bar{z} \wedge dz.$

To see this, we have tacitly observed that ${dv = v_z dz + v_{\bar{z}} d\bar{z}}$.

Admittedly this isn’t the best place to stop.  Next time, I’ll discuss more about how one can manipulate 1- and 2-forms.  Pretty soon we can then get to some big theorems.