Examples of Riemann Surfaces: Part 1 (Updated: 21/09/2025)

Definition of Riemann surface: a complex manifold of dimension 1.

What is a complex manifold?

It is a Hausdorff and 2nd countable topological space X (this is the same as when we discussed differentiable manifolds from before) and the coordinate charts need to not only satisfy the usual consistency condition, but the transition maps must also be holomorphic (i.e. the map preserves angles and orientations).

Let’s look at some examples.

Example 1 (an easy one): the Riemann sphere or $\mathbb{CP}^1$ (space of lines passing through the origin of $\mathbb{C}^2$ )

Aside: how can we visualize a line, set of points $\{ (z_1, z_2) \in \mathbb{C}^2 : \frac{z_1}{z_2} \in \mathbb{C} or \frac{z_2}{z_1} \in \mathbb{C} \}$ , in $\mathbb{C}^2$ using 2 copies of the 2-dimensional plane?

$\mathbb{CP}^1 = \mathbb{C} \cup \{ \infty \}$ .

The holomorphic structure on the Riemann sphere is given by the following charts:

$\Omega_1 = \mathbb{CP}^1 \backslash \{ \infty \}, \phi_1 : \Omega_1 \rightarrow \mathbb{C}$ given by $\phi_1(z) = z$

$\Omega_2 = \mathbb{CP}^1 \backslash \{ 0 \}, \phi_2 : \Omega_2 \rightarrow \mathbb{C}$ given by $\phi_2(z) = \frac{1}{z}$ .

The transition map is $z \mapsto \frac{1}{z}$ restricted to the domain $\mathbb{C}^{*}$ (because this is the image of $\Omega_1 \cap \Omega_2 = \mathbb{CP}^1 \backslash \{ \infty, 0 \}$ for both $\phi_1, \phi_2$ ).

How do we know that there does not exist other non-equivalent atlases on the Riemann sphere?

Example 2 (an important one): this example will be the first Riemann surface we look at which is constructed as the quotient of another simpler Riemann surface under the action of a group.

Aside: there are 2 ways of constructing Riemann surfaces.

As quotients of other Riemann surfaces under the action of a group
As complex algebraic curves (will prove in a later post that Riemann surfaces are in fact the same thing as complex algebraic curves)

Consider $(\mathbb{Z}, +)$ as a subgroup of $(\mathbb{C}, +)$ . Then, the quotient of $\mathbb{C}$ under $(\mathbb{Z}, +)$ is $\mathbb{C}/\mathbb{Z} = \{ x + iy \in \mathbb{C} : x \text{mod 1} \}$ , which is topologically a cylinder.

Let’s define a Riemann structure on this surface by defining the following charts:

Let z \in \mathbb{C}$. Consider the disk $\mathbb{D}_{z_p}$ centered at $z_p$ with radius $\frac{1}{4}$ . Let $\pi : \mathbb{C} \rightarrow \mathbb{C}/\mathbb{Z}$ be the natural projection map.

Then, $\pi$ maps $\mathbb{D}_{z_p}$ bijectively to $\mathbb{C}/\mathbb{Z}$ . Let $\pi(z_p) = p$ and $\Omega_p = \pi(\mathbb{D}_{z_p})$ . It follows that $\phi_z: \Omega_p \rightarrow \mathbb{C}$ given by $\phi_z = \pi^{-1}|_{\Omega_p}$ is well-defined and injective.

We can cover $\mathbb{C}/\mathbb{Z}$ by charts of this form.

Claim: the transition maps are either identity or translation by an integer.

This can be shown in the diagram I have attached below:

The coordinate map corresponding to a chart depends on our choice of $z_p$ ‘s.

Prove: the Riemann surface $\mathbb{C}/\mathbb{Z}$ is biholomorphic to $\mathbb{C} \backslash \{ 0 \}$ .

Step 1: Find the continuous map from $\mathbb{C}/\mathbb{Z} \rightarrow \mathbb{C}^{*}$ .

A point in $\mathbb{C} \backslash \mathbb{Z}$ is given by $\{ (\tilde{x} + k) + iy: \tilde{x}, y \in \mathbb{R}, |\tilde{x}| < 1, \text{and } k \in \mathbb{Z} \}$ .

We take the exponential map given by:

$f: z \mapsto e^{2\pi i z}$

(it maps the vertical strip $\{ z : |\text{Re}(z)| \leq 1 \text{ and with the pairs of points } z_1, z_2 \text{ satisfying } \text{Re}(z_1) = - \text{Re}(z_2) = 1, \text{Im}(z_1) = \text{Im}(z_2) \text{ identified as one point} \}$ (which is topologically the same as $\mathbb{C}/\mathbb{Z}$ ) to the entire complex plane with the origin removed).

The inverse map is given by:

$f^{-1}: z \mapsto \frac{1}{2 \pi i} \{ ln(|z|) + i(arg(z) + 2\pi k) : k \in \mathbb{Z} \}$

Since

$\frac{1}{2 \pi i}\{ ln(|z|) + i(arg(z) + 2\pi k) : k \in \mathbb{Z} \}$

$= \{ (\frac{arg(z)}{2 \pi} + k) - \frac{i}{2\pi} ln|z| : k \in \mathbb{z} \}$

$= [\frac{arg(z)}{2 \pi} - \frac{i}{2 \pi} ln(|z|)] \in \mathbb{C}/\mathbb{Z}$

is a well-defined and unique point, it follows that the inverse map is well-defined and is injective.

Step 2: Show that this map is holomorphic.

On $\mathbb{C}^{*}$ , we take the collection of charts $(U_k, \text{id})$ for $k \in \mathcal{K}$ (where the $U_k$ ‘s are open simply-connected subsets of $\mathbb{C}^{*}$ and $\cup U_{k \in \mathcal{K}} = \mathbb{C}^{*}$ ) to be our atlas.

It suffices to show that $F = id \circ f: f^{-1}(U_k) = \{ [\frac{arg(z)}{2 \pi} - \frac{i}{2 \pi} ln(|z|)]: z \in U_k \} \rightarrow \mathbb{C}$ is holomorphic for any $U_k$ .

Let $p \in \mathbb{C}/\mathbb{Z}$ be arbitrary and $\pi$ be the natural projection map from $\mathbb{C} \rightarrow \mathbb{C}/\mathbb{Z}$ . Then $\pi^{-1}(p) = \{ p + k : k \in \mathbb{Z} \}$ . We define the inverse of $\pi|_{\mathbb{D}_p}$ , say $\phi_p$ (where $\mathbb{D}_p$ is the disk of radius 1/4 centered at p) to be the local coordinate map at $p \in \mathbb{C}/\mathbb{Z}$ (since the radius is small, this map is well-defined).

To show $F$ is holomorphic is to show $F \circ \phi_p^{-1}: \mathbb{D}_p \rightarrow \mathbb{C}$ is holomorphic, but this is just $z = x + iy \mapsto \{ (x+k) + iy : k \in \mathbb{z} \} = [\tilde{x} + iy] \mapsto e^{2\pi I (\tilde{x} + iy)} = e^{2\pi i(x+iy)} = e^{2\pi i z}$ which is holomorphic.

Step 3: Show that the inverse map is also holomorphic.

Here, we consider the map $\phi_p \circ f^{-1} \circ id: U \rightarrow \mathbb{C}$ (where $U$ is an open simply connected subset of $\mathbb{C}^{*}$ ).

Since $0 \notin U$ and $U$ is simply-connected, the logarithm map is defined on $U$ .

Hence, it follows that:

$\phi_p \circ f^{-1}: z \mapsto [\frac{arg(z)}{2 \pi} - \frac{i}{2 \pi} ln(|z|)] \mapsto \frac{arg(z)}{2 \pi} - \frac{i}{2 \pi} ln|z| \in \mathbb{C}$ (which is a well-defined holomorphic map on a simply-connected open subset not containing 0). $\square$

Examples of Riemann Surfaces: Part 1 (Updated: 21/09/2025)

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