Riemann Surfaces — Mathematics Knowledge System

Overview

A Riemann surface is a one-dimensional complex manifold — a surface on which complex analysis can be performed globally, removing the ambiguities of multi-valued functions. The complex logarithm, square root, and algebraic functions all become single-valued on appropriately constructed Riemann surfaces.

Intuition

Imagine unrolling the infinitely winding staircase of $\log z$ onto a flat surface — that surface is the Riemann surface of the logarithm. Each sheet of the staircase is a branch of $\log$, and they are glued together so that the function becomes single-valued and holomorphic everywhere on this new surface. Riemann surfaces give multi-valued complex functions a natural home.

Formal Definition

Definition

A Riemann surface is a connected Hausdorff topological space $X$ equipped with a complex atlas: a collection of charts $\{(U_\alpha, \phi_\alpha)\}$ where each $\phi_\alpha: U_\alpha \to \mathbb{C}$ is a homeomorphism, and the transition maps $\phi_\beta \circ \phi_\alpha^{-1}$ are holomorphic wherever defined.

\phi_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U_\beta) \to \phi_\beta(U_\alpha \cap U_\beta) \text{ is holomorphic}

Holomorphic transition maps

\chi(X) = 2 - 2g

Euler characteristic of genus-g compact surface

Worked Examples

The function $\sqrt{z}$ is two-valued: each nonzero $z$ has two square roots. We introduce a branch cut along the negative real axis and take two copies of $\mathbb{C}$ cut along this ray.
$\mathbb{C}_1 = \mathbb{C} \setminus (-\infty,0], \quad \mathbb{C}_2 = \mathbb{C} \setminus (-\infty,0]$
On sheet 1, take $\sqrt{z} = r^{1/2}e^{i\theta/2}$ with $\theta \in (-\pi,\pi)$. On sheet 2, take $\sqrt{z} = r^{1/2}e^{i(\theta+2\pi)/2} = -r^{1/2}e^{i\theta/2}$.
$w_1 = \sqrt{r}\,e^{i\theta/2}, \quad w_2 = -\sqrt{r}\,e^{i\theta/2}$
Glue the upper edge of the cut on sheet 1 to the lower edge on sheet 2 and vice versa. The resulting surface is connected and simply connected — topologically a sphere minus a point.
$S \cong \mathbb{P}^1(\mathbb{C})$

Answer: The Riemann surface of $\sqrt{z}$ is a two-sheeted cover of $\mathbb{C}$ branched at $z=0$ and $z=\infty$, homeomorphic to $\mathbb{P}^1$.

Practice Problems

Difficulty 7/10

Describe the Riemann surface of $\log z$ and explain why it is not compact.

Difficulty 8/10

Use Riemann–Hurwitz to find the genus of the hyperelliptic curve $w^2 = (z^2-1)(z^2-4)$.

Difficulty 9/10

Explain the uniformization theorem and its consequence for simply connected Riemann surfaces.

Common Mistakes

Common Mistake

A Riemann surface is just an ordinary real surface.

A Riemann surface is a 1-dimensional complex manifold (hence 2-dimensional over $\mathbb{R}$) with holomorphic transition maps, giving it a richer structure than a mere real surface.

Common Mistake

Every Riemann surface is simply connected.

Tori (genus $\geq 1$), punctured planes, and many other Riemann surfaces have nontrivial fundamental groups. The uniformization theorem classifies the simply connected ones.

Quiz

Summary

A Riemann surface is a 1-dimensional complex manifold with a holomorphic atlas.
They provide the natural domain for multi-valued complex functions, making them single-valued.
The Riemann–Hurwitz formula computes the genus of a covering surface from ramification data.
Compact Riemann surfaces are classified up to homeomorphism by their genus $g \geq 0$.
The uniformization theorem classifies simply connected Riemann surfaces as the sphere, plane, or upper half-plane.

References

WebsiteWikipedia — Riemann surface