---
tag: 5K0N8QMV6Q
kind: concept
name: Riemann surface
msc: [30F10, 30F20, 14H55]
sources:
  - url: https://en.wikipedia.org/wiki/Riemann_surface
same_as:
  wikidata: Q753035
  wikipedia: Riemann surface
---

## Summary

A Riemann surface is a connected space that looks locally like an open piece of the complex plane, with changes of coordinates that are holomorphic.
It is a complex manifold of complex dimension one.
On a Riemann surface one can speak of [[holomorphic|0DG0Z72CQD]] and [[meromorphic|0NSD1TPHR0]] functions.
This makes it the natural home of multivalued functions such as $\sqrt{z}$ and $\log z$.
Compact Riemann surfaces are the same objects as smooth projective complex algebraic curves.

## Explanations

### L1 human

Imagine a surface built by gluing pieces of the complex plane, where the gluing maps are holomorphic.
Because the gluing respects holomorphy, you can tell whether a function on the surface is holomorphic by looking at it in any piece.
The plane itself, the [[Riemann sphere|0CDMXGPSES]] and the torus $\C/(\Z + i\Z)$ are examples.
The graph of $w^2 = z$ is a Riemann surface on which $\sqrt{z}$ becomes a single-valued function, namely $w$.