---
tag: GTVQFG2SH9
kind: concept
name: Conformal geometry
msc: [53C18, 51B10]
sources:
  - url: https://en.wikipedia.org/wiki/Conformal_geometry
same_as:
  wikidata: Q513175
  wikipedia: Conformal geometry
---

## Summary

Conformal geometry studies spaces in which angles can be measured but lengths cannot.
There, a Riemannian metric is known only up to multiplication by a positive function.
Its maps are the [[conformal maps|0YYT36R4YD]], those that preserve angles.
In dimension $2$ there are very many conformal maps, such as all holomorphic functions with nonzero derivative.
In dimension at least $3$, [[Liouville's theorem|NH7J4YQMQC]] reduces them to the Möbius transformations.
The conformal geometry of the sphere is Möbius geometry, the subject of [[inversive geometry|HFBW3SXQVM]].

## Explanations

### L1 human

Imagine a map of the world that may stretch some regions more than others, but always keeps the angles between roads correct.
A conformal geometry remembers exactly this: angles, not distances.
The Mercator map is an example: it enlarges Greenland a lot, yet every angle between two streets is drawn true.
Two shapes count as the same in conformal geometry when one can be bent onto the other by such an angle-keeping map.