---
tag: PZCJV40M4K
kind: concept
name: Systolic geometry
msc: [53C23, 53C20]
sources:
  - url: https://en.wikipedia.org/wiki/Systolic_geometry
same_as:
  wikidata: Q2454208
  wikipedia: Systolic geometry
---

## Summary

Systolic geometry studies the systole of a Riemannian manifold, the length of its shortest closed curve that cannot be shrunk to a point.
Its main questions compare the systole with the volume of the manifold.
Typical results say that a manifold with a long systole must have a large volume, as in [[Loewner's torus inequality|1Y9X1YW6GH]] and [[Gromov's systolic inequality|HB2K72PSZ6]].
The field links Riemannian geometry with topology and with the geometry of numbers.

## Explanations

### L1 human

On a doughnut surface some loops cannot be pulled tight to a point, such as a loop around the hole.
The systole is the length of the shortest such loop.
If all these loops are long, the surface must be large: it cannot have small area.
For a flat square torus made from a unit square, the systole is $1$ and the area is $1$.
Systolic geometry asks how small the area can be once the systole is fixed.