---
tag: TCNY8VKQ61
kind: definition
name: Systolic geometry
msc: [53C23, 53C20]
defines: PZCJV40M4K
uses: [7CBT4SM83Y, 0TVJZYB6SH]
sources:
  - url: https://en.wikipedia.org/wiki/Systolic_geometry
---

## Definition

Let $(M, g)$ be a [[compact|0TVJZYB6SH]] connected [[Riemannian manifold|7CBT4SM83Y]] that is not simply connected.
A closed curve in $M$ is *noncontractible* if it is not freely homotopic to a constant curve.
The *systole* $\operatorname{sys}(M, g)$ is the infimum of the lengths of the noncontractible closed curves in $M$; it is attained and positive.
If $M$ has dimension $n$, the *systolic ratio* is $\operatorname{sys}(M, g)^n / \operatorname{vol}(M, g)$.
*Systolic geometry* is the study of the systole and of upper bounds for the systolic ratio over all metrics $g$ on a fixed manifold $M$.

## Remarks

Example: the flat torus $\R^2 / \Z^2$ has systole $1$, attained by the images of the coordinate segments, and area $1$, so its systolic ratio is $1$.
[[Loewner's torus inequality|1Y9X1YW6GH]] says that the systolic ratio of any metric on the torus is at most $2/\sqrt{3}$.

Non-example: on the round sphere $S^2$ every closed curve is contractible, so the systole is not defined.

The systolic ratio does not change when $g$ is multiplied by a positive constant.
The same words are used for higher systoles, defined with cycles of higher dimension instead of curves.