Systolic geometry

Builds on Riemannian manifold Compact space

Definition

Let $(M, g)$ be a compact connected Riemannian manifold 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 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.

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Tag	`TCNY8VKQ61`
URI	`m2://TCNY8VKQ61`
Kind	definition
Writing	W0 stub
MSC	53C23, 53C20
Defines	Systolic geometry
Uses	Riemannian manifold, Compact space
Changed	2 November 2026

Systolic geometry

Definition

Remarks

Reading path

Further reading

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