Differential geometry Metric and synthetic methods in global geometry Theorem

Gromov's systolic inequality

Builds on Systolic geometry Riemannian metric Loewner's torus inequality

Statement

Let $n \ge 1$ be an integer. Let $M$ be a closed connected manifold of dimension $n$ with fundamental group $\pi$ . Let $f \colon M \to K(\pi, 1)$ be a map that induces an isomorphism of fundamental groups. Let $A = \Z$ if $M$ is orientable and $A = \Z/2\Z$ otherwise. Call $M$ essential if $f$ sends the fundamental class of $M$ to a nonzero class of $H_n(K(\pi, 1); A)$ .

There is a constant $C_n > 0$ , depending only on $n$ , with the following property. For every essential $M$ of dimension $n$ and every Riemannian metric $g$ on $M$ , $\operatorname{sys}(M, g)^n \le C_n \operatorname{vol}(M, g),$ where $\operatorname{sys}$ is the systole.

Remarks

Gromov's proof goes through the filling radius. One embeds $M$ isometrically into the Banach space $L^\infty(M)$ by the Kuratowski embedding. For essential $M$ the filling radius is at least $\operatorname{sys}(M, g)/6$ . On the other hand, a filling inequality bounds the filling radius by a constant times $\operatorname{vol}(M, g)^{1/n}$ .

The tori $T^n$ , the real projective spaces $\R P^n$ and all closed aspherical manifolds are essential. For $T^2$ the inequality is a weak form of Loewner's torus inequality.

The hypothesis that $M$ is essential cannot be dropped. On $S^2 \times S^1$ take the product of a round metric of small area $\varepsilon$ on $S^2$ and a circle of length $1$ . Every noncontractible loop winds around the circle factor, so the systole is $1$ , while the volume $\varepsilon$ tends to $0$ .

Going deeper

For a student

This is the higher-dimensional extension of Loewner's and Pu's inequalities in systolic geometry. The constant $C_n$ from the proof is not optimal, and optimal constants are unknown in most cases. The notion of essential manifold is topological: it asks that the fundamental class survive in the homology of the classifying space of the fundamental group.

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Tag	`HB2K72PSZ6`
URI	`m2://HB2K72PSZ6`
Kind	statement, theorem
Writing	W0 stub
MSC	53C23, 53C20
Status	This copy of the site has no build data. Run `m2 status HB2K72PSZ6`.
About	Systolic geometry
Uses	Systolic geometry, Riemannian metric, Loewner's torus inequality
Same as	Wikidata Q5610194 Wikipedia Gromov's systolic inequality for essential manifolds
Changed	2 November 2026

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