# Gromov's inequality for complex projective space

In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality

$\mathrm{stsys}_2{}^n \leq n! \;\mathrm{vol}_{2n}(\mathbb{CP}^n)$,

valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained by the symmetric Fubini-Study metric, providing a natural geometrisation of quantum mechanics. Here $\operatorname{stsys_2}$ is the stable 2-systole, which in this case can be defined as the infimum of the areas of rational 2-cycles representing the class of the complex projective line $\mathbb{CP}^1 \subset \mathbb{CP}^n$ in 2-dimensional homology.

The inequality first appeared in Gromov's 1981 book entitled Structures métriques pour les variétés riemanniennes (Theorem 4.36).

The proof of Gromov's inequality relies on the Wirtinger inequality for exterior 2-forms.

## Projective planes over division algebras $\mathbb{R,C,H}$

In the special case n=2, Gromov's inequality becomes $\mathrm{stsys}_2{}^2 \leq 2 \mathrm{vol}_4(\mathbb{CP}^2)$. This inequality can be thought of as an analog of Pu's inequality for the real projective plane $\mathbb{RP}^2$. In both cases, the boundary case of equality is attained by the symmetric metric of the projective plane. Meanwhile, in the quaternionic case, the symmetric metric on $\mathbb{HP}^2$ is not its systolically optimal metric. In other words, the manifold $\mathbb{HP}^2$ admits Riemannian metrics with higher systolic ratio $\mathrm{stsys}_4{}^2/\mathrm{vol}_8$ than for its symmetric metric, see Bangert et al. (2009).