Gromov's inequality for complex projective space

${\displaystyle {\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{y}\mathrm{s}}_2{}^n\le n!{\mathrm{v}\mathrm{o}\mathrm{l}}_{2n}({\mathbb{C}\mathbb{P}}^n)}$,

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 ${\displaystyle \mathrm{s}\mathrm{t}\mathrm{s}\mathrm{y}{\mathrm{s}}_2}$ 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 ${\displaystyle {\mathbb{C}\mathbb{P}}^1\subset {\mathbb{C}\mathbb{P}}^n}$ in 2-dimensional homology.

The inequality first appeared in Gromov (1981) as Theorem 4.36.

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

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

In the special case n=2, Gromov's inequality becomes ${\displaystyle {\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{y}\mathrm{s}}_2{}^2\le 2{\mathrm{v}\mathrm{o}\mathrm{l}}_4({\mathbb{C}\mathbb{P}}^2)}$. This inequality can be thought of as an analog of Pu's inequality for the real projective plane ${\displaystyle {\mathbb{R}\mathbb{P}}^2}$. 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 ${\displaystyle {\mathbb{H}\mathbb{P}}^2}$ is not its systolically optimal metric. In other words, the manifold ${\displaystyle {\mathbb{H}\mathbb{P}}^2}$ admits Riemannian metrics with higher systolic ratio ${\displaystyle {\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{y}\mathrm{s}}_4{}^2/{\mathrm{v}\mathrm{o}\mathrm{l}}_8}$ than for its symmetric metric (Bangert et al. 2009).