Gromov's inequality for complex projective space

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

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

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 ${\displaystyle \operatorname {stsys_{2}} }$ 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 ${\displaystyle \mathbb {CP} ^{1}\subset \mathbb {CP} ^{n}}$ 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 ${\displaystyle \mathbb {R,C,H} }$

In the special case n=2, Gromov's inequality becomes ${\displaystyle \mathrm {stsys} _{2}{}^{2}\leq 2\mathrm {vol} _{4}(\mathbb {CP} ^{2})}$. This inequality can be thought of as an analog of Pu's inequality for the real projective plane ${\displaystyle \mathbb {RP} ^{2}}$. 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 ${\displaystyle \mathbb {HP} ^{2}}$ is not its systolically optimal metric. In other words, the manifold ${\displaystyle \mathbb {HP} ^{2}}$ admits Riemannian metrics with higher systolic ratio ${\displaystyle \mathrm {stsys} _{4}{}^{2}/\mathrm {vol} _{8}}$ than for its symmetric metric (Bangert et al. 2009).