In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality
s t s y s 2 n ≤ n ! v o l 2 n ( C 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 s t s y 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 C P 1 ⊂ C 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.
In the special case n=2, Gromov's inequality becomes s t s y s 2 2 ≤ 2 v o l 4 ( C P 2 ) . This inequality can be thought of as an analog of Pu's inequality for the real projective plane R 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 H P 2 is not its systolically optimal metric. In other words, the manifold H P 2 admits Riemannian metrics with higher systolic ratio s t s y s 4 2 / v o l 8 than for its symmetric metric (Bangert et al. 2009).
