Gromov's inequality for complex projective space
In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality 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 formula_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 formula_3 in 2-dimensional homology. The inequality first appeared in as Theorem 4.36. The proof of Gromov's  inequality relies on the Wirtinger inequality for exterior 2-forms. Projective planes over division algebras formula_4. In the special case n=2, Gromov's inequality becomes formula_5. This inequality can be thought of as an analog of Pu's inequality for the real projective plane formula_6. 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 formula_7 is not its systolically optimal metric. In other words, the manifold formula_7 admits Riemannian metrics with higher systolic ratio formula_9 than for its symmetric metric . Gromov's