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.
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
The inequality first appeared in Gromov (1981) as Theorem 4.36.
In the special case n=2, Gromov's inequality becomes
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
admits Riemannian metrics with higher systolic ratio
than for its symmetric metric (Bangert et al. 2009).
This Riemannian geometry-related article is a stub.