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 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 in 2-dimensional homology.
The inequality first appeared in Gromov's 1981 book entitled Structures métriques pour les variétés riemanniennes (Theorem 4.36).
The proof of Gromov's inequality relies on the Wirtinger inequality for exterior 2-forms.
Read more about Gromov's Inequality For Complex Projective Space: Projective Planes Over Division Algebras
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