Gromov's inequality for complex projective space
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.
Projective planes over division algebras 
In the special case n=2, Gromov's inequality becomes . This inequality can be thought of as an analog of Pu's inequality for the real projective plane . 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 is not its systolically optimal metric. In other words, the manifold admits Riemannian metrics with higher systolic ratio than for its symmetric metric, see Bangert et al. (2009).
- Loewner's torus inequality
- Pu's inequality
- Gromov's inequality
- Gromov's systolic inequality for essential manifolds
- Systolic geometry
- Bangert, V; Katz, M.; Shnider, S.; Weinberger, S.: E_7, Wirtinger inequalities, Cayley 4-form, and homotopy. Duke Mathematical Journal 146 (2009), no. 1, 35-70. See arXiv:math.DG/0608006
- Gromov, M.: Structures métriques pour les variétés riemanniennes. Edited by J. Lafontaine and P. Pansu. Textes Mathématiques, 1. CEDIC, Paris, 1981 (first edition of Metric Structures for Riemannian and Non-Riemannian Spaces).
- Katz, Mikhail G. (2007), Systolic geometry and topology, Mathematical Surveys and Monographs 137, Providence, R.I.: American Mathematical Society, p. 19, ISBN 978-0-8218-4177-8