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In [[Riemannian geometry]], [[Mikhail Gromov (mathematician)|Gromov]]'s optimal stable 2-[[systolic geometry|systolic]] inequality is the inequality

: <math>\mathrm{stsys}_2{}^n \leq n!
\;\mathrm{vol}_{2n}(\mathbb{CP}^n)</math>,

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 <math>\operatorname{stsys_2}</math> 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 <math>\mathbb{CP}^1 \subset \mathbb{CP}^n</math> 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 (2-forms)|Wirtinger inequality for exterior 2-forms]].

==Projective planes over division algebras <math> \mathbb{R,C,H}</math>==

In the special case n=2, Gromov's inequality becomes <math>\mathrm{stsys}_2{}^2 \leq 2 \mathrm{vol}_4(\mathbb{CP}^2)</math>. This inequality can be thought of as an analog of [[Pu's inequality|Pu's inequality for the real projective plane]] <math>\mathbb{RP}^2</math>. 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 <math>\mathbb{HP}^2</math> is not its systolically optimal metric. In other words, the manifold <math>\mathbb{HP}^2</math> admits Riemannian metrics with higher systolic ratio <math>\mathrm{stsys}_4{}^2/\mathrm{vol}_8</math> than for its symmetric metric, see Bangert et al. (2009).

==See also==
*[[Loewner's torus inequality]]
*[[Pu's inequality]]
*[[Gromov's inequality]]
*[[Gromov's systolic inequality for essential manifolds]]
*[[Systolic geometry]]

==References==
*[[Victor Bangert|Bangert, V]]; [[Mikhail Katz|Katz, M.]]; [[Steve Shnider|Shnider, S.]]; [[Shmuel Weinberger|Weinberger, S.]]: [[E7 (mathematics)|E_7]], [[Wirtinger inequality|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]]).
*{{Citation | last1=[[Mikhail Katz|Katz]] | first1=Mikhail G. | title=Systolic geometry and topology|pages=19 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-4177-8 | year=2007 | volume=137}}

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