Gromov's inequality for complex projective space

Short description: Optimal stable 2-systolic inequality

In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality

${\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{y}\mathrm{s}}_2{}^n\le n!{\mathrm{v}\mathrm{o}\mathrm{l}}_{2n}({\mathbb{ℂ}\mathbb{\mathbb{P}}}^n)$,

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 $\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{y}{\mathrm{s}}_{\mathrm{2}}$ 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 ${\mathbb{ℂ}\mathbb{\mathbb{P}}}^1\subset {\mathbb{ℂ}\mathbb{\mathbb{P}}}^n$ in 2-dimensional homology.

The inequality first appeared in (Gromov 1981) as Theorem 4.36.

The proof of Gromov's inequality relies on the Wirtinger inequality for exterior 2-forms.

Projective planes over division algebras $\mathbb{ℝ},\mathbb{ℂ},\mathbb{ℍ}$

In the special case n=2, Gromov's inequality becomes ${\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{y}\mathrm{s}}_2{}^2\le 2{\mathrm{v}\mathrm{o}\mathrm{l}}_4({\mathbb{ℂ}\mathbb{\mathbb{P}}}^2)$. This inequality can be thought of as an analog of Pu's inequality for the real projective plane ${\mathbb{ℝ}\mathbb{\mathbb{P}}}^2$. 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 ${\mathbb{ℍ}\mathbb{\mathbb{P}}}^2$ is not its systolically optimal metric. In other words, the manifold ${\mathbb{ℍ}\mathbb{\mathbb{P}}}^2$ admits Riemannian metrics with higher systolic ratio ${\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{y}\mathrm{s}}_4{}^2/{\mathrm{v}\mathrm{o}\mathrm{l}}_8$ than for its symmetric metric (Bangert Katz).

See also

• Loewner's torus inequality
• Pu's inequality
• Gromov's inequality (disambiguation)
• Gromov's systolic inequality for essential manifolds
• Systolic geometry

References

• Bangert, Victor; Katz, Mikhail G.; Shnider, Steve; Weinberger, Shmuel (2009). "E₇, Wirtinger inequalities, Cayley 4-form, and homotopy". Duke Mathematical Journal 146 (1): 35–70. doi:10.1215/00127094-2008-061. 
• Gromov, Mikhail (1981) (in fr). Structures métriques pour les variétés riemanniennes. Textes Mathématiques. 1. Paris: CEDIC. ISBN 2-7124-0714-8. 
• Katz, Mikhail G. (2007). Systolic geometry and topology. Mathematical Surveys and Monographs. 137. With an appendix by Jake P. Solomon.. Providence, R.I.: American Mathematical Society. pp. 19. doi:10.1090/surv/137. ISBN 978-0-8218-4177-8.