Gromov's inequality for complex projective space

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

${\mathrm {stsys}}_{2}{}^{n}\leq n!\;{\mathrm {vol}}_{{2n}}({\mathbb {CP}}^{n})$,

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 $\operatorname {stsys_{2}}$ 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 ${\mathbb {CP}}^{1}\subset {\mathbb {CP}}^{n}$ 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 ${\mathbb {R,C,H}}$

In the special case n=2, Gromov's inequality becomes ${\mathrm {stsys}}_{2}{}^{2}\leq 2{\mathrm {vol}}_{4}({\mathbb {CP}}^{2})$. This inequality can be thought of as an analog of Pu's inequality for the real projective plane ${\mathbb {RP}}^{2}$. 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 ${\mathbb {HP}}^{2}$ is not its systolically optimal metric. In other words, the manifold ${\mathbb {HP}}^{2}$ admits Riemannian metrics with higher systolic ratio ${\mathrm {stsys}}_{4}{}^{2}/{\mathrm {vol}}_{8}$ than for its symmetric metric (Bangert et al. 2009).

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. arXiv:math.DG/0608006. doi:10.1215/00127094-2008-061. MR 2475399.
• Gromov, Mikhail (1981). J. Lafontaine; P. Pansu., eds. Structures métriques pour les variétés riemanniennes [Metric structures for Riemann manifolds]. Textes Mathématiques (in French). 1. Paris: CEDIC. ISBN 2-7124-0714-8. MR 0682063.
• 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. p. 19. doi:10.1090/surv/137. ISBN 978-0-8218-4177-8. MR 2292367.