Gromov's inequality for complex projective space

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

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

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

In the special case n=2, Gromov's inequality becomes ${\displaystyle \mathrm {stsys} _{2}{}^{2}\leq 2\mathrm {vol} _{4}(\mathbb {CP} ^{2})}$. This inequality can be thought of as an analog of Pu's inequality for the real projective plane ${\displaystyle \mathbb {RP} ^{2}}$. 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 ${\displaystyle \mathbb {HP} ^{2}}$ is not its systolically optimal metric. In other words, the manifold ${\displaystyle \mathbb {HP} ^{2}}$ admits Riemannian metrics with higher systolic ratio ${\displaystyle \mathrm {stsys} _{4}{}^{2}/\mathrm {vol} _{8}}$ 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. S2CID 2575584.
• Gromov, Mikhail (1981). J. Lafontaine; P. Pansu. (eds.). Structures métriques pour les variétés riemanniennes [Metric structures for Riemann manifolds]. Textes Mathématiques (in French). Vol. 1. Paris: CEDIC. ISBN 2-7124-0714-8. MR 0682063.
• Katz, Mikhail G. (2007). Systolic geometry and topology. Mathematical Surveys and Monographs. Vol. 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.

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