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).

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.CS1 maint: ref=harv (link)
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.CS1 maint: ref=harv (link)
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.

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Systolic geometry
1-systoles of surfaces	Loewner's torus inequality Pu's inequality Filling area conjecture Bolza surface Systoles of surfaces Eisenstein integers
1-systoles of manifolds	Gromov's systolic inequality for essential manifolds Essential manifold Filling radius Hermite constant
Higher systoles	Gromov's inequality for complex projective space Systolic freedom Systolic category

Gromov's inequality for complex projective space

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

See also

References

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