Gromov's inequality for complex projective space
In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality
- [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \mathbb{CP}^1 \subset \mathbb{CP}^n }[/math] 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 [math]\displaystyle{ \mathbb{R,C,H} }[/math]
In the special case n=2, Gromov's inequality becomes [math]\displaystyle{ \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 for the real projective plane [math]\displaystyle{ \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]\displaystyle{ \mathbb{HP}^2 }[/math] is not its systolically optimal metric. In other words, the manifold [math]\displaystyle{ \mathbb{HP}^2 }[/math] admits Riemannian metrics with higher systolic ratio [math]\displaystyle{ \mathrm{stsys}_4{}^2/\mathrm{vol}_8 }[/math] 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.
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