Gromov's inequality for complex projective space

Template:Short description In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality

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

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 ${ℂ\mathbb{P}}^1\subset {ℂ\mathbb{P}}^n$ in 2-dimensional homology.

The inequality first appeared in Template:Harvtxt as Theorem 4.36.

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

Projective planes over division algebras $ℝ,ℂ,ℍ$

In the special case n=2, Gromov's inequality becomes ${\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{y}\mathrm{s}}_2{}^2\le 2{\mathrm{v}\mathrm{o}\mathrm{l}}_4({ℂ\mathbb{P}}^2)$. This inequality can be thought of as an analog of Pu's inequality for the real projective plane ${ℝ\mathbb{P}}^2$. 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 ${ℍ\mathbb{P}}^2$ is not its systolically optimal metric. In other words, the manifold ${ℍ\mathbb{P}}^2$ 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 Template:Harv.

See also

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

References

• Template:Cite journal
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