Английская Википедия:Gromov's inequality for complex projective space

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

<math>\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>\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>\mathbb{CP}^1 \subset \mathbb{CP}^n</math> in 2-dimensional homology.

The inequality first appeared in Шаблон: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 <math> \mathbb{R,C,H}</math>

In the special case n=2, Gromov's inequality becomes <math>\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>\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>\mathbb{HP}^2</math> is not its systolically optimal metric. In other words, the manifold <math>\mathbb{HP}^2</math> admits Riemannian metrics with higher systolic ratio <math>\mathrm{stsys}_4{}^2/\mathrm{vol}_8</math> than for its symmetric metric Шаблон:Harv.

See also

References

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