Gromov's inequality for complex projective space

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

: $\mathrm_2^n \leq n! \;\mathrm_(\mathbb^n)$,

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 $\operatorname$ 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 $\mathbb^1 \subset \mathbb^n$ in 2-dimensional homology.

The inequality first appeared in as Theorem 4.36.

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

Projective planes over division algebras $\mathbb$

In the special case n=2, Gromov's inequality becomes $\mathrm_2^2 \leq 2 \mathrm_4(\mathbb^2)$. This inequality can be thought of as an analog of Pu's inequality for the real projective plane $\mathbb^2$. 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 $\mathbb^2$ is not its systolically optimal metric. In other words, the manifold $\mathbb^2$ admits Riemannian metrics with higher systolic ratio $\mathrm_4^2/\mathrm_8$ than for its symmetric metric .

See also

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

References

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Geometric inequalities
Differential geometry
Riemannian geometry
Systolic geometry