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 {{harvtxt|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> \mathbb{R,C,H}</math>

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 {{harv|Bangert|Katz|Shnider|Weinberger|2009}}.

References

{{cite journal | first1=Victor | last1=Bangert | first2=Mikhail G. | last2=Katz | first3=Steve | last3=Shnider | first4=Shmuel | last4=Weinberger | title=E₇, Wirtinger inequalities, Cayley 4-form, and homotopy| journal=Duke Mathematical Journal | volume=146 | year=2009 | issue=1 | pages=35–70 | arxiv=math.DG/0608006 | mr=2475399 | doi=10.1215/00127094-2008-061 | s2cid=2575584 }}
{{cite book | last=Gromov | first=Mikhail | title=Structures métriques pour les variétés riemanniennes | trans-title=Metric structures for Riemann manifolds | language=fr | editor1=J. Lafontaine | editor2=P. Pansu. | series=Textes Mathématiques | volume=1 | publisher=CEDIC | location=Paris | year=1981 | mr=0682063 | isbn=2-7124-0714-8 }}
{{cite book | last1=Katz | first1=Mikhail G. | title=Systolic geometry and topology|pages=19 | publisher=American Mathematical Society | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-4177-8 | year=2007 | volume=137 | others=With an appendix by Jake P. Solomon. | mr=2292367 | doi=10.1090/surv/137| url=http://www.gbv.de/dms/goettingen/522828477.pdf }}

Category:Geometric inequalities

Category:Differential geometry

Category:Riemannian geometry

Category:Systolic geometry

Gromov's inequality for complex projective space

Projective planes over division algebras <math> \mathbb{R,C,H}</math>

See also

References