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| In [[Riemannian geometry]], [[Mikhail Gromov (mathematician)|Gromov]]'s optimal stable 2-[[systolic geometry|systolic]] inequality is the inequality
| | I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. For many years she's been living in Kentucky but her spouse wants them to move. Doing ballet is some thing she would by no means give up. Distributing production has been his occupation for some time.<br><br>My web site :: clairvoyants ([http://Galab-Work.Cs.Pusan.Ac.kr/Sol09B/?document_srl=1489804 Suggested Web site]) |
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| : <math>\mathrm{stsys}_2{}^n \leq n!
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| \;\mathrm{vol}_{2n}(\mathbb{CP}^n)</math>,
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| valid for an arbitrary Riemannian metric on the [[complex projective space]], where the optimal bound is attained
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| 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.
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| The inequality first appeared in Gromov's 1981 book entitled ''Structures métriques pour les variétés riemanniennes'' (Theorem 4.36).
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| The proof of Gromov's inequality relies on the [[Wirtinger inequality (2-forms)|Wirtinger inequality for exterior 2-forms]].
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| ==Projective planes over division algebras <math> \mathbb{R,C,H}</math>==
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| 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|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, see Bangert et al. (2009).
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| ==See also==
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| *[[Loewner's torus inequality]]
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| *[[Pu's inequality]]
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| *[[Gromov's inequality]]
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| *[[Gromov's systolic inequality for essential manifolds]]
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| *[[Systolic geometry]]
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| ==References==
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| *[[Victor Bangert|Bangert, V]]; [[Mikhail Katz|Katz, M.]]; [[Steve Shnider|Shnider, S.]]; [[Shmuel Weinberger|Weinberger, S.]]: [[E7 (mathematics)|E_7]], [[Wirtinger inequality|Wirtinger inequalities]], Cayley 4-form, and homotopy. [[Duke Mathematical Journal]] 146 (2009), no. 1, 35-70. See arXiv:math.DG/0608006
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| *Gromov, M.: Structures métriques pour les variétés riemanniennes. Edited by J. Lafontaine and P. Pansu. Textes Mathématiques, 1. CEDIC, Paris, 1981 (first edition of [[Metric Structures for Riemannian and Non-Riemannian Spaces]]).
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| *{{Citation | last1=[[Mikhail Katz|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}}
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I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. For many years she's been living in Kentucky but her spouse wants them to move. Doing ballet is some thing she would by no means give up. Distributing production has been his occupation for some time.
My web site :: clairvoyants (Suggested Web site)