Counterexamples to isosystolic inequalities
type: Item
instance of: scholarly article
title: Counterexamples to isosystolic inequalities
zbMATH Open document ID: 0842.53031
DOI: 10.1007/BF01264937
published in: Geometriae Dedicata
publication date: 1996-07-28T00:00:00Z
review text: In the present paper the author explores M. Gromov's counterexamples to systolic inequalities. Does the manifold \(S^2\times S^2\) admit metrics of arbitrarily small volume such that every noncontractible surface inside has at least unit area? This question is still open, but the author proves that the answer is affirmative for its analogue in the case of \(S^n\times S^n\), \(n\geq 3\). More precisely, combining some results of M. Gromov on \(S^1\times S^3\) and some results of C. Pittet on \(S^1\times S^n\), the author proves the following result: The manifold \(S^n\times S^n\) admits for all \(n\geq 3\) metrics with arbitrarily small ratio \(\text{vol/sys}^2_n\).
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 806282
zbMATH Keywords: product of two \(n\)-spheres
zbMATH Keywords: counterexamples
zbMATH Keywords: systolic inequalities
zbMATH Keywords: \(S^ n\times S^ n\)
author: Mikhail G. Katz
reviewed by: Neculai Papaghiuc
MaRDI profile type: Publication
cites work: ASYMPTOTIC INVARIANTS OF SMOOTH MANIFOLDS
cites work: On intersystolic inequalities in dimension 3
cites work: Du côté de chez Pu
cites work: Q4039749
cites work: Normal and integral currents
cites work: Q5622916
cites work: Filling Riemannian manifolds
cites work: Systoles on \(S^ 1\times S^ n\)
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description: scientific article; zbMATH DE number 806282