On two dimensional isosystolic inequalities
type: Item
instance of: scholarly article
title: On two dimensional isosystolic inequalities
zbMATH Open document ID: 0642.53042
author: Shigeru Kodani
published in: Kodai Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Let M be a closed surface. For any Riemannian metric g on M, let Area(g) be its area, and let sys(g) by the length of its shortest noncontractible closed geodesic. The isosystolic constant of M is defined to be \(\Sigma_ M=\inf Area(g)/sys(g)^ 2\) where the infimum is taken over all Riemannian metrics on M. The exact value of \(\Sigma_ M\) has been calculated when M is a projective plane, a torus, and a Klein bottle. Otherwise, Gromov has found estimates from below for \(\Sigma_ M\) in terms of the first Betti number of M. The techniques of this paper produce both upper and lower bounds for \(\Sigma_ M\) in terms of an isosystolic constant for finite graphs whose Euler number is -\(\gamma\) where \(\gamma\) is the genus of M. They also produce a lower bound on \(\Sigma_ M\) in terms of \(\gamma\) which unfortunately is not as good as Gromov's lower bound.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C65
zbMATH DE Number: 4046580
zbMATH Keywords: isosystolic inequality
zbMATH Keywords: closed surface
zbMATH Keywords: area
zbMATH Keywords: closed geodesic
zbMATH Keywords: isosystolic constant
zbMATH Keywords: lower bound
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138037462
OpenAlex ID: W1971384303
cites work: Inégalité isosystolique pour la bouteille de Klein. (Isosystolic inequalities for the Klein bottle)
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DOI: 10.2996/KMJ/1138037462
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description: scientific article; zbMATH DE number 4046580