Length minimizing geodesics and the length spectrum of Riemannian two-step nilmanifolds
type: Item
instance of: scholarly article
title: Length minimizing geodesics and the length spectrum of Riemannian two-step nilmanifolds
zbMATH Open document ID: 1038.53042
DOI: 10.1007/BF02931000
author: Ruth Gornet
author: Maura B. Mast
published in: The Journal of Geometric Analysis
publication date: 2003-12-04T00:00:00Z
review text: The interest of the authors is focused on two-step nilpotent manifolds as closest to the Euclidean one and having clear algebraic structure quite determined by a \(J\)-operator. Due to the Gordon and Wilson technique of constructing isospectral manifolds of this type, which are not locally isometric, we have the possibility to examine the following key question: to what extent does the spectrum of the Laplacian determine the geometry? A major open question is whether there can exist examples of isospectral manifolds with different periods in the length spectrum. The authors extend their earlier result on the singular \(J\)-operator case and show against all hope that in the case the isospectrality necessarily yields the length spectrum coincidence. Relying on the work of \textit{C. Jang} and \textit{K. Park} [Geom. Dedicata 79, 65--80 (2000; Zbl 0943.22005)] they succeeded in calculating a lower bound on the injectivity radius of a general simply connected manifold of the above type. Investigating conjugate and cut points the authors find that they do not coincide for most geodesics in contrast to the one-dimensional center and Heisenberg type cases.
reviewed by: V. S. Kalnitsky
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 58J53
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 2013179
zbMATH Keywords: Laplacian
zbMATH Keywords: isospectral manifolds
zbMATH Keywords: length spectrum
zbMATH Keywords: nilmanifolds
Wikidata QID: Q115391069
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02931000
OpenAlex ID: W1970848937
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description: scientific article; zbMATH DE number 2013179