Orbifold lens spaces that are isospectral but not isometric
type: Item
instance of: scholarly article
title: Orbifold lens spaces that are isospectral but not isometric
zbMATH Open document ID: 1229.58027
author: Naveed Shams Ul Bari
published in: Osaka Journal of Mathematics
publication date: 2011-04-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/0902.2441
full work available at URL: https://projecteuclid.org/euclid.ojm/1300802702
review text: Two Riemannian manifolds are said to be isospectral if they have the same eigenvalues of the Laplace operator. The notion of isospectrality can be extended to Riemannian orbifolds, in which coordinate charts are modeled on quotients of Riemannian manifolds by finite groups of isometries; a standard inverse spectral problem in this setting is whether there exist isospectral non-isometric orbifolds. Examples of such orbifolds of various flavors have been constructed, including arbitrarily large finite families of isospectral orbifolds [\textit{N. Shams, E. Stanhope} and \textit{D. L. Webb}, Arch. Math. 87, No.~4, 375--385 (2006; Zbl 1111.58029)] and isospectral orbifolds with maximal isotropy groups of different orders [\textit{J. P. Rossetti, D. Schueth} and \textit{M. Weilandt}, Ann. Global Anal. Geom. 34, No.~4, 351--366 (2008; Zbl 1166.58016)]. In [Ann. Sci. Éc. Norm. Supér. (4) 13, 303--315 (1980; Zbl 0451.58037)], \textit{A. Ikeda} constructed isospectral non-isometric lens spaces; the paper under review extends this work to the orbifold setting. An orbifold lens space is a quotient of the round sphere by a cyclic group of orthogonal transformations, where the group is permitted to have fixed points. The author shows that, for every dimension greater than 8, there exist pairs of isospectral non-isometric orbifold lens spaces. The proof relies on giving a convenient formulation of the generating functions of orbifold lens spaces and showing that, under certain conditions, there are sufficiently many generating functions to guarantee the existence of isospectral non-isometric orbifold lens spaces.
Mathematics Subject Classification ID: 58J53
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 5882323
zbMATH Keywords: isospectral
zbMATH Keywords: orbifold
zbMATH Keywords: lens space
reviewed by: Emily B. Dryden
MaRDI profile type: Publication
arXiv ID: 0902.2441
cites work: Some planar isospectral domains
cites work: Q4854757
cites work: Q3137109
cites work: Asymptotic expansion of the heat kernel for orbifolds
cites work: Isospectral hyperbolic surfaces have matching geodesics
cites work: Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal.
cites work: Q3723385
cites work: Lens Spaces, Isospectral on Forms but not on Functions
cites work: On lens spaces which are isospectral but not isometric
cites work: On the spectrum of a Riemannian manifold of positive constant curvature
cites work: On the spectra of 3-dimensional lens spaces
cites work: Can One Hear the Shape of a Drum?
cites work: EIGENVALUES OF THE LAPLACE OPERATOR ON CERTAIN MANIFOLDS
cites work: An Isospectral Deformation on an Infranil-Orbifold
cites work: Isospectral orbifolds with different maximal isotropy orders
cites work: ON A GENERALIZATION OF THE NOTION OF MANIFOLD
cites work: The Geometries of 3-Manifolds
cites work: One cannot hear orbifold isotropy type
cites work: Spectral bounds on orbifold isotropy
cites work: Isospectral and non-isometric Riemannian manifolds
description: scientific article