Locally non-isometric yet super isospectral spaces
type: Item
instance of: scholarly article
title: Locally non-isometric yet super isospectral spaces
zbMATH Open document ID: 0964.53026
published in: Geometric and Functional Analysis. GAFA
publication date: 2001-06-28T00:00:00Z
review text: The Laplace-Beltrami operator acting on functions on a closed Riemannian manifold has discrete real spectrum. It was a famous question whether or not this spectrum determines the manifold up to isometry. The question was answered to the negative by \textit{J. W. Milnor} [Proc. Nat. Acad. Sci. USA 51, 542 (1964; Zbl 0124.31202)] and since then many mathematicians have constructed pairs of isospectral non-isometric Riemannian manifolds and even families of such (e.g. \textit{M.-F. Vignéras} [Ann. Math., II. Ser. 112, 21-32 (1980; Zbl 0445.53026)], \textit{C. Gordon} and \textit{E. Wilson} [J. Differ. Geom. 19, 241-256 (1984; Zbl 0523.58043)], \textit{T. Sunada} [Ann. Math., II. Ser. 121, 169-186 (1985; Zbl 0585.58047)]). All these examples are locally isometric. The first locally non-isometric examples were published by \textit{C. Gordon} [J. Differ. Geom. 37, No. 3, 639-649 (1993; Zbl 0792.53037)]. In the present paper the author constructs further pairs of locally non-isometric but isospectral compact Riemannian manifolds with and without boundary. The construction goes as follows: Let \(a\) and \(b\) be natural numbers, let \({\mathbb H} \cong {\mathbb R}^4\) denote the quaternions. If we identify \(\mathbb R^3\) with the imaginary quaternions, then \({\mathbb R}^3\) acts on \({\mathbb H}^a\) by quaternionic multiplication from the right and on \({\mathbb H}^b\) from the left. From this one constructs a Heisenberg type Lie algebra structure on \({\mathbb H}^a \times {\mathbb H}^b \times {\mathbb R}^3\) and hence by exponentiating a Lie group structure. By pushing forward a natural invariant metric to the quotient of \({\mathbb H}^a \times {\mathbb H}^b \times {\mathbb R}^3\) by the subgroup \(\{0\} \times \{0\} \times {\mathbb Z}^3\) one obtains a Riemannian manifold \({\mathbb CH}^{(a,b)}_3\) diffeomorphic to \({\mathbb R}^{4(a+b)}\times T^3\). Let \(B\subset {\mathbb R}^{4(a+b)}\) be a ball about the origin and let \(S\) be its boundary. It turns out that for dif…
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J53
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 1286130
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: isospectral manifolds
zbMATH Keywords: Heisenberg type Lie algebra
author: Zoltán Imre Szabó
reviewed by: Christian Bär
MaRDI profile type: Publication
full work available at URL: s000390050084
OpenAlex ID: W2142933983
DOI: 10.1007/S000390050084
description: scientific article; zbMATH DE number 1286130