Isospectrality in the FIO category
type: Item
instance of: scholarly article
title: Isospectrality in the FIO category
zbMATH Open document ID: 0769.53026
author: Steve Zelditch
published in: Journal of Differential Geometry
publication date: 1993-02-18T00:00:00Z
review text: Let \((M_ i,g_ i)\) be compact Riemannian manifolds and let \(\Delta_ i\) be their Laplacians. They are said to be isospectral if \(\text{spec}(\Delta_ 1) = \text{spec}(\Delta_ 2)\) where each eigenvalue is counted with multiplicity. This means that there is a unitary operator \(U: L_ 2(M_ 1) \to L_ 2(M_ 2)\) which intertwines \(\Delta_ 1\) and \(\Delta_ 2\). If the intertwining operator is restricted to be a Fourier integral operator, the author shows the following geometric consequences hold: (a) If \((M_ 1,g_ 1)\) is nonpositively curved, then the \((M_ i,g_ i)\) have a common finite Riemannian cover. (b) If \((M_ 1,g_ 1)\) is a negatively curved surface with simple length spectrum, then the \((M_ i,g_ i)\) are isometric.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 120218
zbMATH Keywords: Fourier integral operator
zbMATH Keywords: Riemannian cover
zbMATH Keywords: simple length spectrum
Wikidata QID: Q115180976
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214448263
OpenAlex ID: W1516418562
DOI: 10.4310/JDG/1214448263
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description: scientific article; zbMATH DE number 120218