On manifolds of negative curvature with isospectral potentials
type: Item
instance of: scholarly article
title: On manifolds of negative curvature with isospectral potentials
zbMATH Open document ID: 0617.53048
DOI: 10.1016/0040-9383(87)90021-8
author: Robert Brooks
published in: Topology
publication date: 1987-01-01T00:00:00Z
review text: Let M be a manifold of negative curvature and let \(\Delta\) be the Laplacian. Let \(\Phi\) be a potential. One wishes to known when it is possible to recover \(\Phi\) from \(spec(\Delta +\Phi)\). M has simple length spectrum if no two closed geodesics on M have the same length. Guillemin and Kazhdan show that if M has simple length spectrum and if the curvature of M is suitably pinched, then \(\Phi\) is determined by \(spec(\Delta +\Phi)\). The author uses the method of Sunada to construct a manifold M of negative curvature (which does not have simple length spectrum) and two potentials \(\Phi_ 1\) and \(\Phi_ 2\) (which are not equivalent under the action of the group of isometries) so that \(spec(\Delta +\Phi_ 1)=spec(\Delta +\Phi_ 2)\). The author also constructs isospectral potentials on manifolds of all ranks with nonpositive curvature.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 4000926
zbMATH Keywords: manifold of negative curvature
zbMATH Keywords: Laplacian
zbMATH Keywords: closed geodesics
zbMATH Keywords: simple length spectrum
zbMATH Keywords: isospectral potentials
MaRDI profile type: Publication
full work available at URL: 0040-9383(87)90021-8
OpenAlex ID: W2031765029
description: scientific article; zbMATH DE number 4000926