Isospectral deformations II: Trace formulas, metrics, and potentials

DeTurck, Dennis; Gordon, Carolyn S.; Lee, Kyung Bai

doi:10.1002/cpa.3160420803

Cited by 72 publications

(33 citation statements)

References 27 publications

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“…However, until recently, all known isospectral manifolds were at least locally isometric; in particular, all isospectral closed manifolds had a common Riemannian cover. This was due primarily to the fact that most examples could be explained by Sunada's method [Su] or its generalizations [DG2]. The Sunada methods rely almost exclusively on representation theory, with the result that isospectral manifolds constructed using these methods must be locally isometric.…”

mentioning

confidence: 99%

Isospectral deformations of closed riemannian manifolds with different scalar curvature

Gordon¹,

Gornet²,

Schueth³

et al. 1998

Annales De L’institut Fourier

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We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds , more precisely, on S n × T m , where T m is a torus of dimension m ≥ 2 and S n is a sphere of dimension n ≥ 4. These metrics are not locally homogeneous; in particular, the scalar curvature of each metric is nonconstant. For some of the deformations, the maximum scalar curvature changes during the deformation.

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mentioning

confidence: 99%

Isospectral deformations of closed riemannian manifolds with different scalar curvature

Gordon¹,

Gornet²,

Schueth³

et al. 1998

Annales De L’institut Fourier

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“…For Φ ∈ Aut(N ) the metric Φ * g is again left invariant and thus descends to the compact manifold Γ\N. ([GW], [DG2], [Go]). If Φ ∈ AIA(N ; Γ) and g is left invariant, then the Riemannian nilmanifolds (Γ\N, g) and (Γ\N , Φ * g) are isospectral.…”

Section: Definition (I)mentioning

confidence: 99%

“…Moreover, such a deformation is nontrivial (i.e., the manifolds involved are not pairwise isometric) except in the case where the Φ t differ from each other by inner automorphisms. By results of D. DeTurck and C. Gordon ([DG1], [DG2]), the (Γ\N, Φ * t g) are even strongly isospectral ; i.e., they are isospectral also for every other natural elliptic operator, for example for the Laplacian acting on p-forms.…”

Section: Introductionmentioning

confidence: 99%

Line bundle Laplacians over isospectral nilmanifolds

Schueth

1997

Trans. Amer. Math. Soc.

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“…Two compact Riemannian manifolds will be said to be p-isospectral if their p-spectra coincide and in the literature, one abbreviates 0-isospectral to isospectral. It is well known that Spec p (M, g) does not determine the geometry of (M, g), as shown by many examples of nonisometric p-isospectral manifolds, via the so called generalized Sunada method (see for instance [Mi64], [Vi80], [Su85], [DG89]) working for all p and also with methods working for individual values of p (see for instance [Go86], [Ik88], [Gt00], [MR01], [GM06]). …”

Section: Introductionmentioning

confidence: 99%

“…More recently, Miatello, Rossetti and the author [LMR16a] found families of pairs, in any odd dimension n ≥ 5, of lens spaces that are p-isospectral for all p, but are not strongly isospectral. In particular such pair cannot be constructed by the generalized Sunada method due to DeTurck and Gordon [DG89], which uses representation equivalent discrete subgroups.…”

Section: Introductionmentioning

confidence: 99%