Isospectral manifolds with different local geometries

Schueth, Dorothee

doi:10.1515/crll.2001.035

Cited by 29 publications

(38 citation statements)

References 44 publications

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“…Recently, we have learned that Schueth has also obtained examples of isospectral homogeneous spaces [Sch2]. In fact, she produces a continuous family of pairwise isospectral left-invariant metrics on a simply-connected Lie group.…”

Section: Introductionmentioning

confidence: 99%

Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions

Sutton¹

2002

Comment. Math. Helv.

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Abstract.We generalize Sunada's method to produce new examples of closed, locally nonisometric manifolds which are isospectral. In particular, we produce pairs of isospectral, simplyconnected, locally non-isometric normal homogeneous spaces. These pairs also allow us to see that in general group actions with discrete spectra are not determined up to measurable conjugacy by their spectra. In particular, we show this for lattice actions. Mathematics Subject Classification (2000). 58J50, 53C20, 53C30, 22D99.

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Section: Introductionmentioning

confidence: 99%

Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions

Sutton¹

2002

Comment. Math. Helv.

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“…We compare the spectrum of each such symmetric space with the spectra of arbitrary left-invariant metrics on the Lie group. As a departure point we note that the second author showed that there are no non-trivial continuous isospectral deformations of a biinvariant metric within the class of left-invariant metrics on a compact Lie group [10]. This prompts one to ask whether a bi-invariant metric on a compact Lie group G is spectrally isolated within the class of left-invariant metrics.…”

Section: Annales De L'institut Fouriermentioning

confidence: 98%

Spectral isolation of bi-invariant metrics on compact Lie groups

Gordon¹,

Schueth²,

Sutton³

2010

Ann. inst. Fourier

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We show that a bi-invariant metric on a compact connected Lie group G is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a biinvariant metric g0 on G there is a positive integer N such that, within a neighborhood of g0 in the class of left-invariant metrics of at most the same volume, g0 is uniquely determined by the first N distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities). In the case where G is simple, N can be chosen to be two. R ÉSUM É. Soit G un groupe de Lie compact et connexe, et soit g0 une métrique bi-invariante sur G. On démontre que g0 est isolée spectralement dans la classe des métriques invariantes à gauche: Plus précisément, il existe un entier positif N tel que, dans un voisinage de g0 dans la classe des métriques invariantes à gauche et de volume égal ou inférieur à celui de g0, la métrique g0 est determinée de manière unique par les N premières valeurs propres strictement positives de son Laplacien (sans multiplicités). Si G est simple, on peut choisir N = 2.

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“…The proof of Theorem 1.1 uses the Riemannian submersion method (see, e.g., [10], [9], [13], [30], [31], [32]). The present paper generalizes previous results of Gordon and the first author [11], who constructed continuous families of compactly supported perturbations of the Euclidean metric on R n with the same scattering phase.…”

Section: Remark 13 (I)mentioning

confidence: 99%

Isoscattering deformations for complete manifolds of negative curvature

Perry

Schueth²

2006

J Geom Anal

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Abstract. We construct continuous families of non-isometric metrics on simply connected manifolds of dimension n ≥ 9 which have the same scattering phase, the same resolvent resonances, and strictly negative sectional curvatures. This situation contrasts sharply with the case of compact manifolds of negative curvature, where Guillemin/Kazhdan, Min-Oo, and Croke/Sharafutdinov showed that there are no nontrivial isospectral deformations of such metrics.

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Isospectral manifolds with different local geometries

Cited by 29 publications

References 44 publications

Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions

Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions

Spectral isolation of bi-invariant metrics on compact Lie groups

Isoscattering deformations for complete manifolds of negative curvature

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