Isospectral metrics and potentials on classical compact simple Lie groups

Proctor, Emily

doi:10.1307/mmj/1123090770

Cited by 12 publications

(8 citation statements)

References 21 publications

(15 reference statements)

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“…Indeed, Schueth [16] and Proctor [14] have shown the existence of non-trivial isospectral deformations of left-invariant metrics on every classical simple compact Lie group except for a few lowdimensional ones. On the other hand, the authors, along with Schueth, demonstrate in a forthcoming article [5], that each bi-invariant metric on a connected compact Lie group is spectrally isolated within the class of all left-invariant metrics.…”

Section: Results 1 Within the Class Of Compact Symmetric Spaces Of A Gmentioning

confidence: 99%

Spectral isolation of naturally reductive metrics on simple Lie groups

Gordon

Sutton

2009

Math. Z.

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We show that within the class of left-invariant naturally reductive metrics $\mathcal{M}_{\operatorname{Nat}}(G)$ on a compact simple Lie group $G$, every metric is spectrally isolated. We also observe that any collection of isospectral compact symmetric spaces is finite; this follows from a somewhat stronger statement involving only a finite part of the spectrum.Comment: 19 pages, new title and abstract, revised introduction, new result demonstrating that any collection of isospectral compact symmetric spaces must be finite, to appear Math Z. (published online Dec. 2009

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Section: Results 1 Within the Class Of Compact Symmetric Spaces Of A Gmentioning

confidence: 99%

Spectral isolation of naturally reductive metrics on simple Lie groups

Gordon

Sutton

2009

Math. Z.

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“…Hence, within the class of left-invariant metrics on a compact Lie group G, any metric TOME 60 (2010), FASCICULE 5 g = g 0 that is isospectral to a bi-invariant metric g 0 must be sufficiently far away from g 0 . In contrast, we note that the second author exhibited the first examples of continuous isospectral families of left-invariant metrics on compact simple Lie groups [10]; see also [9].…”

Section: Annales De L'institut Fouriermentioning

confidence: 95%

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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“…However, as we noted previously, in 1993, Carolyn Gordon produced the first examples of closed isospectral manifolds that are not locally isometric [Gor1,Gor2] via a construction inspired by Szabo's approach to building isospectral, yet locally non-isometric manifolds with boundary [Sz]. 1 The ensuing years have seen many more examples of isospectral, yet locally non-isometric closed manifolds [Sc1,Pr,Sc3], including surprising pairs arising from the third named author's generalization of Sunada's method [Sut, AYY]. s examples of isospectral metrics on S 2 × T 2 [Sc2] demonstrated that the local geometry of a closed Riemannian manifold of dimension at least four need not be encoded in its spectrum, leaving open the following problem.…”

Section: Problemmentioning

confidence: 99%

“…Restricting our attention to locally homogeneous spaces, the examples of Gordon [Gor1,Gor2] demonstrate that in dimension 8 and higher isospectral locally homogeneous spaces need not be locally isometric (cf. [GorWil2,Sc2,Sut,Pr]). Furthermore, making use of the third author's generalization of Sunada's method [Sut], An, Yu and Yu [AYY] demonstrate that isospectral locally homogeneous spaces of dimension at least 26 need not share the same model geometry (cf.…”

Section: Introductionmentioning

confidence: 99%

Geometric structures and the Laplace spectrum

Lin¹,

Schmidt²,

Sutton³

2019

Preprint

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Inspired by the role geometric structures play in our understanding of surfaces and three-manifolds, and Berger's observation that a surface of constant sectional curvature is determined up to local isometry by its Laplace spectrum, we explore the extent to which compact locally homogeneous three-manifolds are characterized up to local isometry by their spectra. We observe that there are eight "metrically maximal" three-dimensional geometries on which all compact locally homogeneous three-manifolds are modeled and we demonstrate that for five of these geometries the associated compact locally homogeneous three-manifolds are determined up to local isometry by their spectra within the universe of locally homogeneous three-manifolds. Specifically, we show that among compact locally homogeneous three-manifolds, a Riemannian three-manifold is determined up to local isometry if its universal Riemannian cover is isometric to (1) a symmetric space, (2) R 2 ⋊ R endowed with a left-invariant metric, (3) Nil endowed with a left-invariant metric, or (4) S 3 endowed with a left-invariant metric sufficiently close to a metric of constant sectional curvature. We then deduce that three-dimensional Riemannian nilmanifolds and locally symmetric spaces with universal Riemannian cover S 2 × E are uniquely characterized by their spectra among compact locally homogeneous three-manifolds. Finally, within the collection of closed manifolds covered by Sol equipped with a left-invariant metric, local geometry is "audible." 2010 Mathematics Subject Classification. 58J50, 53C20.

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Isospectral metrics and potentials on classical compact simple Lie groups

Cited by 12 publications

References 21 publications

Spectral isolation of naturally reductive metrics on simple Lie groups

Spectral isolation of naturally reductive metrics on simple Lie groups

Spectral isolation of bi-invariant metrics on compact Lie groups

Geometric structures and the Laplace spectrum

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