Spectral isolation of bi-invariant metrics on compact Lie groups

Gordon, Carolyn S.; Schueth, Dorothee; Sutton, Craig J.

doi:10.5802/aif.2567

Cited by 5 publications

(7 citation statements)

References 25 publications

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“…However, in light of the fact that most counterexamples in spectral geometry exploit metrics with "large" symmetry groups, Theorem 4.1 and the results of [5] lend strong support to this conjecture.…”

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

confidence: 87%

“…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%

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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: 87%

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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“…In the author's opinion, the best results toward providing an answer to Question 1.1 were obtained in [GSS10]. In this article, Gordon, Schueth and Sutton proved that a bi-invariant metric is spectrally isolated in M G .…”

Section: Introductionmentioning

confidence: 84%

“…For A ∈ GL(m, R), set A = tr(A t A) 1/2 = ( i,j a 2 i,j ) 1/2 . Theorem 2.1 (Gordon, Schueth, Sutton [GSS10]). Let G be a compact simple Lie group, let g I be a bi-invariant metric on G and let g A be a left-invariant metric defined as above with A ∈ GL(m, R) and det(A) ≥ 1 (i.e.…”

Section: Spectra Of Left Invariant Metricsmentioning

confidence: 99%

Spectral Uniqueness of Bi-Invariant Metrics on Symplectic Groups

LAURET

2018

Transformation Groups

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In this short note, we prove that a bi-invariant Riemannian metric on Sp(n) is uniquely determined by the spectrum of its Laplace-Beltrami operator within the class of leftinvariant metrics on Sp(n). In other words, on any of these compact simple Lie groups, every left-invariant metric which is not right-invariant cannot be isospectral to a bi-invariant metric. The proof is elementary and uses a very strong spectral obstruction proved by Gordon, Schueth and Sutton.

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“…The local version of this type of rigidity is often referred to as spectral isolation. The spectral isolation of symmetric or locally symmetric metrics seems to be a folklore conjecture that has been around for some time; see [15] for some recent work and history on this problem. Conjecture B implies the stronger global spectral rigidity conjecture immediately for locally symmetric metrics; one might say the locally symmetric metric is spectrally isolated globally in that case.…”

Section: Final Remarks 51 Conjectural Characterization Of Arithmeticitymentioning

confidence: 99%

Primitive geodesic lengths and (almost) arithmetic progressions

Lafont

McReynolds

2019

Publicacions Matemàtiques

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In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsification of arithmetic progressions, and prove that every negatively curved, closed Riemannian manifold has arbitrarily long almost arithmetic progressions in its primitive length spectrum. Concerning genuine arithmetic progressions, we prove that every non-compact, locally symmetric, arithmetic manifold has arbitrarily long arithmetic progressions in its primitive length spectrum. We end with a conjectural characterization of arithmeticity in terms of arithmetic progressions in the primitive length spectrum. We also suggest an approach to a well known spectral rigidity problem based on the scarcity of manifolds with arithmetic progressions. IntroductionGiven a Riemannian manifold M, the associated geodesic length spectrum is an invariant of central importance. When the manifold M is closed and equipped with a negatively curved metric, there are several results that show primitive, closed geodesics on M play the role of primes in Z (or prime ideals in O K ). Prime geodesic theorems like Huber [19], Margulis [25], and Sarnak [36] on growth rates of closed geodesics of length at most t are strong analogs of the prime number theorem (see, for instance, also [8], [30], [39], and [40]). Sunada's construction of length isospectral manifolds [41] was inspired by a similar construction of non-isomorphic number fields with identical Dedekind ζ -functions (see [28]). The Cebotarev density theorem has also been extended in various directions to lifting behavior of closed geodesics on finite covers (see [42]). There are a myriad of additional results, and this article continues to delve deeper into this important theme. Let us start by introducing some basic terminology:Definition. Let (M, g) be a Riemannian orbifold, and [g] a conjugacy class inside the orbifold fundamental group π 1 (M). We let L [g] ⊂ [0, ∞) consist of the lengths of all closed orbifold geodesics in M which represent the conjugacy class [g]. This could be empty if M is non-compact, and if M is a compact manifold (rather than orbifold), then L [g] takes values in R + := (0, ∞). The length spectrum of (M, g) is the multiset L (M, g) obtained by taking the union of all the sets L [g] , where [g] ranges over all conjugacy classes in M.We say a conjugacy class [g] is primitive if the element g is not a proper power of some other element (in particular g must have infinite order). The primitive length spectrum of (M, g) is the multiset L p (M, g) obtained by taking the union of all the sets L [g] , where [g] ranges over all primitive conjugacy classes in M.

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Spectral isolation of bi-invariant metrics on compact Lie groups

Cited by 5 publications

References 25 publications

Spectral isolation of naturally reductive metrics on simple Lie groups

Spectral isolation of naturally reductive metrics on simple Lie groups

Spectral Uniqueness of Bi-Invariant Metrics on Symplectic Groups

Primitive geodesic lengths and (almost) arithmetic progressions

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