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Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal

Carolyn S. Gordon; Juan Pablo Rossetti — 2003

Annales de l'Institut Fourier

Let $M$ be a $2 m$ -dimensional compact Riemannian manifold. We show that the spectrum of the Hodge Laplacian acting on $m$ -forms does not determine whether the manifold has boundary, nor does it determine the lengths of the closed geodesics. Among the many examples are a projective space and a hemisphere that have the same Hodge spectrum on 1- forms, and hyperbolic surfaces, mutually isospectral on 1-forms, with different injectivity radii. The Hodge $m$ -spectrum also does not distinguish orbifolds from manifolds....

Symplectic rigidity of geodesic flows on two-step nilmanifolds

Carolyn S. Gordon; Yiping Mao; Dorothee Schueth — 1997

Annales scientifiques de l'École Normale Supérieure

Inverse spectral results on even dimensional tori

Carolyn S. Gordon; Pierre Guerini; Thomas Kappeler; David L. Webb — 2008

Annales de l’institut Fourier

Given a Hermitian line bundle $L$ over a flat torus $M$ , a connection $\nabla$ on $L$ , and a function $Q$ on $M$ , one associates a Schrödinger operator acting on sections of $L$ ; its spectrum is denoted $S p e c (Q; L, \nabla)$ . Motivated by work of V. Guillemin in dimension two, we consider line bundles over tori of arbitrary even dimension with “translation invariant” connections $\nabla$ , and we address the extent to which the spectrum $S p e c (Q; L, \nabla)$ determines the potential $Q$ . With a genericity condition, we show that if the connection is invariant under...

Spectral isolation of bi-invariant metrics on compact Lie groups

Carolyn S. Gordon; Dorothee Schueth; Craig J. Sutton — 2010

Annales de l’institut Fourier

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 bi-invariant metric $g_{0}$ on $G$ there is a positive integer $N$ such that, within a neighborhood of $g_{0}$ in the class of left-invariant metrics of at most the same volume, $g_{0}$ 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....

Isospectral deformations of closed riemannian manifolds with different scalar curvature

Carolyn S. Gordon; Ruth Gornet; Dorothee Schueth; David L. Webb; Edward N. Wilson — 1998

Annales de l'institut Fourier

We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds , more precisely, on $S^{n} \times T^{m}$ , where $T^{m}$ is a torus of dimension $m \geq 2$ and $S^{n}$ is a sphere of dimension $n \geq 4$ . 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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