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.
In 2004, Sormani and Wei introduced the covering spectrum: a geometric invariant that isolates part of the length spectrum of a Riemannian manifold. In their paper they observed that certain Sunada isospectral manifolds share the same covering spectrum, thus raising the question of whether the covering spectrum is a spectral invariant. In the present paper we describe a group theoretic condition under which Sunada's method gives manifolds with identical covering spectra. When the group theoretic condition of our method is not met, we are able to construct Sunada isospectral manifolds with distinct covering spectra in dimension 3 and higher. Hence, the covering spectrum is not a spectral invariant. The main geometric ingredient of the proof has an interpretation as the minimum-marked-lengthspectrum analogue of Colin de Verdière's classical result on constructing metrics where the first k eigenvalues of the Laplace spectrum have been prescribed.1991 Mathematics Subject Classification. 53C20, 58J50.
Abstract. The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the same marked length spectrum, they necessarily share the same Laplace spectrum on functions. In contrast, we present the first example of a pair of isospectral Riemannian manifolds with the same marked length spectrum but not the same spectrum on one-forms. Outside of the standard spheres vs. the Zoll spheres, which are not even isospectral, this is the only example of a pair of Riemannian manifolds with the same marked length spectrum, but not the same spectrum on forms. This partially extends and partially contrasts the work of Eberlein, who showed that on two-step nilmanifolds, the same marked length spectrum implies the same Laplace spectrum both on functions and on forms.Section 1: Introduction.The spectrum of a closed Riemannian manifold (M, g), denoted spec(M, g), is the collection of eigenvalues with multiplicities of the associated Laplace-Beltrami operator acting on smooth functions. Two Riemannian manifolds (M, g) and (M ′ , g ′ ) are said to be isospectral if spec(M, g) = spec(M ′ , g ′ ). The Laplace-Beltrami operator may be extended to act on smooth p-forms by ∆ = dδ + δd, where δ is the adjoint of d and p is a positive integer. We call its eigenvalue spectrum the p-form spectrum.The length spectrum of a Riemannian manifold is the set of lengths of smoothly closed geodesics, counted with multiplicity. The multiplicity of a length is defined as the number of distinct free homotopy classes that contain a closed geodesic of that length. We denote the length spectrum of (M, g) by [L]-spec(M, g). This is a natural notion, since the geodesic of shortest length in a free homotopy class Two Riemannian manifolds (M 1 , g 1 ) and (M 2 , g 2 ) have the same marked length spectrum if there exists an isomorphism between the fundamental groups of M 1 and M 2 such that corresponding free homotopy classes contain smoothly closed geodesics of the same length. Clearly, manifolds with the same marked length spectrum necessarily have the same length spectrum.The purpose of this paper is to study the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions and the Laplace spectrum on forms on Riemannian nilmanifolds.The relationship between the Laplace spectrum and lengths of closed geodesics arises from the study of the wave equation (see [DGu], [GuU]), and in the case of compact, hyperbolic manifolds, from the Selberg Trace Formula (see [C], Chapter XI). Colin de Verdiere [CdV] has shown that generically, the Laplace spectrum determines the length spectrum. On Riemann surfaces, Huber showed that the length spectrum and the Laplace spectrum are equivalent notions (see [Bu] for an exposition).The Poisson formula gives the relationship between the Lap...
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