The marked length spectrum vs. the Laplace spectrum on forms on Riemannian nilmanifolds

Abstract

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...

Laplacian spectrum on a nilmanifold, truncations and effective theories

“…Obtaining a four-dimensional low energy effective theory from a ten-dimensional one requires knowledge of the spectrum and eigenmodes of the Laplacian operator on M. Indeed, the eigenvalues correspond to masses of four-dimensional states for a free theory, giving access to energy hierarchies, while the eigenmodes provide a basis to expand the ten-dimensional field, prior to the dimensional reduction; this procedure is discussed in more detail in Section 5. Harmonic analysis on nilmanifolds, in particular on the three-dimensional Heisenberg manifold M , 1 has been considered before in the mathematical literature [4][5][6][7][8], however the results are usually not presented in a way familiar to most physicists. Moreover the analyses available typically do not consider the dependence of the spectrum on the metric modulia rather useful piece of information from the physics point of view as it directly affects the masses of the physical fields.…”

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

“…During the past two decades many new non-isometric isospectral spaces have been found (e.g., [GW], [BT], [BG], [Gt1], [Gt2], [Gor1], [Sza1] and [GGSWW]). 1 The first examples of topological significance were produced by Vignéras and Ikeda. In [Vig] examples of 3-dimensional hyperbolic spaces with non-isomorphic fundamental groups were constructed and in [Ike] isospectral lens spaces were produced.…”

Length spectra andp-spectra of compact flat manifolds

“…This is the case since the ( We notice that all the pairs in the table above are not marked length isospectral. Some of the examples in the table have some similar spectral properties as other known examples in the context of nilmanifolds (see [Go] and [Gt1]). …”

“…Examples of manifolds with similar spectral properties are given in [Go,Ex. 2.4(a)] and in [Gt1,Ex.I], by using 2 and 3-step nilmanifolds, respectively. We note that such an example cannot exist for hyperbolic manifolds since strongly isospectral implies [L]-isospectral in this context (see [GoM]).…”