Abstract: This paper examines the length spectrum on two-step nilmanifolds toward determining what, exactly, the wave trace says about isospectral manifolds. In particular, for each length occurring in the length spectrum of a two-step nilmanifold, we compute the leading order term in the associated wave invariant, under the assumption of the clean intersection hypothesis. As an application, we explain certain examples of Heisenberg manifolds, constructed by C. S. Gordon \cite{G1}, that are isospectral on functions, but have different multiplicities in the length spectrum. The multiplicity of a length is defined here as the number of free homotopy classes of loops that can be represented by a closed geodesic of that length.
beamer slides from my talk on the wave trace at the
CUNY Geometric Analysis Conference : The Laplace and Length Spectra ; February 3-5, 2006
- seminar talk, wave invariants of torus
Abstract: The purpose of this paper is to study the $p$-form spectrum of the Laplacian $\Delta_{p}$ acting on lens spaces.
Examples of non-isometric lens spaces that are isospectral for all $p < p_{0}$ and not for $p \ge p_{0}$ (up to duality) were given by Ikeda (\cite{ikeda}). Here we consider the opposite situation: Can there exist non-isometric pairs of lens spaces that are isospectral for some $p_{0} > 0$, and not for any $p < p_{0}?$ We affirmatively answer this question by presenting examples of such pairs that have been found computationally. We discuss the approach to these computations, results of the computations, applications to representation theory, as well as the representation theory underlying the problem.