The Marked Length Spectrum vs. The Laplace Spectrum on Forms on Riemannian Nilmanifolds

TL;DR

This paper investigates the relationship between the marked length spectrum and the Laplace spectrum on functions and forms on Riemannian nilmanifolds, revealing cases where these spectra coincide and differ, especially in three-step nilmanifolds.

Contribution

It demonstrates that for many three-step nilmanifolds, identical marked length spectra imply identical Laplace spectra on functions, and provides the first example where these spectra differ on forms.

Findings

Same marked length spectrum implies same Laplace spectrum on functions for many three-step nilmanifolds.

First example of isospectral manifolds with same marked length spectrum but different spectra on one-forms.

Contrasts previous results by Eberlein on two-step nilmanifolds.

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.