THE MARKED LENGTH SPECTRUM VS THE LAPLACE SPECTRUM ON FORMS ON RIEMANNIAN NILMANIFOLDS

The subject of this paper is the relationships among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particul ar, we show that for a large class of three-step nilmanifolds, if a pa ir of nilmanifoIds in this class has the same marked length spectrum, they necessarily share the same Laplace spectrum on functions. In cont rast, we present the first example of a pair of isospectral Riemannian manifolds with the same marked length spectrum but not the same spect rum 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 o f Riemannian manifolds with the same marked length spectrum, but not t he same spectrum on forms. This partially extends and partially contra sts the work of Eberlein, who showed that on two-step nilmanifolds, th e same marked length spectrum implies the same Laplace spectrum both o n functions and on forms.