R. Gornet, THE MARKED LENGTH SPECTRUM VS THE LAPLACE SPECTRUM ON FORMS ON RIEMANNIAN NILMANIFOLDS, Commentarii mathematici helvetici, 71(2), 1996, pp. 297-329
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.