Riemannian manifolds isospectral on functions but not on 1-forms (Q1070554)

For (M,G) a compact Riemannian manifold, let \(spec^ p (M,g)\) denote the spectrum of the Laplace-Beltrami operator acting on the space of smooth p-forms on M. Two manifolds (M,g) and (M',g') will be said to be p- isospectral if \(spec^ p (M,g)=spec^ p (M',g')\). It would be of interest to determine whether for each k, the collection of all \(spec^ p (M,g)\), \(p=0,1,...,k\) contains more information than does \(spec^ p (M,g)\), \(p=0,1,...,k-1\). We answer the question affirmatively when \(k=1\) by constructing examples of 0-isospectral manifolds which are not 1- isospectral. The manifolds involved are compact quotients of Heisenberg groups.