The spectrum of the Laplacian on Riemannian Heisenberg manifolds (Q1080194)

The spectrum of a Riemannian manifold is the collection of eigenvalues of the Laplacian acting on functions. A Riemannian Heisenberg manifold is a compact manifold of the form \((\Gamma \setminus H_ n,g)\) where \(H_ n\) is the \((2n+1)\)-dimensional Heisenberg group, \(\Gamma\) is a discrete subgroup, and g is a Riemannian metric whose lift to \(H_ n\) is left- invariant. After classifying the Riemannian Heisenberg manifolds and explicitly computing their spectra, we find that any 3-dimensional Heisenberg manifold is uniquely determined by its spectrum. However, in all higher dimensions, there are many pairs of isospectral, non-isometric Riemannian Heisenberg manifolds. Among these examples are pairs with non- isomorphic fundamental groups. A more recent result not included in this paper states that for some of these examples of isospectral manifolds, the Laplacians acting on 1-forms are not isospectra. For others, the Laplacians acting on p-forms are isospectral for all p.