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The spectrum of the Laplacian on Riemannian Heisenberg manifolds. (English) Zbl 0599.53038
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.

MSC:
53C20 Global Riemannian geometry, including pinching
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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