The Journal of Geometric Analysis

, Volume 11, Issue 4, pp 649-667

First online:

Flat manifolds isospectral on p-forms

  • R. J. MiatelloAffiliated withFa.M.A.F., Universidad Nacional de Córdoba
  • , J. P. RossettiAffiliated withFa.M.A.F., Universidad Nacional de Córdoba

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We study isospectrality on p-forms of compact flat manifolds by using the equivariant spectrum of the Hodge-Laplacian on the torus. We give an explicit formula for the multiplicity of eigenvalues and a criterion for isospectrality. We construct a variety of new isospectral pairs, some of which are the first such examples in the context of compact Riemannian manifolds. For instance, we give pairs of flat manifolds of dimension n=2p, p≥2, not homeomorphic to each other, which are isospectral on p-forms but not on q-forms for q∈p, 0≤q≤n. Also, we give manifolds isospectral on p-forms if and only if p is odd, one of them orientable and the other not, and a pair of 0-isospectral flat manifolds, one of them Kähler, and the other not admitting any Kähler structure. We also construct pairs, M, M′ of dimension n≥6, which are isospectral on functions and such that βp(M)<βp(M’), for 0<p<n and pairs isospectral on p-forms for every p odd, and having different holonomy groups, ℤ4 and ℤ 2 2 , respectively.

Math Subject Classifications

primary 58J53 secondary: 20H15

Key Words and Phrases

isospectral Bieberbach group p-spectrum