Abstract
It is known that ifH m is the classical (2m+1)-dimensional Heisenberg group, Γ a cocompact discrete subgroup ofH m andg a left invariant metric, then (Γ/H m, g) is infinitesimally spectrally rigid within the family of left invariant metrics. The purpose of this paper is to show that for everym≥2 and for a certain choice of Γ andg, there is a deformation (Γ/H m, g α) withg=g 1 such that for every α≠1, (Γ/H m, g α)does admit a nontrivial isospectral deformation. For α≠1 the metricsg α will not beH m-left invariant, and the (Γ/H m, gxα) will not be nilmanifolds, but still solvmanifolds.
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Schueth, D. Isospectral deformations on Riemannian manifolds which are diffeomorphic to compact Heisenberg manifolds. Commentarii Mathematici Helvetici 70, 434–454 (1995). https://doi.org/10.1007/BF02566017
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DOI: https://doi.org/10.1007/BF02566017