ISOSPECTRAL DEFORMATIONS ON RIEMANNIAN-MANIFOLDS WHICH ARE DIFFEOMORPHIC TO COMPACT HEISENBERG MANIFOLDS

It is known that if H-m is the classical (2m + 1)-dimensional Heisenbe rg group, Gamma a cocompact discrete subgroup of H-m and g a left inva riant metric, then (Gamma\H-m, g) is infinitesimally spectrally rigid within the family of left invariant metrics. The purpose of this paper is to show that for every m greater than or equal to 2 and for a cert ain choice of Gamma and g, there is a deformation (Gamma\H-m, g(alpha) ) with g = g(1) such that for every alpha not equal 1, (Gamma\H-m, g(a lpha)) does admit a nontrivial isospectral deformation. For alpha not equal 1 the metrics g(alpha) will not be H-m-left invariant, and the ( Gamma\H-m, g(alpha)) will not be nilmanifolds, but still solvmanifolds .