Isospectral deformations on Riemannian manifolds which are diffeomorphic to compact Heisenberg manifolds (Q1906917)

Let \(H_m\) be the classical \((2m + 1)\)-dimensional Heisenberg group, \(\Gamma\) a cocompact discrete subgroup of it, and \(g^t\) a continuous family of left invariant metrics. It is well known that if all the manifolds occurring during the deformations \((\Gamma / H_m, g^t)\) are isospectral, then this deformation must be trivial. In other words, \((\Gamma\setminus H_m, g^t)\) is infinitesimally spectrally rigid within the family of left invariant metrics. The author proves that for every \(m > 2\) there is a cocompact discrete subgroup \(\Gamma\) of \(H_m\) and a continuous 2-parameter family of metrics \(g_\alpha^t(0 < \alpha \leq 1,\;t \in \mathbb{R})\) on \(H_m\) such that for every fixed \(\alpha\), the deformation \((\Gamma / H_m, g^t_\alpha)\) with varying \(t\) is isospectral; for \(\alpha = 1\) the \(g^t_1\) are \(H_m\)-left invariant, and this deformation is trivial. But for every \(0 < \alpha < 1\), the corresponding isospectral deformation is nontrivial. For fixed \(t\) and varying \(\alpha\) the deformation \((\Gamma / H_m, g^t_\alpha)\) is not isospectral. The \(g^t_\alpha\) with \(\alpha < 1\) are not \(H_m\)-left invariant, but left invariant with respect to a different group structure.