Distinguishing isospectral nilmanifolds by bundle Laplacians (Q1357639)

In 1984, \textit{C. S. Gordon} and \textit{E. N. Wilson} [J. Differ. Geom. 19, 241-256 (1984; Zbl 0532.58032)] constructed isospectral deformations on compact solvmanifolds. These deformations are all strongly isospectral; the spectrum of any natural self-adjoint elliptic operator associated with the metric is constant during the deformation. In the present paper, the authors show the deformation in question is also \(\pi_1\)-isospectral; the spectrum of any natural selfadjoint elliptic operator associated with the metric and twisted with a representation of the fundamental group is constant during the deformation. The question then is how to detect the inequivalence of the metrics involved using spectral techniques. The authors study non-flat Hermitian line bundles over the manifold in question to detect the non-triviality of these isospectral deformations.