Inaudibility of naturally reductive property

Abstract In this paper, we use a characterization of naturally reductive 2-step nilponent Lie groups via Ambrose–Singer’s homogeneous structures to prove that one cannot determine if a closed Riemannian manifold is naturally reductive using the information encoded in the spectrum of the Laplace–Beltrami operator. To do that, we consider a new isospectral pair of 2-step nilmanifolds of dimension 9 such that one of them is naturally reductive and the other is not.