Inaudibility of naturally reductive property
Teresa Arias-Marco, José-Manuel Fernández-Barroso
TL;DR
The paper addresses whether the naturally reductive property can be discerned from the spectrum of the Laplace–Beltrami operator on closed manifolds. It constructs an explicit isospectral pair of $9$-dimensional compact $2$-step nilmanifolds, $N^j$ and $N^{j'}$, defined via quaternion-based maps on a $\mathfrak{v}\oplus\mathfrak{z}$ decomposition, and proves that $N^j$ is naturally reductive while $N^{j'}$ is not by employing Ambrose–Singer homogeneous structures and a Type $\mathcal{A}$ analysis. This yields a concrete inaudibility result: the spectrum cannot determine natural reductivity. The work extends prior inaudibility results to a new class of nilmanifolds and highlights limitations in spectral geometry for detecting homogeneous geometric properties.
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.