Inaudibility of naturally reductive property

TL;DR

This paper demonstrates that the naturally reductive property of certain Riemannian manifolds cannot be deduced from spectral data, using a new isospectral pair of 2-step nilmanifolds to illustrate this limitation.

Contribution

It introduces a novel isospectral pair of 2-step nilmanifolds, showing the spectral data does not determine the naturally reductive property.

Findings

Spectral data cannot distinguish naturally reductive manifolds from non-naturally reductive ones.

Constructed a new isospectral pair of 9-dimensional 2-step nilmanifolds.

Proved that the naturally reductive property is not spectrally detectable.

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.