Non-compact inaudibility of Naturally Reductive property
Teresa Arias-Marco, José-Manuel Fernández-Barroso
TL;DR
The paper investigates whether the naturally reductive property is audible in noncompact 2-step nilpotent Lie groups, focusing on generalized Heisenberg groups. It leverages the structure of $\mathfrak{n}=\mathfrak{v}\oplus\mathfrak{z}$ with $j:\mathfrak{z}\to\mathfrak{so}(\mathfrak{v})$ and known classifications to characterize natural reductivity in terms of the center dimension and isotypy of $\mathfrak{v}$. By combining Szabó's explicit isospectral intertwiners with Gordon’s results on isospectral noncompact nilmanifolds, the authors show that natural reductivity is inaudible: isospectral noncompact generalized Heisenberg groups can differ in NR status, exemplified by $N(2,0)$ (NR) vs $N(1,1)$ (not NR) with center of dimension $3$. This demonstrates that the Laplace–Beltrami spectrum alone cannot determine NR status in noncompact settings, highlighting a limitation in spectral geometry for this geometric property.
Abstract
Naturally reductive manifolds are an important class of Riemannian manifolds because they provide examples that generalize the locally symmetric ones. A property is said to be inaudible if there exists a unitary operator which intertwines the Laplace-Beltrami operator of two Riemannian manifolds such that one of them satisfies the property and the other does not. In this paper, we study the relation between 2-step nilpotent Lie groups and the naturally reductive property to prove that this property is inaudible, using a pair of non-compact 11-dimensional generalized Heisenberg groups.