Non-compact inaudibility of Naturally Reductive property

TL;DR

This paper demonstrates that the naturally reductive property of Riemannian manifolds cannot be detected solely from spectral data, by showing it is inaudible through specific examples involving 11-dimensional generalized Heisenberg groups.

Contribution

It proves that the naturally reductive property is inaudible, providing explicit examples with non-compact 11-dimensional generalized Heisenberg groups.

Findings

Naturally reductive property is inaudible.

Constructs explicit examples using 11-dimensional generalized Heisenberg groups.

Shows spectral data cannot distinguish naturally reductive manifolds.

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.