Noncompact Inaudibility of the Naturally Reductive Property

Abstract Naturally reductive manifolds form an important class of Riemannian manifolds because they provide examples that generalize locally symmetric ones. A property is said to be inaudible if there exists a unitary operator intertwining the Laplace–Beltrami operators of two Riemannian manifolds such that one of them enjoys the property whereas the other does not. In this paper we study the relation between $ 2 $ -step nilpotent Lie groups and the naturally reductive property and prove that this property is inaudible by using a pair of noncompact $ 11 $ -dimensional generalized Heisenberg groups.