Continuous families of Riemannian manifolds, Isospectral on functions but not on 1-forms

The purpose of this paper is to present the first continuous families of Riemannian manifolds that are isospectral on functions but not on 1-forms, and, simultaneously, the first continuous families of Riemannian manifolds with the same marked length spectrum but not the same 1-form spectrum. Examples of isospectral manifolds that are not isospectral on forms are sparse, as most examples of isospectral manifolds can be explained by Sunada’s method or its generalizations, hence are strongly isospectral. The examples here are three-step Riemannian nilmanifolds, arising from a general method for constructing isospectral Riemannian nilmanifolds previously presented by the author. Gordon and Wilson constructed the first examples of nontrivial isospectral deformations, continuous families of Riemannian nilmanifolds. Isospectral manifolds constructed using the Gordon-Wilson method, a generalized Sunada method, are strongly isospectral and must have the same marked length spectrum. Conversely, Ouyang and Pesce independently showed that all isospectral deformations of two-step nilmanifolds must arise from the Gordon-Wilson method, and Eberlein showed that all pairs of two-step nilmanifolds with the same marked length spectrum must come from the Gordon-Wilson method. To the memory of Hubert Pesce, a valued friend and colleague.