Continuous Families of Riemannian manifolds, isospectral on functions but not on 1-forms

TL;DR

This paper introduces the first continuous families of Riemannian manifolds that are isospectral on functions but not on 1-forms, providing counterexamples to existing conjectures in higher-step nilmanifolds.

Contribution

It presents new examples of isospectral manifolds that challenge previous assumptions, especially in higher-step nilmanifolds, using a general construction method.

Findings

First continuous families of manifolds isospectral on functions but not on 1-forms.

Counterexamples to the Ouyang-Pesce Conjecture for higher-step nilmanifolds.

Method for constructing isospectral Riemannian nilmanifolds extended to new cases.

Abstract

The purpose of this paper is to present the first continuous families of Riemannian manifolds 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. The examples presented here are Riemannian three-step nilmanifolds and thus provide a counterexample to the Ouyang-Pesce Conjecture for higher-step nilmanifolds. Ouyang and Pesce independently showed that all isospectral deformations of two-step nilmanifolds must arise from the Gordon-Wilson method for constructing isospectral nilmanifolds. In particular, all continuous families of Riemannian two-step nilmanifolds that are isospectral on functions must also be isospectral on p-forms for all p. They conjectured that all isospectral deformations of nilmanifolds must arise in this manner. These examples arise from a general method for constructing isospectral Riemannian nilmanifolds previously introduced by the author.