Continuous families of isospectral Riemannian metrics which are not locally isometric

TL;DR

This paper constructs continuous families of isospectral Riemannian metrics with different local geometries, demonstrating that spectral data alone does not determine Ricci curvature, with new examples in manifolds with boundary and closed manifolds.

Contribution

It provides the first examples of isospectral manifolds with different local geometries, showing Ricci curvature is not spectrally determined, in dimensions greater than 6.

Findings

Existence of continuous isospectral metric families with different local geometry

New examples of isospectral pairs of closed manifolds with different local geometry

Spectral data does not uniquely determine Ricci curvature

Abstract

Two Riemannian manifolds are said to be isospectral if the associated Laplace-Belttrami operators have the same eigenvalue spectrum. If the manifolds have boundary, one specifies DIrichlet or Neumann isospectrality depending on the boundary conditions imposed on the eigenfunctions. We construct continuous families of (Neumann and Dirichlet) isospectral metrics which have different local geometry on manifolds with boundary in every dimension greater than 6 and also new examples of pairs of closed isospectral manifolds with different local geometry. These examples illustrate for the first time that the Ricci curvature of a Riemannian manifold is not spectrally determined.