Isospectral deformations of negatively curved Riemannian manifolds with boundary which are not locally isometric

Abstract

To what extent does the eigenvalue spectrum of the Laplace-Beltrami operator on a compact Riemannian manifold determine the geometry of the manifold? We present a method for constructing isospectral manifolds with different local geometry, generalizing an earlier technique. Examples include continuous families of isospectral negatively curved manifolds with boundary as well as various pairs of isospectral manifolds. The latter illustrate that the spectrum does not determine whether a manifold with boundary has negative curvature, whether it has constant Ricci curvature, and whether it has parallel curvature tensor, and the spectrum does not determine whether a closed manifold has constant scalar curvature.