Isospectral pairs of metrics on balls, spheres, and other manifolds with different local geometries

Abstract

The first isospectral pairs of metrics are constructed on balls and spheres. This long standing problem, concerning the existence of such pairs, has been solved by a new method called "Anticommutator Technique." Among the wide range of such pairs, the most striking examples are provided on (4k-1)-dimensional spheres, where k > 2. One of these metrics is homogeneous (since it is the metric on the geodesic sphere of a 2-point homogeneous space), while the other is locally inhomogeneous. These examples demonstrate the surprising fact that no information about the isometries is encoded in the spectrum of Laplacian acting on functions. In other words, "The group of isometries, even the local homogeneity property, is lost to the "Non-Audible" in the debate of "Audible versus Non-Audible Geometry"."