  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"."
