Abstract
The first isospectral pairs are constructed on the most simple simply connected domains, namely, on balls and spheres. This long-standing problem, concerning the existence of such pairs, has been solved by a new method call “anticommutator technique”. Among the wide range of such pairs, the most striking examples are provided on the spheres $S^{4k-1}$, where $k\ge 3$. 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 the Laplacian acting on functions. In other words, the group of isometries, even the local homogeneity property, is lost to the nonaudible in the debate of audible versus nonaudible geometry.
