Isospectral Pairs of Metrics on Balls, Spheres, and other Manifolds with Different Local Geometries

Abstract

The first isospectral pairs of metrics 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 called Among the wide range of such pairs, the most striking examples are provided on the spheres S4k-1, where k > 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. In this article we settle a long-standing and intensively studied problem of spectral geometry by constructing the first examples of nontrivial isospectral metrics on the simplest simply connected manifolds, namely, on balls and spheres. This result is achieved by a new isospectral construction technique called anticommutator technique. It provides a whole cornucopia of such examples, not just on balls and spheres, but on other types of manifolds too. The most striking examples are constructed on the geodesic spheres of the hyperbolic quaternionic 2-point homogeneous spaces. The induced metrics on these spheres are obviously homogeneous. While the metrics constructed isospectrally to these are locally inhomogeneous. This shows the weakness of the information about the local geometry encoded in the spectrum of Laplacian acting on functions.