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

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 ”Anticommutator Technique”. 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 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”.” To the memory of my son Dániel (1976-1999) Research in Spectral Geometry has been booming in the last two decades. Since Milnor’s [M,1964] first example of a pair of 16 dimensional isospectral but non-isometric flat tori, many new examples were constructed. See some of the most important results in [V,1980]; [I,1980]; [GW,1984,1986]; [S,1985]; [Bu,1986]; [BT,1987] [DG,1989]; [GWW,1992], etc. However, all of these previously constructed isospectral pairs consist of locally isometric spaces and they differ from each other just in global shape; global isometries do not exist in these cases but local isometries do. 1991 Mathematics Subject Classification. Primary 58G25; Secondary 53C20, 22E25.