A cornucopia of isospectral pairs of metrics on spheres with different local geometries. (English) Zbl 1087.53037

Research in spectral geometry started in the early 60’s. This field can be called audible versus nonaudible geometry and the fundamental question is the following: To what extent is the geometry of compact Riemannian manifolds encoded in the spectrum of the Laplacian acting on functions?
In this paper the author gives a review of results leading to isospectral pairs of metrics with different local geometries. These examples show how little information on the geometry is encoded in the spectrum of the Laplacian acting on functions. The reviewed paper concludes the comprehensive study started in the previous paper of the author [Ann. Math. (2) 154, 437–475 (2001; Zbl 1012.53034)].
The investigations concern four different cases, since the author performs them on the ball- and sphere-type domains both of 2-step nilpotent Lie groups and their solvable extensions. The details are shared between these two papers and the important details concerning the sphere-type domains and the solvable extensions are described in the reviewed paper. A new so called anticommutator technique is developed for these constructions. The author proves two main isospectrality theorems and a few nonisometry theorems. A cornucopia announced in the title concerns the final Cornucopia Theorem, which is a combination of the isospectrality theorems and of the nonisometry theorems. This theorem describes the abundance of the isospectral pairs of metrics constructed by the anticommutator technique on spheres with different local geometries.


53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J53 Isospectrality


Zbl 1012.53034
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