TAILIEUCHUNG - Đề tài " A cornucopia of isospectral pairs of metrics on spheres with different local geometries "

This article concludes the comprehensive study started in [Sz5], where the first nontrivial isospectral pairs of metrics are constructed on balls and spheres. These investigations incorporate four different cases since these balls and spheres are considered both on 2-step nilpotent Lie groups and on their solvable extensions. In [Sz5] the considerations are completely concluded in the ball-case and in the nilpotent-case. The other cases were mostly outlined. In this paper the isospectrality theorems are completely established on spheres. . | Annals of Mathematics A cornucopia of isospectralpairs of metrics on spheres with different local geometries By Z. I. Szab o Annals of Mathematics 161 2005 343 395 A cornucopia of isospectral pairs of metrics on spheres with different local geometries By Z. I. Szabo Abstract This article concludes the comprehensive study started in Sz5 where the first nontrivial isospectral pairs of metrics are constructed on balls and spheres. These investigations incorporate four different cases since these balls and spheres are considered both on 2-step nilpotent Lie groups and on their solvable extensions. In Sz5 the considerations are completely concluded in the ball-case and in the nilpotent-case. The other cases were mostly outlined. In this paper the isospectrality theorems are completely established on spheres. Also the important details required about the solvable extensions are concluded in this paper. A new so called anticommutator technique is developed for these constructions. This tool is completely different from the other methods applied on the field so far. It brought a wide range of new isospectrality examples. Those constructed on geodesic spheres of certain harmonic manifolds are particularly striking. One of these spheres is homogeneous since it is the geodesic sphere of a 2-point homogeneous space while the other spheres although isospectral to the previous one are not even locally homogeneous. This shows that how little information is encoded about the geometry for instance about the isometries in the spectrum of Laplacian acting on functions. Research in spectral geometry started out in the early 60 s. This field might as well be called audible versus nonaudible geometry. This designation much more readily suggests the fundamental question of the field To what extent is the geometry of compact Riemann manifolds encoded in the spectrum of the Laplacian acting on functions It started booming in the 80 s however all the isospectral metrics constructed until .

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