具有不同局部几何的球面上等谱度量对的聚宝盆
A cornucopia of isospectral pairs of metrics on spheres with different local geometries
摘要：
本文总结了在[Sz5]中开始的全面研究，其中第一个非平凡的等谱度量对被构造在球和球体上。这些调查纳入4个不同的情况下，因为这些球和领域都被认为是2步幂零李群和他们的可解扩展。在[Sz5]的考虑是完全结束在球的情况下，并在幂零的情况下。其他案件主要是概述。本文完全建立了球面上的等谱性定理。本文还总结了可解扩张所需的重要细节。一种新的所谓的反交换子技术是为这些建设。该工具与迄今为止在该领域应用的其他方法完全不同。它带来了一系列新的等光谱性的例子。在某些调和流形的测地球面上构造的那些特别引人注目。其中一个球面是齐次的（因为它是2点齐性空间的测地球面），而其他球面虽然与前一个球面是等谱的，但甚至不是局部齐次的。这表明在作用于函数的拉普拉斯算子谱中，关于几何（例如，关于等距）的信息是多么的少。
英文摘要：
This article concludes the comprehensive study started in [Sz5], where the first non-trivial isospectral pairs of metrics are constructed on balls and spheres. These investigations incorporate 4 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 in 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 how little information is encoded about the geometry (for instance, about the isometries) in the spectrum of Laplacian acting on functions.
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