Isospectral and isotonal metrics with different local geometries

本项目的主要部分描述了一种新的等谱构造技术(反换位器技术)，它在最简单的流形上提供了第一对等谱度量对：球和球面上的度量。最引人注目的例子是在合适的球面上构造，其中等谱派的一个成员是齐次度规，而另一个成员是局部非齐次的。这证明了一个令人惊讶的事实，即没有关于等距线的信息编码在作用于函数的拉普拉斯量的频谱中。这些研究也扩展到拉普拉斯的形式谱。相关问题也被考虑在内。其中之一是构造布朗运动等价空间(等温度量)。这种等价关系比等谱性质强得多，但它不决定局部几何。同样的说法也适用于具有等价密度函数的度量学(Isodasyc Metrics)。在这些问题的深度中，很容易发现相对论和量子物理学之间的旧争论。在相对论中，整个物理学都是从一个弯曲的空间派生出来的。实际上，物理学等同于这个弯曲空间的完整几何。爱因斯坦是这样说的：“没有物理学这回事，一切都是几何学。”与相对论相反，量子物理学只使用几何学的特定方面，如几个算符的光谱或由度量空间定义的布朗运动。然而，爱因斯坦提出，布朗运动可能决定了完整的局部几何。这反映了人们对以下问题的困惑程度：量子物理使用了多少几何？拟议中的调查首次表明，量子物理使用的关于几何的信息是多么少。例如，量子物理完全忽略了空间的等距关系，否则，空间的等距关系就构成了在《几何学》中发展的一个理论的核心部分。

Abstract for DMS - 0104361The main part of the project describes a new isospectralconstruction technique (Anticommutator Technique), which providesthe first isospectral pairs of metrics on the most simple manifolds: on balls and spheres. The most striking examples are constructedon suitable spheres, where one of the members of an isospectral pairis a homogeneous metric, while the other is locally inhomogeneous. This demonstrates the surprising fact that no information about the isometries is encoded in the spectrum of the Laplacian acting on functions. These investigations also extend to the Laplacian spectrum of forms. Related questions are also considered. One of them is construction of Brownian-motion-equivalent spaces (Isothermal Metrics). This equivalence relation is much stronger then the isospectrality property, yet it does not determine the local geometry. The same statement is true regarding the metrics with equivalent density functions (Isodasyc Metrics).The old argument between Relativity and Quantum Physics is easily discovered in the depth of these questions. In Relativity, the whole Physics is derived from a curved space. Actually, Physics is identified with the complete Geometry of this curved space. Einstein put his idea this way: "There is no such thing as Physics. Everything is Geometry." Contrary to Relativity, the Quantum Physics uses only particular aspects of Geometry such as the spectra of several operators or the Brownian Motion defined by a metric space. Einstein suggested, however, that the Brownian Motion may determine the complete local geometry. This reflects the extent of the confusion about thefollowing question: "How much Geometry is used by the Quantum Physics?"The proposed investigations demonstrate, for the first time, how littleinformation about Geometry is used by Quantum Physics. For instance,Quantum Physics completely ignores the isometries of the spaces, which otherwise form the central piece of a theory developed in Geometry.