Isospectral and isotonal metrics with different local geometries

具有不同局部几何形状的等谱和等调度量

• 批准号: 0104361
• 负责人: Zoltan Szabo
• 金额: $ 10万
• 依托单位: Research Foundation Of The City University Of New York (Lehman)
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104361&HistoricalAwards=false
• 关键词: Isospectral; isotonal; metrics; different; local

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.

DMS -0104361摘要该项目的主要部分描述了一种新的等谱构造技术（反换向器技术），它提供了最简单流形上的第一个等谱度量对：球和球体。最引人注目的例子是在合适的球面上构造的，其中一个等谱对的成员是齐次度量，而另一个是局部非齐次的。这证明了一个令人惊讶的事实，即没有关于等距的信息被编码在作用于函数的拉普拉斯算子的谱中。这些调查也延伸到拉普拉斯频谱的形式。相关的问题也被考虑。其中之一是布朗运动等价空间（Isothermal空间）的构造。这种等价关系比等谱性强得多，但它并不决定局部几何。同样的陈述也适用于具有等效密度函数的度规（Isodasyc）。在这些问题的深处，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，整个物理学都是从弯曲的空间中推导出来的。实际上，物理学等同于这个弯曲空间的完整几何学。爱因斯坦是这样表述他的观点的：“没有物理学这样的东西。一切都是几何。与相对论相反，量子物理学只使用几何学的特定方面，例如几个算子的光谱或由度量空间定义的布朗运动。然而，爱因斯坦认为布朗运动可以决定完整的局部几何。 这反映了对以下问题的混淆程度：“量子物理学使用了多少几何？“拟议中的调查表明，第一次，量子物理学使用几何学的信息是多么少。例如，量子物理学完全忽略了空间的等距性，否则它将构成几何学中发展的理论的核心部分。