Mathematical Sciences: Spectral Geometry & Representation Theory on Higher Step Riemannian Nilmanifolds

数学科学：谱几何

• 批准号: 9409209
• 负责人: Ruth Gornet
• 金额: $ 1.8万
• 依托单位: Texas Tech University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1995-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9409209&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Spectral; Geometry; &amp

9409209 GORNET For (M,g) a closed Riemannian manifold, denote by spec(M,g) the collection of eigenvalues with multiplicities of the associated Laplace-Beltrami operator acting on smooth functions on M. Gornet's research focuses on the question "What geometric information is contained in the spectrum of a manifold?" In particular, she has developed a new construction for producing pairs of isospectral Riemannian nilmanifolds. This construction has produced new pairs of isospectral manifolds exhibiting many properties not previously found. Such examples are the only way to discover properties not determined by the spectrum. Geometric properties Gornet is currently studying include the length spectrum (the collection of lengths of closed geodesics) and the marked length spectrum (which also includes information on the free homotopy classes of the closed geodesics.) Techniques used in Gornet's research come from Riemannian geometry, representation theory, and Lie groups. Gornet proposes to use the Research Planning Grant to explore new techniques in representation theory and in the study of closed geodesics to supplement her current background in spectral and Riemannian geometry. ***

小行星9409209 对于（M，g）闭黎曼流形，用spec（M，g）表示作用在M上光滑函数上的Laplace-Beltrami算子的重数特征值的集合. Gornet的研究集中在“流形的谱中包含什么几何信息？”这个问题上。特别是，她开发了一种新的构造来产生等谱黎曼流形对。这种构造产生了新的等谱流形对，它们表现出许多以前没有发现的性质，这些例子是发现谱不确定的性质的唯一途径。 Gornet目前正在研究的几何性质包括长度谱（闭合测地线的长度集合）和标记长度谱（其中还包括闭合测地线的自由同伦类的信息）。Gornet研究中使用的技术来自黎曼几何、表示论和李群。戈尔内建议使用研究规划补助金，以探索新的技术在代表性理论和研究封闭测地线，以补充她目前的背景谱和黎曼几何。 ***