Mathematical Sciences: Inverse Spectral Problems in Riemannian Geometry

数学科学：黎曼几何中的逆谱问题

• 批准号: 9101355
• 负责人: Carolyn Gordon
• 金额: $ 4.26万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-07-01 至 1992-07-01
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9101355&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Inverse; Spectral; Problems

The principal investigator will continue her study of the Laplace-Beltrami operator on compact Riemannian manifolds. She will try to determine the extent to which the spectrum of the Laplacian determines the geometry of the manifold. One goal of the research is to study examples of isospectral manifolds and identify geometric invariants which are not determined by the spectra. In addition she will try to identify other natural operators whose spectra might distinguish the metrics. Two manifolds or surfaces are said to be isospectral if the corresponding eigenvalues of the Laplacians on the manifolds are equal. If one thought of the two surfaces as surfaces of drums this would mean that the two drums would sound the same when struck. The principal investigator will study situations in which the two surfaces "sound the same" but have different shapes.

首席研究人员将继续研究紧凑型riemannian歧管的拉普拉斯 - 贝特拉米操作员。她将尝试确定拉普拉斯（Laplacian）的光谱确定歧管的几何形状的程度。该研究的一个目标是研究同一歧管的实例，并确定未由光谱确定的几何不变剂。此外，她将尝试识别其他自然操作员的光谱可能会区分指标。 如果laplacians在歧管上的相应特征值相等，则认为两个歧管或表面是同一的。如果一个人认为这两个表面是鼓的表面，那将意味着两个鼓在击中时听起来相同。首席研究人员将研究两个表面“听起来相同”但形状不同的情况。