Spaces of Nonpositive Curvature and Geometric Rigidity

非正曲率和几何刚度空间

• 批准号: 9971749
• 负责人: Christopher Croke
• 金额: $ 26.07万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971749&HistoricalAwards=false
• 关键词: Spaces; Nonpositive; Curvature; Geometric; Rigidity

AbstractAward: DMS-9971749Principal Investigator: Christopher CrokeThis project concerns two major themes. The first, joint workwith Bruce Kleiner, considers infinite groups G actingcocompactly on nonpositively curved spaces H (in the sense ofAlexandrov), and treats the relationship between the geometry ofH and the induced action of G on the ideal boundary of H. Thiscan be considered an aspect of geometric group theory and ispartially motivated by some questions of Gromov. The other theme,involving Kleiner and Sharafutdinov as coauthors (on differentaspects), is the study of rigidity theorems (i.e. metricuniqueness) on compact manifolds. Here for example we considerisospectral problems: to what extent must spaces with the samespectra (e.g. eigenvalues of the Laplace Beltrami operator, orlengths of closed geodesics) be isometric. This also includesquestions about metric rigidity induced by conjugacy of geodesicflows, as well as inverse scattering problems.The second theme of the project concerns the question of whethera space can be determined by a certain set of data. One part ofthis relates to questions of remote sensing. For example: canyou determine the density of an object (say a persons body or themoon) from measurements taken "from the outside"? The CAT scanis a practical example where one determines the mass density ofan object from measurements of the total mass along straightlines. An alternative set of measurements is the set of times ittakes for sound to travel between any two points on the boundary(this is a special case of the boundary rigidity question dealtwith in the proposal). The thrust of the proposed study is todetermine under which circumstances certain sets of data(e.g. eigenvalues, lengths of closed geodesics, distances betweenboundary points) are sufficient to completely determine thegeometry of the spaces in question. Groups show up naturally assymmetries of various spaces. The first theme of this projectconcerns a class of infinite groups which are symmetries ofHadamard spaces H (which include spaces of nonpositivecurvature.) As such they induce symmetries on a naturalboundary, B, of H. Bruce Kleiner and the PI have found anunexpected relationship between the action of the group on B andthe geometry of H. It had been suspected that in this settingthat the action on B would be determined only by the nature ofthe group (as is the case in the related setting of hyperbolicgroups acting on negatively curved spaces). This study intendsto determine the precise nature of this relationship and use itto study properties of the group and space.

摘要奖：DMS-9971749首席研究员：克里斯托弗克罗克这个项目涉及两个主要主题。 第一部分是与布鲁斯·克莱纳（Kleiner）的合作，考虑了在非正曲空间H（在Alexandrov意义下）上作用于余紧的无限群G，并讨论了H的几何与G在H的理想边界上的诱导作用之间的关系。 这可以被认为是几何群论的一个方面，部分是由格罗莫夫的一些问题所激发的。另一个主题，涉及Kleiner和Sharafutdinov作为合著者（在不同方面），是研究刚性定理（即度量唯一性）紧流形。 例如，我们在这里讨论等谱问题：在何种程度上必须空间的相同谱（例如特征值的拉普拉斯贝尔特拉米算子，或长度封闭测地线）是等距的。 这也包括有关度量刚性引起的共轭的geodisflows问题，以及逆散射问题。该项目的第二个主题涉及的问题，whether一个空间可以确定的一组特定的数据。 其中一部分涉及遥感问题。 例如：你能从“外部”测量来确定一个物体（比如一个人的身体或月亮）的密度吗？ CAT扫描是一个实际的例子，其中一个确定的质量密度ofan对象的总质量的测量沿着直线。 另一组测量值是声音在边界上任何两点之间传播所需的时间（这是提案中处理的边界刚性问题的特殊情况）。 所提出的研究的主旨是确定在何种情况下某些数据集（例如特征值，闭合测地线的长度，边界点之间的距离）足以完全确定所讨论的空间的几何形状。 群自然地表现出各种空间的不对称性。 本课题的第一个主题是关于一类无限群，它们是Hadamard空间H（包括非正曲率空间）的对称群。 因此，它们在H的自然边界B上诱导对称性。 布鲁斯克莱纳和PI发现了一个意想不到的关系之间的行动集团对B和几何形状的H。 人们曾怀疑，在这种情况下，对B的作用将仅由群的性质决定（就像双曲群作用于负弯曲空间的相关情况一样）。 本研究试图确定这种关系的确切性质，并利用它来研究群和空间的性质。