Spaces of Nonpositive Curvature and Geometric Rigidity

项目编号：dms -9971749首席研究员：Christopher croke本项目涉及两个主要主题。第一篇论文是与Bruce Kleiner合著的，研究了无穷群G紧作用于非正弯曲空间H（在alexandrov意义上），并讨论了H的几何形状与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Gromov的一些问题所激发的。另一个主题，包括Kleiner和Sharafutdinov作为合作者（在不同方面），是紧流形上的刚性定理（即度量唯一性）的研究。例如，这里我们考虑等谱问题：具有相同谱的空间（例如拉普拉斯贝尔特拉米算子的特征值，或封闭测地线的长度）在多大程度上必须是等距的。这也包括由测地线流动的共轭性引起的度量刚性问题，以及逆散射问题。该项目的第二个主题是关于空间是否可以由一组特定数据确定的问题。其中一部分与遥感问题有关。例如：你能通过“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，其中人们通过测量沿直线的总质量来确定物体的质量密度。另一组测量是声音在边界上任意两点之间传播所需的时间（这是建议中处理的边界刚性问题的一个特殊情况）。拟议研究的主旨是确定在何种情况下某些数据集(例如：特征值，封闭测地线的长度，边界点之间的距离)足以完全确定所讨论空间的几何形状。群体自然表现出各种空间的不对称性。本课题的第一个主题涉及一类无限群，它们是hadamard空间H（包括非正曲率空间）的对称。因此，它们在h的自然边界B上产生对称性。Bruce Kleiner和PI发现群在B上的作用与h的几何形状之间存在意想不到的关系。人们曾怀疑，在这种情况下，B上的作用仅由群的性质决定（就像作用于负弯曲空间的双曲群的相关设置中的情况一样）。本研究旨在确定这种关系的确切性质，并利用它来研究群体和空间的性质。

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.