Nonpositive Curvature and Geometric Rigidity

摘要DMS-0202536本项目涉及两个主要主题。第一个是对紧致流形上的刚性定理(即度量唯一性)的研究。例如，我们考虑等谱问题：具有相同谱的空间(例如，Laplace Beltrami运算符的本征值或闭测地线的长度)在多大程度上必须是等距的。这也包括由测地线流的共轭引起的度量刚性问题，以及逆散射问题。另一类是在非正曲空间H(Alexandrov意义下)上紧作用的无限群G。这个项目是研究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Gromov的一些问题引起的。除了这两个主要主题外，该提案还涉及到作者就各种等周不等长问题继续开展的工作。这些群表现为Hadamard空间H(包括非正曲率空间)的对称性。该项目的第一个主题涉及是否可以通过一组特定的数据来确定空间的问题。这其中的一部分与遥感问题有关。例如：你能从“外部”的测量结果中确定一个物体(比如人、物体或月球)的密度吗？CAT扫描是一个实际的例子，人们可以通过测量直线上的总质量来确定物体的质量密度(或者更准确地说，吸收系数)。另一种测量方法是声音在边界上任何两点之间传播所需的时间集(这是提案中涉及的边界刚性问题的一个特例)。拟议研究的主旨是确定在哪些情况下某些数据集(例如特征值、闭合测地线的长度、边界点之间的距离)足以完全确定有关空间的几何形状。群体自然而然地表现为各种空间的对称性。这个项目的第二个主题是关于研究一类无限群.

ABSTRACT DMS - 0202536.This project concerns two major themes. The first is the study ofrigidity theorems (i.e. metric uniqueness) on compact manifolds. Herefor example we consider isospectral problems: to what extent must spaceswith the same spectra (e.g. eigenvalues of the Laplace Beltramioperator, or Lengths of closed geodesics) be isometric. This alsoincludes questions about metric rigidity induced by conjugacy ofgeodesic flows, as well as inverse scattering problems. The other themeconsiders infinite groups G acting cocompactly on nonpositively curvedspaces H (in the sense of Alexandrov). The project is to study therelationship between the geometry of H and the induced action of G onthe ideal boundary of H. This can be considered an aspect of geometricgroup theory and is partially motivated by some questions of Gromov. As well as these two major themes the proposal concerns the authorscontinuing work on various isoperimetric inequalities. These groups show up as symmetries of Hadamard spaces H (which include spaces of nonpositive curvature.) The first theme of the project concerns the question of whether a spacecan be determined by a certain set of data. One part of this relates toquestions of remote sensing. For example: can you determine the densityof an object (say a persons body or the moon) from measurements taken"from the outside"? The CAT scan is a practical example where onedetermines the mass density (or more accurately the absorptioncoefficient) of an object from measurements of the total mass alongstraight lines. An alternative set of measurements is the set of timesit takes for sound to travel between any two points on the boundary(this is a special case of the boundary rigidity question dealt with inthe proposal). The thrust of the proposed study is to determine underwhich circumstances certain sets of data (e.g. eigenvalues, lengths ofclosed geodesics, distances between boundary points) are sufficient tocompletely determine the geometry of the spaces in question. Groups show up naturally as symmetries of various spaces. The second theme of this project concerns the study a certain class of infinite groups.