Nonpositive Curvature and Geometric Rigidity

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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.