Nonpositive Curvature and Geometric Rigidity

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

ABSTRACT DMS - 0202536. This project concerns two major themes. The first is the study of rigidity theorems (i.e. metric uniqueness) on compact manifolds. Here for example we consider isospectral problems: to what extent must spaces with the same spectra (e.g. eigenvalues of the Laplace Beltrami operator, or Lengths of closed geodesics) be isometric. This also includes questions about metric rigidity induced by conjugacy of geodesic flows, as well as inverse scattering problems. The other theme considers infinite groups G acting cocompactly on nonpositively curved spaces H (in the sense of Alexandrov). The project is to study the relationship between the geometry of H and the induced action of G on the ideal boundary of H. This can be considered an aspect of geometric group theory and is partially motivated by some questions of Gromov. As well as these two major themes the proposal concerns the authors continuing 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 space can be determined by a certain set of data. One part of this relates to questions of remote sensing. For example: can you determine the density of an object (say a persons body or the moon) from measurements taken "from the outside"? The CAT scan is a practical example where one determines the mass density (or more accurately the absorption coefficient) of an object from measurements of the total mass along straight lines. An alternative set of measurements is the set of times it takes for sound to travel between any two points on the boundary (this is a special case of the boundary rigidity question dealt with in the proposal). The thrust of the proposed study is to determine under which circumstances certain sets of data (e.g. eigenvalues, lengths of closed geodesics, distances between boundary points) are sufficient to completely 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.