Principal Investigator: Christopher Croke
This project concerns two major themes. The first, joint work with Bruce Kleiner, considers infinite groups G acting cocompactly on nonpositively curved spaces H (in the sense of Alexandrov), and treats 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. The other theme, involving Kleiner and Sharafutdinov as coauthors (on different aspects), 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 second 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 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 first theme of this project concerns a class of infinite groups which are symmetries of Hadamard spaces H (which include spaces of nonpositive curvature.) As such they induce symmetries on a natural boundary, B, of H. Bruce Kleiner and the PI have found an unexpected relationship between the action of the group on B and the geometry of H. It had been suspected that in this setting that the action on B would be determined only by the nature of the group (as is the case in the related setting of hyperbolic groups acting on negatively curved spaces). This study intends to determine the precise nature of this relationship and use it to study properties of the group and space.