Spaces of Nonpositive Curvature and Geometric Rigidity

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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.