Mathematical Sciences: Groups Actions and Rigidity in Riemannian Geometry

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9626232 Croke The proposed research lies in the area of Riemannian Geometry. More specifically, the investigator plans to study the induced action of fundamental groups on the boundary of the universal covering space of manifolds of non-positive curvature; to pursue spectral rigidity problems for compact manifolds of negative sectional curvature; to continue his study of isoperimetric inequalities. Riemannian manifolds are an abstraction of spaces possessing a notion of distance. These are abstract spaces in that a Riemannian manifold does not have to sit inside an ambient space, unlike curves and surfaces in 3-space. The overall shape of such a space is then determined by its so called topology as well as the distance function. The proposed research has to do with understanding this interaction between the topology and distance function.