Mathematical Sciences: Geometry and Topology of Riemannian Manifolds

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Two investigators will continue their studies of global problems in differential geometry and related areas. One will investigate the Gromov-Hausdorff closure of the class of Riemannian manifolds with curvature bounded from below and diameter from above. In this setting, structure problems, finiteness problems, and uniqueness problems will be investigated in terms of volume and new metric invariants such as excess and triameter. The second postdoctoral investigator will study the relationship between geometry and topology of certain classes of manifolds. Finiteness, Riemannian fibrations, injectivity radius, and rigidity will be among the aspects of these spaces to be investigated. For many years geometers have worked to understand the relationship between the curvature of generalized surfaces and their shape. These investigators have proven that such surfaces which have curvature bounded below and diameter bounded above, must have certain specified shapes. In this project they will expand this investigation to include surfaces with other desireable properties.