Mathematical Sciences: Collapsed Riemannian Manifolds and Applications

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9626252 Rong The proposed research lies in the general area of Riemannian geometry. More specifically, it deals with various applications of collapsed Riemannian manifolds with bounded sectional curvature to global Riemannian geometry. These applications include structures of fundamental groups of compact manifolds, and Gauss-Bonnet-Chern type theorems for noncompact manifolds. 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 lie in an ambient space, unlike curves and surfaces in 3-space. Collapsed Riemannian manifolds have certain 'singularities' which make them harder to study, but they arise naturally as limits of certain families of nonsingular Riemannian manifolds.