Mathematical Sciences: Geometry and Topology of Riemannian Manifolds

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Karsten Grove's research will be directed towards the important questions of finding the relationships between the geometry and topology of Riemannian manifolds. The main emphasis will be devoted to problems relating geometry and/or transformation groups to homotopy theory. The general areas to be considered are finiteness problems for manifolds with bounds on curvature, diameter and volume; the structure of manifolds with non-negative curvature; Riemannian foliations and isoparametric submanifolds; Morse theory for maps. In the context of finiteness theorems he is seeking a result which would show that for manifolds with curvature bounded below, diameter bounded above and volume bounded below, there are only finitely many possible homotopy types. The focus of attention in the case of manifolds of non-negative curvature will be the conjecture that such manifolds are rationally elliptic. Finally he will investigate relations between the homotopy and cohomogeneity of a manifold and the geometry of orbit foliations of space forms.