9625452 Eberlein The proposed research lies in the general area of Riemannian geometry. The investigator plans to study the geometry of complete Riemannian manifolds of nonpositive curvature and rank 1, a condition implied by but weaker than strictly negative sectional curvature. Particular emphasis will be given to manifolds that are compact or are simply connected and homogeneous. More specifically, the investigator will pursue the following three topics: geometry of compact rank 1 manifolds; geometry of simply connected 2-step nilpotent Lie groups with a left invariant Riemannian metric; homogeneous spaces of strictly negative sectional curvature. Riemannian manifolds are an abstraction of curved spaces possessing a distance function. Unlike curves and surfaces in 3-space these are abstract spaces in that they do not necessarily lie in an ambient space. The proposed research has to do with understanding the large scale structure of certain of these spaces, and looking at various examples coming from Lie group theory.