9504135 Brin The proposed research lies in singular Riemannian geometry. The investigator in collaboration with W. Ballman proposes to investigate orbihedra of nonpositive curvature. An orbihedron is a simply connected simplicial complex with a group of homeomorphisms acting properly discontinuously and preserving the simplicial structure. An orbihedron with an invariant metric is a singular Riemannian manifold in the sense of Alexandrov. Riemannian manifolds are abstract versions of surfaces in space and their higher dimensional analogs. Originally, Riemannian manifolds were restricted to be smooth without sharp edges. In singular Riemannain geometry, the smoothness requirement is slightly weakened to allow certain singularities. In some sense, the concept of a singular Riemannain manifold is more useful as it includes polyhedral surfaces.
