Mathematical Sciences: Topics in Spaces and Manifolds of Nonpositive Curvature

小行星9504135 所提出的研究在于奇异黎曼几何。研究者与W. Ballman建议研究非正曲率的轨道面。一个正多面体是一个单连通的单纯复形，它具有一组不连续的同胚，并保持单纯结构。一个具有不变度量的半球面是亚历山德罗夫意义下的奇异黎曼流形。 黎曼流形是空间中曲面的抽象形式及其高维类似物。最初，黎曼流形被限制为光滑而没有尖锐的边缘。在奇异黎曼几何中，光滑性要求稍微减弱，以允许某些奇异性。在某种意义上，奇异黎曼流形的概念更有用，因为它包括多面体曲面。

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.