Mathematical Sciences: Topics in Spaces and Manifolds of Nonpositive Curvature

[9504135]布林提出的研究在于奇异黎曼几何。与W. Ballman合作的研究者提议研究非正曲率的轨道面体。椭圆面体是单连通的单形复合体，具有一组适当不连续的同胚作用，并保持了单形结构。具有不变度规的圆面体是亚历山德罗夫意义上的奇异黎曼流形。黎曼流形是空间中曲面及其高维类似物的抽象版本。最初，黎曼流形被限制为光滑，没有锋利的边缘。在奇异黎曼几何中，为了允许某些奇异性，对平滑性的要求被略微削弱。在某种意义上，奇异黎曼流形的概念更有用，因为它包含了多面体曲面。

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.