Mathematical Sciences: Collapsed Riemannian Manifolds and Applications

9626252 Rong建议的研究属于黎曼几何的一般领域。更具体地说，它讨论了截面曲率有界的折叠黎曼流形在整体黎曼几何中的各种应用。这些应用包括紧致流形的基本群的结构和非紧致流形的Gauss-Bonnet-Chern型定理。黎曼流形是具有距离概念的空间的抽象。这些都是抽象空间，因为黎曼流形不像三维空间中的曲线和曲面那样必须位于环境空间中。折叠黎曼流形有某些“奇点”，这使得研究它们变得更加困难，但它们是作为某些非奇异黎曼流形的极限自然出现的。

9626252 Rong The proposed research lies in the general area of Riemannian geometry. More specifically, it deals with various applications of collapsed Riemannian manifolds with bounded sectional curvature to global Riemannian geometry. These applications include structures of fundamental groups of compact manifolds, and Gauss-Bonnet-Chern type theorems for noncompact manifolds. Riemannian manifolds are an abstraction of spaces possessing a notion of distance. These are abstract spaces in that a Riemannian manifold does not have to lie in an ambient space, unlike curves and surfaces in 3-space. Collapsed Riemannian manifolds have certain 'singularities' which make them harder to study, but they arise naturally as limits of certain families of nonsingular Riemannian manifolds.