Mathematical Sciences: Geometry and Topology of Riemannian Manifolds

该项目处理黎曼几何中的全局问题。研究奇异空间（如Alexandrov空间和有限度量空间）对黎曼流形的逼近；研究黎曼流形的稳定性和反稳定性问题；通过轨道几何研究黎曼流形的曲率与对称性之间的关系；寻找具有大等距群和正曲率的流形的例子。黎曼流形是具有距离函数的弯曲空间的抽象。与三维空间中的曲线和曲面不同，这些流形不一定位于环境空间中，因此它们是抽象空间。提出的研究旨在了解这些流形的各个整体方面，特别是，研究者对黎曼流形的有限度量空间近似和对称性质感兴趣。有限度量空间与图密切相关，因此该项目可能会影响黎曼几何的离散应用。

9626375 Grove The project deals with global problems in Riemannian geometry. The investigator intends to study approximations of Riemannian manifolds by singular spaces such as Alexandrov spaces and finite metric spaces; to study stability and inverse stability problems for Riemannian manifolds; to investigate the relationship between curvature and symmetry of Riemannian manifolds via the geometry of orbits; to find examples of manifolds with large isometry groups and with positive curvature. Riemannian manifolds are an abstraction of curved spaces possessing a distance function. Unlike curves and surfaces in 3-space these manifolds do not necessarily sit inside an ambient space, hence they are abstract spaces. The proposed research aims to understand various global aspects of these manifolds, in particular, the investigator is interested in approximations of Riemannian manifolds by finite metric spaces and in symmetry properties. Finite metric spaces are closely related to graphs so that the project may impact on discrete applications of Riemannian geometry.