Curvature and Symmetry

全局黎曼几何推广了经典的欧几里得几何、球面几何和双曲几何。这一研究领域的主要挑战之一是了解局部几何不变量(如曲率)如何与全局拓扑不变量(如基本群)有关，如曲率，即所考虑的空间如何“弯曲”，基本群指示空间是否有一维“洞”。自从整体黎曼几何的概念提出以来，具有曲率界的流形得到了广泛的研究。一个相对较新的方法，也是本项目的焦点，研究具有下曲率界限的流形的一个相对较新的方法是引入对称性。研究生将通过研究接受培训。首席研究员将从事一个项目，在该项目中，她将仔细研究和分析具有下曲率下限的黎曼流形和亚历山大空间的对称性，同时考虑截面曲率和Ricci曲率下界以及它们对亚历山大空间的相应推广，以期对这类基本上未知的空间有更深入的了解。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Global Riemannian Geometry generalizes the classical Euclidean, Spherical, and Hyperbolic geometries. One of the major challenges in this area of study is to understand how local geometric invariants such as curvature, that is, how the space under consideration "bends", relate to global topological invariants such as fundamental group, which indicates whether or not the space has 1-dimensional "holes". Manifolds with curvature bounds have been studied intensively since the conception of global Riemannian geometry. One relatively recent approach, which is the focus of this project, to the study of manifolds with lower curvature bounds has been the introduction of symmetries. Graduate students will be trained through research.The principal investigator will pursue a program in which she carefully studies and analyzes symmetries of both Riemannian manifolds with lower curvature bounds and Alexandrov spaces, considering both sectional curvature and Ricci curvature lower bounds and their corresponding generalizations to Alexandrov spaces, with an eye to gaining a deeper understanding of this largely unknown class of spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.