Curvature and Symmetry

整体黎曼几何推广了经典欧几里得几何、球面几何和双曲几何。这一研究领域的主要挑战之一是理解局部几何不变量（如曲率），即所考虑的空间如何“弯曲”，与全局拓扑不变量（如基本群）之间的关系，这表明空间是否存在一维“洞”。自整体黎曼几何概念提出以来，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，这是本项目的重点，研究具有低曲率边界的流形已经引入了对称性。研究生将通过研究进行训练。首席研究员将认真研究和分析具有下曲率边界的黎曼流形和亚历山德罗夫空间的对称性，考虑到截面曲率和里奇曲率下界及其相应的亚历山德罗夫空间的推广，以期对这一很大程度上未知的空间类别有更深入的了解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.