Curvature and Symmetry

整体黎曼几何推广了经典的欧几里德几何、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），即所考虑的空间如何“弯曲”，与全局拓扑不变量（如基本群）有关，这表明空间是否有一维“洞”。自整体黎曼几何提出以来，具有曲率界的流形一直是人们研究的热点。一个相对较新的方法，这是这个项目的重点，研究流形的曲率下限一直是引进对称性。研究生将通过研究进行培训。首席研究员将从事一个程序，她仔细研究和分析两个黎曼流形与曲率下界和亚历山德罗夫空间的对称性，考虑截面曲率和里奇曲率下界及其相应的推广到亚历山德罗夫空间，该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准。

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.