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Lower Curvature Bounds, Symmetries, and Topology

较低的曲率界限、对称性和拓扑

基本信息

批准号：
1611780
负责人：
Catherine Searle
金额：
$ 15万
依托单位：
Wichita State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-09-01 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1611780&HistoricalAwards=false
关键词：
Lower Curvature Bounds Symmetries Topology

项目摘要

Award: DMS 1611780, Principal Investigator: Catherine E. Searle Global Riemannian geometry generalizes the classical Euclidean, Spherical and Hyperbolic geometries to a wide variety of geometric spaces in which the distance between points is described by minimizing the lengths of curves that join those points. Curvature or bending properties of Riemannian spaces generalize the visual sense we have that a sphere is round (positively curved, with the sense that curvature is related to the diameter of the sphere, and that a sphere of smaller diameter is more greatly curved than a sphere of large diameter) or Euclidean space is flat (of zero curvature). Differential geometers construct local ways to measure the curvature or bending properties of a geometry, and a major goal is to relate these local aspects of a Riemannian space to global properties that are much more flexible and are described as topology. For example, if a space has the property that around every point there is a neighborhood that is metrically identical to the arctic region of a sphere of radius 1, must the entire space turn out to be identical to that sphere? (The answer is no, but not by much.) What happens if those neighborhoods are merely close in metric properties to that arctic region - does changing from constant curvature to allowing small variations change that answer? Several versions of the notion of curvature are studied, summarizing local geometry in greater or lesser levels of detail, and manifolds with curvature bounds have been studied intensively since the inception of global Riemannian geometry. The projects supported by this grant will study symmetries of Riemannian manifolds and of some related spaces in the presence of lower bounds on curvature.This research program concerns 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. Basic problems in this agenda concern the following areas: (1) symmetries and topology of positively and non-negatively curved Riemannian manifolds and Alexandrov spaces and (2) symmetries and topology of Riemannian manifolds of positive Ricci curvature and almost non-negative sectional curvature. Classification problems in these regimes are both difficult and intriguing, and touch on several mathematical specialties that are neighbors of differential geometry, including Lie groups and their actions on manifolds, as well as algebraic topology.

奖项：DMS 1611780，首席研究员：Catherine E. Searle 全局黎曼几何将经典的欧几里得几何、球面几何和双曲几何推广到各种几何空间，其中点之间的距离是通过最小化连接这些点的曲线长度来描述的。黎曼空间的曲率或弯曲特性概括了我们的视觉感觉：球体是圆形的（正弯曲，曲率与球体的直径有关，并且较小直径的球体比大直径的球体弯曲得更大）或欧几里得空间是平坦的（曲率为零）。微分几何学家构造局部方法来测量几何的曲率或弯曲特性，主要目标是将黎曼空间的这些局部方面与更加灵活并被描述为拓扑的全局特性联系起来。例如，如果一个空间具有这样的属性，即每个点周围都有一个与半径为 1 的球体的北极区域在度量上相同的邻域，那么整个空间是否必须与该球体相同？（答案是否定的，但相差不大。）如果这些邻域仅在度量属性上与北极地区接近，会发生什么 - 从恒定曲率更改为允许微小变化会改变答案吗？研究了曲率概念的几个版本，以或多或少的细节水平总结了局部几何，并且自全局黎曼几何诞生以来，具有曲率界限的流形已得到深入研究。这笔赠款支持的项目将研究黎曼流形和一些相关空间在存在曲率下界的情况下的对称性。该研究计划涉及截面曲率和里奇曲率下界及其对亚历山德罗夫空间的相应推广，着眼于更深入地了解这一类很大程度上未知的空间。本议程中的基本问题涉及以下领域：（1）正弯曲黎曼流形和非负弯曲黎曼流形以及亚历山德罗夫空间的对称性和拓扑；（2）正里奇曲率和几乎非负截面曲率的黎曼流形的对称性和拓扑。这些体系中的分类问题既困难又有趣，并且涉及与微分几何相邻的几个数学专业，包括李群及其在流形上的作用，以及代数拓扑。