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 Global Riemannian几何形状概括了经典的欧几里得，球形和双曲线几何形状到各种几何学空间，在这些几何学空间中，该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点之间距离之间的距离。 Riemannian空间的弯曲或弯曲特性概括了我们拥有的视觉意义，即一个球体是圆形的（呈呈曲线，曲率与球体的直径相关，并且较小直径的球体比大直径的球体更弯曲的球体更大弯曲）或e纤维直径的球体）或Euclidean Space Flat flat flat flat flat（零曲线）。差异几何图形构建了局部方法来测量几何形状的曲率或弯曲特性，一个主要目标是将Riemannian空间的这些局部方面与更灵活的全球性质联系起来，并将其描述为拓扑。例如，如果一个空间的特性在每个点附近都有一个与半径1的北极区相同的邻域，那么整个空间都必须与该领域相同吗？（答案是否定的，但不是太多。）如果这些社区仅在该北极区域的度量特性中接近度量，会发生什么 - 从恒定曲率变为允许小变化变化会改变答案吗？研究了几种版本的曲率概念，以越来越多的细节概述局部几何形状，并且自全球riemannian几何形状成立以来，就对具有曲率边界的歧管进行了深入研究。这笔赠款支持的项目将研究曲率下范围内的Riemannian歧管和一些相关空间的对称性。该研究计划涉及截面曲率和RICCI曲率下限及其对Alexandrov空间的相应概括，以使人们更深入地了解这个众所周知的众所周知的众所周知的Spaces。该议程中的基本问题涉及以下领域：（1）对对称性和非弯曲的riemannian歧管和亚历山德罗夫空间的对称性和拓扑结构，以及（2）Riemannian积极ricci曲率曲率和几乎非维度的分段曲率的对称和拓扑。这些制度中的分类问题既困难又有趣，并且涉及几个数学专业，这些专业是差异几何的邻居，包括谎言群及其对多种流形的行为以及代数拓扑。