Lower Curvature Bounds, Symmetries, and Topology

全球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各样的几何空间，其中点之间的距离是通过最小化连接这些点的曲线的长度来描述的。黎曼空间的曲率或弯曲特性概括了我们的视觉感觉，即球体是圆的（正弯曲，曲率与球体的直径有关，直径较小的球体比直径较大的球体弯曲得更大）或欧几里德空间是平的（零曲率）。微分几何构建局部方法来测量几何的曲率或弯曲特性，其主要目标是将黎曼空间的这些局部方面与更灵活的全局特性联系起来，并将其描述为拓扑。例如，如果一个空间有这样的性质：在每一点周围都有一个邻域，它与半径为1的球体的北极区域在度量上是相同的，那么整个空间必须与这个球体相同吗？（答案是否定的，但差距不大。）如果这些社区仅仅在度量属性上接近北极地区会发生什么——从恒定曲率到允许小的变化会改变答案吗？几个版本的曲率的概念进行了研究，总结了局部几何或多或少的细节水平，流形与曲率界已经深入研究，因为整体黎曼几何的开始。本基金资助的项目将研究黎曼流形和一些相关空间在曲率下界存在下的对称性。本研究计划涉及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，旨在更深入地了解这类基本上未知的空间。本课题的基本问题涉及以下几个方面：(1)正、非负弯曲黎曼流形和Alexandrov空间的对称性和拓扑；(2)正里奇曲率和几乎非负截面曲率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，并且涉及微分几何的几个数学专业，包括李群及其在流形上的作用，以及代数拓扑。

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.