Lower Curvature Bounds, Symmetries, and Topology

奖：DMS 1611780，首席研究员：凯瑟琳·E·塞尔全球黎曼几何将经典的欧几里德几何、球面几何和双曲几何推广到各种几何空间，其中点之间的距离通过最小化连接这些点的曲线的长度来描述。黎曼空间的曲率或弯曲性质推广了我们的视觉感觉，即球体是圆的(正曲线，曲率与球体的直径有关，并且直径较小的球体比直径大的球体弯曲得更大)或欧几里德空间是平坦的(零曲率)。微分几何构造了测量几何的曲率或弯曲性质的局部方法，其主要目标是将黎曼空间的这些局部方面与更灵活且被描述为拓扑的全局性质联系起来。例如，如果一个空间有这样的性质，即在每个点周围都有一个邻域与半径为1的球体的北极区域度量相同，那么整个空间必须证明与该球体相同吗？(答案是否定的，但幅度不大。)如果这些社区只是在公制性质上接近北极地区，会发生什么？从恒曲率到允许微小变化的变化会改变答案吗？研究了曲率概念的几个版本，或多或少地总结了局部几何的细节，自全局黎曼几何诞生以来，具有曲率界限的流形一直被深入研究。这笔拨款支持的项目将研究黎曼流形和一些相关空间在曲率下界存在时的对称性。这项研究既涉及截面曲率和Ricci曲率下界，也涉及它们对Alexandrov空间的相应推广，以期对这类基本上未知的空间有更深入的了解。这一议程中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空间的对称与拓扑；(2)Ricci曲率为正且几乎非负截曲率的黎曼流形的对称与拓扑。在这些体系中的分类问题既困难又耐人寻味，涉及到几个与微分几何相邻的数学专业，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.