Lower and Upper Curvature Bounds: Topology vs. Geometry

曲率下界和曲率上界：拓扑与几何

• 批准号: 0604557
• 负责人: William Goldman
• 金额: --
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604557&HistoricalAwards=false
• 关键词: Lower; Upper; Curvature; Bounds; Topology

This proposal is concerned with the effects of various lower and upper curvature bounds on the topology of Riemannian manifolds and the structure of Gromov-Hausdorff limits of manifolds with lower curvature bounds. Broadly speaking this proposal has three main parts. The first part deals with the structure of manifolds with lower sectional and Ricci curvature bounds. Together with A.Petrunin and W. Tuschmann the PI plans to continue investingating the structure of the fundamental groups of nonnegatively and almost nonnegatively curved manifolds and also look for new topological obstructions to nonnegative and almost nonnegative curvature for simply connected manifolds. The PI also plans to continue his work with B. Wilking on the fundamental groups of manifolds with lower Ricci curvature bounds. In particular we would like to show that the fundamental group of a manifold of almost nonnegative Ricci curvature contains a nilpotent subgroup of finite index with the bound on the index depending only on the dimension.The second part of the proposal (which is a joint project with A. Lytchak) deals with the notion of submetries which is a generalization of Riemannian submersion to singular spaces and its relation to collapsing under a lower curvature bound. The last part is the joint project with I. Belegradek on continuing our study of ends of open negatively and nonpositively pinched manifolds. This proposal deals with the question of how the local geometric picture of a space (i.e the way it's "curved" or "bent" locally) influences the global properties of the space (such as the number of holes of various dimensions the space might have). One of the ways to measure how a space is curved is given by its Ricci curvature. Understanding the influence of various Ricci curvature bounds on the global properties of a space is not only interesting in its own right but it's also important because Ricci curvature plays a fundamental role in the Einstein general relativity theory.

该建议涉及各种下部和上部曲率边界对Riemannian歧管拓扑的影响以及具有较低曲率边界的歧管的Gromov-Hausdorff限制的结构。从广义上讲，该提议有三个主要部分。第一部分涉及具有较低截面和RICCI曲率边界的歧管结构。 PI与A. -Petrunin和W. Tuschmann一起计划继续投资于非模仿和几乎非弯曲的流形的基本群体的结构，还为简单地连接的流形寻找了非负和非阴性弯曲的新拓扑障碍。 PI还计划继续与B. Wilking一起工作，该基本曲线曲率范围较低。特别是，我们要表明，几乎非负RICCI曲率的一组基本组包含一个有限指数的尼尔属性亚组，仅根据尺寸限制在索引上，取决于提案的第二部分。在较低的曲率结合下。最后一部分是与I. belegradek的联合项目，继续我们对开放式和非态度捏合的末端进行研究。该提案涉及一个问题，即空间的局部几何形状图片（即，在本地“弯曲”或“弯曲”的方式如何影响空间的全局特性（例如空间可能拥有的各个维度的孔的数量）。测量空间弯曲方式的方法之一是其RICCI曲率给出。了解各种RICCI曲率界限对空间全球特性的影响不仅有趣，而且这也很重要，因为RICCI曲率在爱因斯坦一般相对论理论中起着基本作用。