Lower and Upper Curvature Bounds: Topology vs. Geometry

本文讨论了不同的曲率下界对黎曼流形的拓扑结构和曲率下界流形的Gromov-Hausdorff极限结构的影响。大体上讲，这项提议有三个主要部分。第一部分讨论具有下截面界和Ricci曲率界的流形的结构。PI计划与A.Petrunin和W.Tuschmann一起继续研究非负和几乎非负的曲流形的基本群的结构，并寻找对单连通流形的非负和几乎非负曲率的新的拓扑障碍。PI还计划继续他与B.Wilking在具有较低Ricci曲率界的流形的基本群方面的工作。特别地，我们想要证明几乎非负Ricci曲率的流形的基本群包含一个有限指数的幂零子群，其上界仅取决于维度。第二部分(这是与A.Lytchak的联合项目)讨论了次距离的概念，它是黎曼浸没到奇异空间的推广，以及它与曲率下界下的折叠的关系。最后一部分是与I.Belegradek关于继续我们对开负和非正收缩流形末端的研究的联合项目。这项建议涉及空间的局部几何图像(即它在局部“弯曲”或“弯曲”的方式)如何影响空间的全局属性(例如空间可能具有的各种维度的孔洞的数量)的问题。度量一个空间如何弯曲的方法之一是由它的Ricci曲率给出的。了解各种Ricci曲率界限对空间整体性质的影响不仅本身很有趣，而且很重要，因为Ricci曲率在爱因斯坦广义相对论中扮演着重要的角色。

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.