Lower and Upper Curvature Bounds: Topology vs. Geometry

本文讨论了不同的曲率下界和曲率上界对黎曼流形拓扑的影响以及具有曲率下界的流形的Gromov-Hausdorff极限的结构。概括地说，这项建议有三个主要部分。第一部分讨论了具有下截面和Ricci曲率界的流形的结构。Petrunin和W. PI Tuschmann计划继续研究非负和几乎非负曲率流形的基本群的结构，并寻找简单连通流形的非负和几乎非负曲率的新拓扑障碍。PI还计划继续与B合作。Wilking关于具有Ricci曲率下界的流形的基本群。特别地，我们想证明几乎非负Ricci曲率流形的基本群包含一个幂零子群，其指数的界仅取决于维数。Lytchak）处理的概念submetries这是一个推广的黎曼浸没奇异空间和它的关系崩溃下的一个较低的曲率界。最后一部分是与我公司的合作项目。Belegradek继续我们的研究开放的负面和nonpositively捏流形的结束。这个建议处理的问题是空间的局部几何图像（即它局部“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的数量）。度量空间弯曲程度的方法之一是由它的里奇曲率给出。理解各种Ricci曲率边界对空间整体性质的影响不仅本身很有趣，而且也很重要，因为Ricci曲率在爱因斯坦广义相对论中起着重要作用。

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.