Collapsed Riemannian Manifolds and Applications

摘要奖：DMS-9971360主要研究员：荣晓春我们一直在研究截面曲率有界于绝对值的折叠流形的黎曼几何问题，并将其应用于整体黎曼几何和其他方面。我们在理解具有收缩正截面曲率的折叠流形的拓扑方面取得了进展，包括基本群是虚循环的，最大折叠流形与透镜空间的微分同态直到有限覆盖，单连通流形的第二Betti数不为零。我们在非正截面曲率的折叠流形方面取得了一些进展，包括局部分裂结构的存在。我们对非紧流形的Gauss-Bonnet-Chern型定理以及与某些折叠度量序列相关的拓扑不变量进行了改进。在不久的将来，我们将继续在这些领域开展我们的项目。我们还将开始研究直径有界、曲率有界的折叠流形的结构。这代表了三年前为NSF Grant DMS 9626252提出的工作的继续。在过去的二十年里，黎曼几何中最重要的发展之一是对黎曼流形的理解，它似乎比它们的实际维度小。例如，一个非常薄的甜甜圈的表面看起来像一个圆。我的主要研究集中在这样的流形上，它们的曲率也在不同半径的两个球面的曲率之间。我们发现了对这些流形的拓扑结构的限制。我们的进步被一个令人惊讶的事实放大了：在所有曲率位于不同半径的两个球体的曲率之间的流形中，除了有限的许多流形似乎都小于它们的实际维度。因此，我们在这方面的研究取得了长足的进展：曲率介于两个不同半径球面的曲率之间的流形。

AbstractAward: DMS-9971360Principal Investigator: Xiaochun RongWe have been studying problems in Riemannian geometry related tocollapsed manifolds with sectional curvature bounded in absolutevalue and applications to global Riemannian geometry and others.We have made progress on understanding the topology of collapsedmanifolds with pinched positive sectional curvature, includingthe fundamental group is virtually cyclic, the maximallycollapsed manifolds are diffeomorphic to lens spaces up to afinite covering, the nonvanishing of the second Betti number forsimply connected manifolds. We have made progress on collapsedmanifolds of nonpositive sectional curvature, including theexistence of the local splitting structure. We have made progresson Gauss-Bonnet-Chern type theorems for noncompact manifolds aswell as the topological invariants associated to a sequence ofcertain collapsing metrics. We will continue our programs inthese areas in the near future. We will also begin to investigatestructures of collapsed manifolds with bounded diameter and alower curvature bound. This represents a continuation of workproposed three years ago for NSF Grant DMS 9626252.One of the most important developments in Riemannian geometry inthe last two decades is in the understanding for the Riemannianmanifolds which appear to be smaller than their actualdimension. For instance, the surface of a very thin donut lookslike a circle. A major part of my research focus on suchmanifolds whose curvature are also between the curvature of twospheres of different radius. We have found restrictions on thetopological structure of these manifolds. Our progress isamplified by the amazing fact that among all manifolds whosecurvature are between the curvature of two spheres of differentradius, all but finitely many appear to be smaller than theiractual dimension. Therefore, we have made progress on the studywith a long history: the manifolds whose curvature are betweenthe curvature of two spheres of different radius.