Collapsed Riemannian Manifolds and Applications

AbstractAward：DMS-9971360主要研究者：荣晓春我们一直在研究黎曼几何中与截面曲率绝对有界的坍缩流形有关的问题以及在整体黎曼几何等方面的应用，在对截面曲率为pinched正的坍缩流形的拓扑结构的理解上取得了一些进展，包括基本群是虚循环的，最大塌陷流形在有限覆盖下同构于透镜空间，即单连通流形的第二Betti数不为零。我们在非正截面曲率的流形上取得了一些进展，包括局部分裂结构的存在性。我们在非紧流形的Gauss-Bonnet-Chern型定理以及与某些塌缩度量序列相关的拓扑不变量方面取得了进展。我们将在不久的将来继续我们在这些领域的计划。我们也将开始研究具有有界直径和较低曲率界的坍缩流形的结构。这是三年前NSF Grant DMS 9626252提出的工作的继续。在过去的二十年里，黎曼几何最重要的发展之一是对黎曼流形的理解，这些流形看起来比它们的实际维数要小。例如，一个非常薄的甜甜圈的表面看起来像一个圆。我的研究主要集中在曲率介于两个不同半径球面曲率之间的流形上。我们发现了这些流形的拓扑结构的限制。我们的进展被一个惊人的事实放大了，在所有曲率在两个不同半径的球面曲率之间的流形中，除了100%之外，所有的流形看起来都小于它们的实际尺寸。因此，我们对曲率介于两个不同半径球面曲率之间的流形的研究取得了进展。

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.