[论文解读] Collapsed Riemannian Manifolds with Bounded Sectional Curvature

Xiaochun Rongt|arXiv (Cornell University)|Jan 1, 2002

Geometric Analysis and Curvature Flows参考文献 46被引用 3

一句话总结

本文综述了黎曼几何中截面曲率有界的流形的坍塌理论的发展，确立了当 e 足够小时，e-坍塌流形（e-collapsed manifolds）具有几乎平坦纤维的奇异纤维丛结构。其主要贡献在于通过纤维丛对坍塌流形进行结构表征，揭示了在非正曲率及正曲率夹紧情形下的深层几何与拓扑约束。

ABSTRACT

One of the most important developments in Riemannian geometry over the last two decades is the structure theory of Cheeger-Fukay a-Gromov, for Mn of bounded sectional curvature, say |KM»| e _1. For ex­ ample, a very thin cylinder is very collapsed (although its curvature vanishes identically). If one fixes e and in addition, a bound, d, on the diameter, then in each dimension, there are only finitely many manifolds, which are not e-collapsed. The basic result of collapsing theory states the existence of a constant, e(n) > 0, such that a manifold which is e-collapsed, for e < e(n), has a particular kind of singular fibration structure with (or almost flat ) fibers. The fibers lie in the e-collapsed directions. The first nontrivial collapsing with bounded curvature, arose in a se­ quence of metrics on the 3-sphere constructed by M. Berger. The first major result on the collapsed (still a corner stone of the theory) is M. Gromov's description of almost manifolds i.e. admitting a se­ quence of metrics with curvature and diameter going to zero. Gromov showed that such are infranilmanifolds. We will survey the main development of the collapsing theory and its applications to Riemannian geometry since 1990. The common starting point is the existence of the above mentioned singular fibration structure. Many new geometrical and topological constraints of collapsed metrics have been discovered that are accompanied with new ideas and techniques as well as tools from related fields, and light has been shed on some classical problems and conjectures, which do not, on the face of it, involve collapsing. Substantial progress has been made on with non-positive curvature, on positively pinched manifolds, collapsed with an a priori diameter bound, and subclasses whose members satisfy additional topological conditions e.g. 2connectedness.

研究动机与目标

• 系统化并总结自1990年以来，截面曲率有界的黎曼流形坍塌理论的主要进展。
• 阐明在 e 小于与维度相关的阈值 e(n) 的条件下，e-坍塌流形中奇异纤维丛的结构性作用。
• 探讨坍塌理论如何揭示非正曲率、正曲率夹紧以及额外拓扑条件（如2-连通性）下流形的新几何与拓扑约束。
• 将坍塌现象与不显式涉及坍塌的经典黎曼几何问题联系起来，从而拓展该理论的应用范围。

提出的方法

• 利用基础结果：当 e < e(n) 时，e-坍塌流形（e-collapsed manifolds）具有几乎平坦纤维的奇异纤维丛结构，其中纤维对应于坍塌方向。
• 应用度量几何与格罗莫夫-豪斯多夫收敛技术，分析坍塌序列的极限空间。
• 以格罗莫夫的几乎平坦流形理论为核心，特别是几乎平坦流形被表征为仿李群流形（infranilmanifolds）的结果。
• 分析在直径有界条件下，特定曲率类（如非正曲率和正曲率夹紧）中的坍塌序列。
• 利用几何群论与幂零群作用的工具，理解纤维及其单值性（holonomy）的结构。
• 研究具有额外拓扑约束（如2-连通性）的坍塌流形子类，以得出更强的结构性结论。

实验结果

研究问题

• RQ1 在截面曲率有界的 e-坍塌黎曼流形中，精确的纤维丛结构是什么？
• RQ2 此类纤维丛的存在如何约束坍塌流形的拓扑与几何结构？
• RQ3 坍塌理论在何种方式下揭示了不显式涉及坍塌的经典黎曼几何猜想？
• RQ4 在附加假设（如非正曲率或2-连通性）下，坍塌流形中会涌现出哪些结构性性质？
• RQ5 直径有界与曲率有界如何共同影响固定维度下可能的拓扑类型数量为有限？

主要发现

• 对每个维度 n，当 e < e(n) 时，若 e-坍塌流形的直径有界于 d，则其微分同胚类型仅有有限多种。
• 满足 |KM| ≤ 1 的 e-坍塌流形具有奇异纤维丛结构，其纤维为几乎平坦的，对应于坍塌方向。
• 具有有界曲率的坍塌现象的首个非平凡例子，源于 M. 伯杰在3-球面上构造的一系列度量。
• 格罗莫夫关于几乎平坦流形即为仿李群流形的结果，仍是坍塌理论的基石性成果。
• 在理解非正曲率坍塌流形及正曲率夹紧流形方面，已取得重大进展。
• 额外的拓扑约束（如2-连通性）在坍塌情形下可导出更强的结构性结论。