Topological and equivariant rigidity in the presence of lower curvature bounds

存在曲率下限时的拓扑刚度和等变刚度

• 批准号: 339994903
• 负责人: Dr. Fernando Galaz-García
• 金额: --
• 依托单位: Department of Mathematics
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/339994903?language=en
• 关键词: Topological; equivariant; rigidity; presence; lower

The theory of compact groups acting on manifolds has a long tradition and is quite well understood nowadays. The more general case of groups acting on metric and/or singular spaces is today mostly pursued in the context of non-positively curved metric spaces. However, fundamental to other parts of modern geometry are Alexandrov spaces (of curvature bounded below) equipped with an isometric action of a compact Lie group G. These metric spaces include, for example, Riemannian manifolds with a lower sectional curvature bound and an isometric G-action. Of particular interest to Riemannian geometers have been Riemannian manifolds with positive or non-negative curvature equipped with isometric compact Lie group actions. This interest is in great part propelled by the so-called Grove program, which proposes to classify Riemannian manifolds of positive or non-negative sectional curvature with a large isometry group.The simplest groups that one may consider acting on a manifold are compact abelian Lie groups of positive dimension, i.e., tori. By now the theory of smooth torus actions is well developed. In the context of Riemannian geometry and, in particular, of the Grove program, positively curved Riemannian manifolds with torus actions have been extensively studied. Thanks to the work of several authors, most notably Grove and Searle, Fang and Rong, and Wilking, fairly complete classification results are available, provided that either the manifold, or the torus acting upon it, has sufficiently large dimension. For the action of a circle and a 2-torus in dimensions 5 and 6, respectively, the usual methods have so far failed to yield topological and equivariant classification results.Taking as departure point the Grove program and the well-developed theory of cohomological methods for smooth torus actions on smooth manifolds, the present proposal aims, on the one hand, at applying and developing equivariant topological methods in the context of Riemannian manifolds with a lower sectional curvature bound and, on the other hand, at studying closed Alexandrov spaces of cohomogeneity one. The primary goals are, respectively, to obtain a topological and equivariant classification of closed, simply connected 6-manifolds with an effective, isometric 2-torus action, and to classify closed, positively curved Alexandrov spaces of cohomogeneity one. The solution of these problems entails solving many self-contained problems which are of interest in their own right, including the consideration of singular spaces (e.g. orbifolds and Alexandrov spaces) with torus actions.

紧群作用于流形的理论有着悠久的传统，并且现在已经被很好地理解了。群作用于度量空间和/或奇异空间的更一般的情况，现在大多是在非正曲度量空间的背景下进行的。然而，现代几何的其他部分的基础是亚历山德罗夫空间（曲率有界）配备了一个紧李群G的等距作用。 这些度量空间包括，例如，具有下截面曲率界和等距G作用的黎曼流形。特别感兴趣的黎曼几何学家一直黎曼流形与积极或非负曲率配备等距紧李群行动。这种兴趣在很大程度上是由所谓的格罗夫计划推动的，该计划提出用一个大的等距群对正或非负截面曲率的黎曼流形进行分类。人们可以考虑作用于流形的最简单的群是正维数的紧致交换李群，即，桃丽目前，光滑环面作用量理论已发展得相当成熟。在黎曼几何的背景下，特别是格罗夫计划，积极弯曲黎曼流形与环面行动已被广泛研究。 由于工作的几个作者，最显着的格罗夫和塞尔，方和荣，和威尔金，相当完整的分类结果是可用的，只要无论是流形，或环面作用于它，有足够大的尺寸。 对于5维和6维的圆和2-环面的作用，通常的方法至今未能给出拓扑和等变的分类结果.本文以光滑流形上光滑环面作用的格罗夫程序和成熟的上同调方法理论为出发点，一方面，在应用和发展等变拓扑方法的上下文中的黎曼流形与较低的截面曲率界，另一方面，在研究封闭的亚历山德罗夫空间的cohomogeneity之一。 的主要目标是，分别获得一个拓扑和等变分类的封闭，单连通6-流形与有效的，等距2-环面行动，并分类封闭，积极弯曲的亚历山德罗夫空间的上齐性。 这些问题的解决需要解决许多独立的问题，这些问题本身就很有趣，包括考虑具有环面作用的奇异空间（例如orbifolds和Alexandrov空间）。