Group actions and curvature

群体行动和曲率

• 批准号: 1112913
• 负责人: Wolfgang Ziller
• 金额: $ 28.7万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1112913&HistoricalAwards=false
• 关键词: Group; actions; curvature

The overall description of the proposal is to study manifolds with positive or more generally non-negative sectional curvature under the assumption of a large isometry group. In past proposals the principal investigator has used this approach to produce many new examples of non-negative curvature, including some on exotic spheres. Recently he also constructed a new example of positive curvature among an infinite family of candidates he discovered in previous proposals, and found an obstruction to one of these candidates as well. The principal investigator plans to study these candidates in more detail in the hope of finding more examples of positive curvature. The new obstruction is quite general and needs to be explored in more detail for actions with large isometry group, for example polar actions. Its implications without the presence of an isometric group action will be explored as well. There are many questions of a more general nature within this subject of ``non-negative sectional curvature with large isometry groups" that the principal investigator plans to study. Finally, as was done in past proposals with success, studying topological properties of new and known examples can be very difficult but also very rewarding.Manifolds with positive sectional curvature can be defined by the property that the sum of the 3 angles in any triangle is larger than 180 degrees, i.e. their geometry is similar to that of the round sphere. Global Riemannian geometry can be described as relating local stretching and bending to the global shape of space. Since the beginning of global Riemannian geometry, manifolds with positive or more generally non-negative curvature have been an important part of this subject. Nevertheless one still has few obstructions to the existence of such geometries, especially if one wants to distinguish between those manifolds that admit non-negative sectional curvature and those that admit positive sectional curvature. Unfortunately one also has few examples with non-negative curvature and even rarer are examples with positive curvature. It is thus of paramount importance to construct new examples, one of goals of this proposal.

该建议的总体描述是在大等距群的假设下研究具有正或更一般的非负截面曲率的流形。在过去的提案中，首席研究员已经使用这种方法产生了许多非负曲率的新例子，包括一些在奇异球体上的例子。最近，他还在他在先前的建议中发现的无限候选者中构造了一个新的正曲率例子，并发现了其中一个候选者的障碍。首席研究员计划更详细地研究这些候选者，希望找到更多正曲率的例子。新的障碍是相当普遍的，需要更详细地探索具有大等距群的作用，例如极性作用。在不存在等距群作用的情况下，也将探讨其含义。在首席研究员计划研究的“具有大等距群的非负截面曲率”这一主题中，有许多更一般性质的问题。最后，正如在过去的成功建议中所做的那样，研究新的和已知示例的拓扑性质可能非常困难，但也非常有益。具有正截面曲率的流形可以定义为任意三角形的3个角之和大于180度，即其几何形状类似于圆球。整体黎曼几何可以被描述为将局部拉伸和弯曲与空间的整体形状联系起来。自整体黎曼几何开始以来，具有正曲率或更一般的非负曲率的流形一直是该学科的重要组成部分。然而，对于这种几何的存在，人们仍然有一些障碍，特别是当人们想要区分那些承认非负截面曲率的流形和那些承认正截面曲率的流形时。不幸的是，很少有非负曲率的例子，而正曲率的例子就更少了。因此，构建新的例子是至关重要的，这也是本建议的目标之一。