Group actions and curvature

该建议的总体描述是在大等距群的假设下研究截面曲率为正或更一般的非负的流形。在过去的提案中，首席研究人员使用这种方法产生了许多非负曲率的新例子，包括一些关于奇异球体的例子。最近，他还构建了一个新的例子，在他在以前的提案中发现的无限候选者家庭中存在正曲率，并发现了对其中一个候选者的阻碍。首席研究员计划对这些候选者进行更详细的研究，希望找到更多正曲率的例子。新的障碍物非常普遍，需要更详细地研究具有大等轴测群的作用，例如极作用。在不存在等距群体行动的情况下，其影响也将被探索。在“具有大等轴测群的非负截面曲率”这一主题中，有许多更一般性质的问题，首席研究员计划研究。最后，正如在过去的建议中成功地完成的那样，研究新的和已知的例子的拓扑性质可能是非常困难的，但也是非常有益的。具有正截面曲率的流形可以通过任意三角形中的三个角的和大于180度的性质来定义，即它们的几何类似于圆球的几何。全球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。自从整体黎曼几何诞生以来，具有正或更一般的非负曲率的流形一直是这一课题的重要组成部分。然而，对于这种几何的存在，人们仍然没有什么障碍，特别是如果想要区分那些允许非负截面曲率的流形和那些允许正截面曲率的流形。不幸的是，非负曲率的例子也很少，更少见的是正曲率的例子。因此，构建新的例子至关重要，这是本提案的目标之一。

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.