Group actions and curvature

该提案的总体描述是在大等距群的假设下研究具有正或更一般非负截面曲率的流形。在过去的提案中，首席研究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，他还在之前提案中发现的无限候选族中构建了一个新的正曲率示例，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。新的障碍非常普遍，需要更详细地探索具有大等距组的动作，例如极性动作。还将探讨在不存在等距群体动作的情况下其含义。首席研究员计划研究的“具有大等距群的非负截面曲率”这一主题中存在许多更一般性的问题。最后，正如过去成功的提案所做的那样，研究新的和已知的例子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通过任何三角形中的 3 个角度之和大于 180 度的属性来定义，即它们的几何形状与圆形球体的几何形状相似。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，自从全局黎曼几何开始以来，具有正曲率或更普遍的非负曲率的流形一直是这一主题的重要组成部分，然而，人们对这种几何的存在仍然没有什么障碍，特别是如果人们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，具有非负曲率的流形也很少。因此，构建新的示例至关重要，这也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.