Group actions and curvature

该提案的总体描述是研究流形的积极或更一般的非负截面曲率的假设下，一个大的等距群。在过去的提案中，主要研究者已经使用这种方法产生了许多新的非负曲率的例子，包括一些奇异的球体。最近，他还构建了一个新的例子，正曲率之间的无限家庭的候选人，他发现在以前的建议，并发现了障碍，这些候选人之一，以及。首席研究员计划更详细地研究这些候选人，希望找到更多正曲率的例子。新的障碍是相当普遍的，需要更详细地探讨具有大等距群的作用，例如极性作用。它的影响，而不存在的等距组的行动将被探讨以及。主要研究者计划研究的“具有大等距组的非负截面曲率”这一主题中有许多更一般性的问题。最后，就像过去的一些成功的建议一样，研究新的和已知的例子的拓扑性质是非常困难的，但也是非常有益的。具有正截面曲率的流形可以定义为任意三角形中的3个角之和大于180度，即它们的几何形状类似于圆球的几何形状。整体黎曼几何可以描述为将局部拉伸和弯曲与空间的整体形状相关联。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这门学科的重要组成部分。然而，人们仍然有一些障碍，存在这样的几何形状，特别是如果一个人想要区分那些流形，承认非负截面曲率和那些承认正截面曲率。不幸的是，也有一些例子具有非负曲率，甚至更罕见的是具有正曲率的例子。因此，建立新的范例至关重要，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.