Global Riemannian Geometry

摘要DMS-0204177。PI：Peter Petersen该PI建议研究具有积分曲率界的流形，而不是通常的逐点曲率界。许多传统的结果延续了下来，但人们并不完整地知道较弱的曲率假设会出什么问题。提案的第二部分也是主要部分解决了具体但非常困难的问题，即4球面上的3球面丛具有正曲率。这些空间中只有一个是已知的正曲率的，即7-球面。我们期望几个这种类型的奇异球体具有正曲率，并且我们将从这个项目中得到一些有趣的6维和7维流形的无穷族。PI建议在两个不同的项目上工作。第一种是从物体中收集信息，在这些物体中，人们对物体的曲线有模糊的假设，而不是具体的假设。PI已经在这个方向上建立了一些有趣的结果，但仍有许多未知之处。第二个也是更雄心勃勃的部分是关于一个人可以制造出多圆的扭曲物体的调查。在这里，人们想象一个球体是完美的圆形和无扭曲的。在可以可视化的维度中没有直观的例子，但在维度7中，似乎有几个物体不是球体，但仍然可能是相当圆的。对这一问题的积极解决将解决一个非常突出的问题和两个不太为人所知的问题。

ABSTRACT DMS - 0204177. PI: Peter PetersenThe PI proposes to investigate manifolds with integral curvature bounds as opposed to the usual pointwise curvature bounds. Many traditional results carry over, but people do not have a complete picture of what can go wrong with the weaker curvature assumptions. The second and main part of the proposal addresses the specific but very hard question of which 3-sphere bundles over the 4-sphere have positive curvature. Only one of these spaces, namely the 7-sphere, is know to have positive curvature. We expect that several exotic spheres of this type have positive curvature and also that we will get some interesting infinite families of manifolds in dimensions 6 and 7 out ofthis project.The PI is proposing to work on two separate projects. The first deals with gleaning information from objects where one has fuzzy rather than specific assumptions about how they curve. The PI has already established some interesting results in this direction, but much is still unknown. The second and more ambitious part is an investigation of how round one can make twisted objects. Here one imagines a sphere as perfectly round and untwisted. There are no intuitive examples in the dimensions that can be visualized, but in dimension 7 there apear to be several objects which are not spheres but still could be fairly round. A positive solution to this problem would solve one very prominent problem and two lesser known problems.