Global Riemannian Geometry

摘要DMS - 0204177。 PI：Peter Petersen PI建议研究具有积分曲率边界的流形，而不是通常的逐点曲率边界。许多传统的结果继续下去，但人们没有一个完整的图片，什么可以出错的较弱的曲率假设。该提案的第二部分和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。其中只有一个空间，即7-球面，已知具有正曲率。我们希望这种类型的几个奇异球具有正曲率，并且我们将在这个项目中得到一些有趣的6维和7维流形的无限族。第一个处理从对象中收集信息，其中人们对它们如何弯曲有模糊而不是具体的假设。PI已经在这个方向上建立了一些有趣的结果，但还有很多未知数。第二个也是更雄心勃勃的部分是研究人们可以制造出多圆的扭曲物体。在这里，人们想象一个完美的圆形和无扭曲的球体。没有直观的例子，在维度，可以可视化，但在第7维似乎有几个对象，这不是球体，但仍然可以是相当圆。这个问题的积极解决方案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.