Non-negative curvature and group actions

摘要-DMS-0806070该提案的总体描述是在一个大等距群的假设下研究具有正截面曲率或更一般的非负截面曲率的流形。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。本建议研究了一类特定的流形，承认等距群行动与一维商，他认为是优秀的候选人新的例子，正曲率。这些候选人在以前的提案中获得的分类定理的一部分。首席研究员计划研究这些流形上的一类具体度量，并在其曲率性质方面已经获得了相当多的专业知识。这个项目是非常困难的，预计将需要长期的时间投资。有很多问题的一个更普遍的性质在这个主题的``\nnc\与大型等距群”，主要研究者计划研究，并承诺更快的回报。最后，正如过去成功的建议所做的那样，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何三角形中的3个角之和大于180度的性质，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不变量（如曲率）与整体拓扑不变量相关联。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。一个基本的未解决的问题是奇异球，即看起来像球但普通微积分完全不同的流形，是否可以携带正弯曲的度量。对称性是许多几何问题的一个重要方面，首席研究员计划研究在大对称群存在下具有正曲率或更一般的非负曲率的流形。这次调查的目的之一是寻找新的例子。

Abstract-DMS-0806070The overall description of the proposal is to study manifolds with positive or more generally nonnegative 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 nonnegative curvature, including some on exotic spheres. The present proposal studies a specific class of manifolds that admit an isometric group action with one dimensional quotient and which he considers to be excellent candidates for new examples with positive curvature. These candidates were obtained in a previous proposal as part of a classification theorem. The principal investigator plans to study a concrete class of metrics on these manifolds and has obtained considerable expertise in their curvature properties already. This project is extremely difficult and is expected to require a long term time investment.There are many questions of a more general nature within this subject of ``\nnc\ with large isometry groups" that the principal investigator plans to study, and which promise a much quicker return. 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 curvaturecan 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 invariants like curvature to global topological invariants. Since the beginning of global Riemannian geometry, manifolds with positive or more generally non-negative curvature have been an important part of this subject. A basic unsolved question is whether exotic spheres, i.e. manifolds that look like spheres but on which ordinary calculus is quite different, can carry positively curved metrics. Symmetries are an important aspect of many geometric questions and the principal investigator plans to study manifolds with positive or more generally non-negative curvature under the presence of a large symmetry group. One of the goals of this investigation is the search for new examples.