Non-negative curvature and group actions

摘要-DMS-0806070本方案的总体描述是在大等距群的假设下研究截面曲率为正或更一般的非负的流形。在过去的提案中，首席研究人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。本提案研究了一类特殊的流形，这些流形允许一维商的等距群作用，并且他认为这类流形是具有正曲率的新例子的极好候选者。作为分类定理的一部分，这些候选者是在先前的提议中获得的。首席研究员计划研究这些流形上的一类具体度量，并已经在它们的曲率性质方面获得了相当多的专业知识。这个项目非常困难，预计需要长期的时间投资。在这一主题中，有许多更一般的问题，首席研究人员计划研究，这些问题有望更快地得到回报。最后，正如在过去的建议中成功地完成的那样，研究新的和已知的例子的拓扑性质可能是非常困难的，但也是非常有益的。具有正截面曲率的流形可以通过任意三角形中的三个角的和大于180度的性质来定义，即它们的几何类似于圆球的几何。整体黎曼几何可以描述为将曲率等局部不变量与整体拓扑不变量联系起来。自从整体黎曼几何诞生以来，具有正或更一般的非负曲率的流形一直是这一课题的重要组成部分。一个基本的悬而未决的问题是，奇异球体，即看起来像球体但其上的普通微积分有很大不同的流形，是否可以携带正曲线度量。对称性是许多几何问题的一个重要方面，主要研究者计划在一个大的对称群的存在下研究具有正的或更一般的非负曲率的流形。这项调查的目标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.