Non-negative curvature and group actions

该方案的总体描述是在大等距群的假设下研究具有正或更一般的非负截面曲率的流形。在过去的提案中，首席研究员已经使用这种方法产生了许多非负曲率的新例子，包括一些在奇异球体上的例子。本文研究了一类特殊的流形，它们具有一维商的等距群作用，他认为这类流形是具有正曲率的新例子的优秀候选者。这些候选者在之前的提议中作为分类定理的一部分得到。首席研究员计划在这些流形上研究一类具体的度量，并且已经在曲率特性方面获得了相当多的专业知识。这个项目是非常困难的，预计需要长期的时间投资。在首席研究员计划研究的“具有大等距群的\nnc\”这一主题中，有许多更普遍的问题，这些问题有望更快地得到回报。最后，正如在过去的成功建议中所做的那样，研究新的和已知示例的拓扑性质可能非常困难，但也非常有益。具有正截面曲率的流形可以用任意三角形的3个角之和大于180度的性质来定义，即它们的几何形状类似于圆球。全局黎曼几何可以被描述为将局部不变量如曲率与全局拓扑不变量联系起来。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该学科的重要组成部分。一个基本的未解决的问题是，奇异的球体，即看起来像球体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，首席研究员计划在一个大对称群的存在下研究具有正曲率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.