Group Actions on Manifolds with Positive Sectional Curvature

黎曼几何中的一个普遍问题是寻找和描述具有正截面曲率的完备黎曼度量的流形，如果曲率没有正下界，则根据Cheeger-Gromoll-MeyerSoul定理，该流形与欧氏空间同构。在闭的正曲流形类中，有一些限制，其中大部分是经典的，如Bonnet-Myers定理和Synge定理。对于封闭的单连通流形，在给定的维数上，基本上只有格罗莫夫定理限制总贝蒂数。由于已知的障碍物很少，令人沮丧的是，已知的例子虽然有很多，但相对较少。我的研究是关于理解已知例子的几何和拓扑结构。更具体地说，目标是：1）试图通过研究双曲流形上更一般的度量来寻找正弯曲流形的新例子（与J. - H.Eschenburg），2）计算正曲率流形的等距群，3）观察7维Berger空间是否同构于4-球面上的3-球面丛（与N.黎曼几何起源于试图理解曲率。凭直觉，我们知道桌面是平的，而篮球和马鞍是弯曲的。 几何能够精确地量化曲率，它提供了一个数值不变量，有助于区分对象。 例如，甜甜圈的表面和咖啡杯的表面具有相同的性质，即，它们都是有一个孔的表面，但是它们的形状不同。另一方面，球的表面（通常被称为球体）在形状和性质上不同于坚果的表面（通常被称为环面）。我们如何才能确保情况总是如此？人们可能会想，是否有可能使球体适当变形，以便我们最终得到环面。一个球面到处都有正曲率，而可以证明的是，无论一个环面是什么形状，它在某个地方总是曲率为零。这告诉我们，这两个物体在某种程度上是根本不同的。 微分几何也是用来表达广义相对论的语言，广义相对论是我们对引力及其对宇宙影响的最佳理论描述。在广义相对论中，一个真空的时空宇宙本质上是平坦的。这种理想化的状态会因质量或能量的存在而扭曲，因此，引力是时空的曲率，通过理解洛伦兹时空的几何形状，有朝一日人们可能会理解宇宙的形状。我的工作包括研究更高维度的正弯曲物体。这是试图理解曲率（几何）所施加的结构如何对理解物体的性质（拓扑）至关重要，反之亦然。

Abstract for DMS - 0103993A general problem in Riemannian geometry is to find and describe manifoldsthat admit a complete Riemannian metric of positive sectional curvature.If there is no positive lower bound on the curvature, then the manifold isknown to be diffeomorphic to Euclidean space, by the Cheeger-Gromoll-MeyerSoul theorem. In the class of closed, positively curved manifolds, thereare few restrictions, most of which are classical, such as theBonnet-Myers and Synge theorems. For closed, simply connected manifolds,there is essentially just Gromov's theorem bounding the total Betti numberin a given dimension. Given that there are few known obstructions, it isfrustrating that the set of known examples, although infinite, isrelatively small. My research is concerned with understanding the geometryand topology of the known examples. More specifically, the goals are: 1)to attempt to find new examples of positively curved manifolds by studyingmore general metrics on biquotients (in collaboration with J.-H.Eschenburg), 2) to compute the isometry groups for the known cohomogeneityone manifolds of positive curvature and 3) to see whether the7-dimensional Berger space is diffeomorphic to a 3-sphere bundle over the4-sphere (in collaboration with N. Kitchloo).Riemannian geometry arose from trying to understand curvature. Intuitively, we know that tabletops are flat while basketballs and saddlesare curved. Geometers are able to quantify curvature precisely and itprovides a numerical invariant that helps distinguish objects. Forinstance, the surface of a doughnut and the surface of a coffee cup havethe same nature i.e., they are both surfaces with one hole, but they areshaped differently. On the other hand, the surface of a ball (usuallycalled a sphere) is different in shape and nature from the surface of adoughnut (usually called a torus). How can we be sure that this is alwaysthe case? One may wonder if it is possible to deform the sphere suitablyso that we might end up with the torus. A sphere has positive curvatureeverywhere while it can be shown that no matter what shape a torus takes,it will always have zero curvature somewhere. This tells us that the twoobjects are somehow fundamentally different from each other. Differentialgeometry is also the language used to express the general theory ofrelativity, our best theoretical description of gravity and its effects onthe universe. In general relativity, a vacuous space-time universe wouldbe inherently flat. This idealized state is warped by the presence ofmasses or energy, Thus, gravity is the curvature in space-time, and byunderstanding the geometry of Lorentzian space-time, one may some dayunderstand the shape of the universe. My work involves the study ofpositively curved objects in higher dimensions. This is part of trying tounderstand how the structure imposed by curvature (geometry) is essentialto understanding the nature (topology) of an object and vice versa.