Group Actions on Manifolds with Positive Sectional Curvature

正截面曲率流形上的群作用

• 批准号: 0103993
• 负责人: Krishnan Shankar
• 金额: $ 7.8万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2003-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103993&HistoricalAwards=false
• 关键词: Group; Actions; Manifolds; Positive; Sectional

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.

摘要：对于DMS - 0103993A，黎曼几何中的一般问题是寻找和描述允许具有正截面曲率的完全黎曼度规的流形。如果曲率上没有正的下界，那么根据Cheeger-Gromoll-MeyerSoul定理，流形对于欧几里德空间是微分同态的。在闭正弯曲流形中，限制条件很少，大多数是经典的，如bonnet - myers定理和Synge定理。对于封闭的单连通流形，在给定的维度上，本质上只有限定总贝蒂数的Gromov定理。考虑到已知的障碍很少，令人沮丧的是，已知的例子集虽然无限，但相对较小。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)试图通过研究双商上更一般的度量来寻找正弯曲流形的新例子(与j.h。Eschenburg), 2)计算已知的正曲率同质流形的等距群，以及3)观察7维Berger空间是否在4球上与3球束微分同态（与N. Kitchloo合作）。黎曼几何起源于对曲率的理解。直觉上，我们知道桌面是平的，而篮球和马鞍是弯曲的。几何学家能够精确地量化曲率，它提供了一个有助于区分物体的数值不变量。例如，甜甜圈的表面和咖啡杯的表面具有相同的性质，即它们都是带有一个孔的表面，但它们的形状不同。另一方面，球（通常称为球体）的表面与甜甜圈（通常称为环面）的表面在形状和性质上都不同。我们怎么能确定情况总是如此呢？有人可能会想，是否有可能适当地变形球体，使我们最终得到环面。球体在任何地方都有正曲率，但可以证明，无论环面呈何种形状，它在某些地方的曲率总是为零。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。这种理想状态被质量或能量的存在所扭曲，因此，引力是时空的曲率，通过理解洛伦兹时空的几何，人们可能有一天会理解宇宙的形状。我的工作涉及高维正弯曲物体的研究。这是试图理解曲率（几何）所施加的结构如何对理解物体的本质（拓扑）至关重要的一部分，反之亦然。