Rigidity theorems in geometry and topology

几何和拓扑中的刚性定理

• 批准号: 1104352
• 负责人: Krishnan Shankar
• 金额: $ 14.16万
• 依托单位: University of Oklahoma Norman Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-10-01 至 2015-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104352&HistoricalAwards=false
• 关键词: Rigidity; theorems; geometry; topology

The proving of rigidity theorems in geometry and topology is ubiquitous not just in these areas but in the modern approach to mathematics. Two of the finest examples of such theorems are the Sphere theorem of Rauch, Klingenberg and Berger (which restricts the global topology under geometric constraints) and the Mostow Rigidity theorem (which restricts the geometry under topological constraints). To this end the PI proposes to study more general spaces and investigate rigidity phenomena for them. In previous work with R. Spatzier and B. Wilking we showed that a manifold with sectional curvature less than or equal to 1 and a conjugate point along every geodesic at t equal to pi; is (locally) isometric to a compact, rank one symmetric space. This led to a natural question about other possible rigidity phenomena along these lines: Is there rigidity if we have the same condition on conjugate points but assume sectional curvature greater than or equal to 1? There has been significant progress on this question in collaboration with Ben Schmidt and Ralf Spatzier. In joint work with C. Sormani we extended the notion of conjugate points and proved classical theorems for complete length spaces. This leads to the question of extending rigidity theorems to spaces with two sided curvature bounds in the sense of Toponogov. We have some preliminary results in this more general setting. Several other projects are proposed including the study of Riemannian submersions from compact Lie groups (continuing recent work in this area) and studying the geometry of low cohomogeneity actions on round spheres.Most of us have an intuitive understanding of the term, "curvature". Tabletops and desktops are flat while basketballs and saddles are curved. The surface of our planet is curved as well, and we know the controversy that generated in Columbus' time! The mathematical study of the curvature of objects is the purview of differential geometry. Geometers are able to quantify curvature precisely and it provides a numerical invariant that helps distinguish objects. One may distinguish objects by shape or nature. For instance, the surface of a doughnut and the surface of a coffee cup have the same nature i.e., they are both surfaces with one hole, but they are shaped differently. On the other hand, the surface of a ball (usually called a sphere) is different in shape and nature from the surface of a doughnut (usually called a torus). How can we be sure that this is always the case? One may wonder if it is possible to deform the sphere suitably so that we might end up with the torus. This is the kind of problem that differential geometry can tackle with relative ease. A sphere has positive curvature everywhere 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 two objects are somehow fundamentally different from each other. Results of this kind are of interest to geometers as well as physicists. Differential geometry is the language used to express the general theory of relativity, our best theoretical description of gravity and its effects on the universe. In general relativity, a vacuous space-time universe would be inherently flat. This idealized state is warped by the presence of masses or energy, due to their gravitational effects. Thus, gravity is the curvature in space-time, and this is used to explain phenomena such as black holes and gravitational lensing. My are of research is the study of objects in higher dimensions (just as spacetime is four dimensional) that have positive curvature and exhibit rigidity properties under suitable additional geometric restrictions. In higher dimensions, matters are far less visually apparent. One deals almost exclusively with equations and sophisticated geometrical techniques that describe the curvature of manifolds, a term that refers to objects that, roughly speaking, have no sharp edges. Manifolds of bounded size are called compact manifolds. One of the great mysteries in the study of positive curvature is the dearth of examples of (compact) manifolds that have positive curvature at every point. The techniques at hand are few and the number of known examples remains relatively small. Therefore, any light that one can shed on the structure of such manifolds would be valuable. Structure theorems such as these are what might lead us some day to determine the precise shape of the universe.

几何和拓扑学中刚性定理的证明不仅在这些领域中无处不在，而且在现代数学方法中也是如此。这类定理的两个最好的例子是Rauch、Klingenberg和Berger的球面定理(它在几何约束下限制全局拓扑)和Mostow刚性定理(它在拓扑约束下限制几何)。为此，PI建议研究更一般的空间，并研究它们的刚性现象。在以前与R.Spatzier和B.Wilking的工作中，我们证明了截面曲率小于或等于1且沿每条测地线在t等于pi处的共轭点的流形是(局部)等距于紧致的秩一对称空间的。这就引出了一个关于这些线上其他可能的刚性现象的自然问题：如果我们在共轭点上有相同的条件，但假设截面曲率大于或等于1，那么是否存在刚性？在与本·施密特和拉尔夫·斯帕齐尔的合作下，在这个问题上取得了重大进展。在与C.Sormani的联合工作中，我们推广了共轭点的概念，并证明了完备长度空间的经典定理。这就引出了将刚性定理推广到Toponogov意义下具有双边曲率界的空间的问题。在这个更一般的背景下，我们已经有了一些初步结果。还提出了其他几个项目，包括研究紧李群的黎曼浸没(继续这一领域的最新工作)，以及研究圆球上低同调作用的几何。我们中的大多数人都对“曲率”这个术语有一个直观的理解。桌面和桌面是平的，而篮球和鞍子是弧形的。我们星球的表面也是弯曲的，我们知道在哥伦布时代引起的争议！物体曲率的数学研究是微分几何的范畴。几何仪能够精确地量化曲率，并提供帮助区分对象的数值不变量。人们可以根据形状或性质来区分物体。例如，甜甜圈的表面和咖啡杯的表面具有相同的性质，即它们都是只有一个孔的表面，但它们的形状不同。另一方面，球的表面(通常称为球体)在形状和性质上与甜甜圈(通常称为环面)的表面不同。我们怎么能确定情况总是如此呢？人们可能会想，是否有可能适当地使球体变形，这样我们就可以得到环面。这是微分几何可以相对轻松地解决的那种问题。球面在任何地方都有正曲率，但可以证明，无论环面是什么形状，它在某个地方总是有零曲率的。这告诉我们，这两个物体在某种程度上是根本不同的。几何学家和物理学家都对这类结果感兴趣。微分几何是用来表达广义相对论的语言，这是我们对重力及其对宇宙影响的最好理论描述。在广义相对论中，真空的时空宇宙本质上是平坦的。这种理想化的状态由于质量或能量的存在而扭曲，因为它们的引力效应。因此，引力是时空中的曲率，这被用来解释黑洞和引力透镜等现象。我的研究是研究高维物体(就像时空是四维的一样)，这些物体具有正曲率，并在适当的附加几何限制下表现出刚性性质。在更高的维度上，事情在视觉上要明显得多。其中一个几乎完全涉及描述流形曲率的方程和复杂的几何技术，这个术语指的是粗略地说，没有锋利边缘的物体。有界大小的流形称为紧流形。研究正曲率的最大谜团之一是缺乏在每一点都有正曲率的(紧致)流形的例子。手头的技术很少，已知的例子数量仍然相对较少。因此，人们对这种流形结构的任何了解都将是有价值的。有一天，这样的结构定理可能会引导我们确定宇宙的精确形状。