Rigidity theorems in geometry and topology

几何和拓扑学中刚性定理的证明不仅在这些领域而且在现代数学方法中无处不在。这类定理的两个最好的例子是Rauch， Klingenberg和Berger的球体定理（在几何约束下限制全局拓扑）和Mostow刚性定理（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。在之前与R. Spatzier和B. Wilking的合作中我们证明了截面曲率小于或等于1的流形在t =时每个测地线上都有一个共轭点；是（局部地）等距于紧致的，秩一的对称空间。这就引出了一个关于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等于1，那么是否存在刚性？在与本·施密特（Ben Schmidt）和拉尔夫·斯帕齐尔（Ralf Spatzier）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，并证明了完全长度空间的经典定理。这就引出了将刚性定理扩展到托波诺戈夫意义上的具有双面曲率边界的空间的问题。在这种更普遍的情况下，我们有一些初步的结果。提出了其他几个项目，包括紧李群的黎曼淹没研究（继续该领域的近期工作）和研究圆球上低齐性作用的几何。我们大多数人对“曲率”这个术语有一个直观的理解。桌面和桌面是平的，而篮球和马鞍是弯曲的。我们星球的表面也是弯曲的，我们知道在哥伦布时代产生的争议！对物体曲率的数学研究是微分几何的研究范围。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。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？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。这是微分几何相对容易解决的一类问题。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。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，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。由于质量或能量的引力作用，这种理想状态会被扭曲。因此，引力是时空的曲率，这被用来解释黑洞和引力透镜等现象。我的研究方向是研究高维（就像时空是四维一样）具有正曲率并在适当的附加几何限制下表现出刚性特性的物体。在更高的维度中，物质在视觉上就不那么明显了。其中一门几乎专门研究描述流形曲率的方程和复杂的几何技术，流形指的是，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。正曲率研究中最大的谜团之一是缺乏在每个点都有正曲率的（紧）流形的例子。手头的技术很少，已知示例的数量仍然相对较少。因此，任何能揭示这种流形结构的光都是有价值的。诸如此类的结构定理可能会在某一天引导我们确定宇宙的精确形状。

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.