Group Actions and Curvature

AbstractAward：DMS-0513981主要研究者：Krishnan Shankar非负曲率黎曼流形的研究是一个丰富的课题，有许多开放的问题。PI在这一领域提出了两个研究项目。第一个与R. Spatzier是R. Spatzier和B.威尔金;我们证明了一个上曲率有界且沿每条测地线沿着具有球面Jacobi场的流形必局部等距于一个紧的秩为1的对称空间。这引出了更多有趣的问题。第二个项目提出了在连续对称的存在下，在正弯曲流形的基本群上找到障碍物;除了经典的Synge定理，没有已知的障碍物。第三个项目是在geometricgroup理论领域。在与N。布雷迪，M. Bridson和M. Forester我们通过构造所谓的雪花群构造了许多一阶和二阶Dehn函数的新例子。我们希望进一步探讨其他类的非线性群（如CAT（0）群、高阶Dehn函数等）的Dehn函数问题。我们大多数人对曲率项有一个直观的理解。桌面和桌面是平的，而篮球和马鞍是弯曲的。我的研究是关于高维物体的非负曲率的研究。这属于微分几何的范畴，微分几何是爱因斯坦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好的理论描述。直觉上，一个正弯曲的物体具有这样的特性，即所有画在它上面的三角形都比画在桌面上的三角形粗。类似地，负曲率对应于薄三角形。因此，篮球的表面有正曲率，而鞍状物的表面有负曲率。在更高的维度上，物质在视觉上不那么明显，人们使用方程和复杂的几何技术来研究流形的曲率，粗略地说，流形是没有尖锐边缘的物体。微分几何中最大的问题之一是缺乏非负曲流形的例子，也没有多少结构定理。我的工作涉及试图理解流形的结构，在某些约束的存在下，如非负曲率或对称性。

AbstractAward: DMS-0513981Principal Investigator: Krishnan ShankarThe study of non-negatively curved Riemannian manifolds is a richsubject with many open problems. The PI proposes two researchprojects in this area. The first project in collaboration withR. Spatzier is continuation of recent work with R. Spatzier andB. Wilking; we showed that a manifold with upper curvature bound1 and spherical Jacobi fields along every geodesic must belocally isometric to a compact, rank one symmetric space. Thishas led to further interesting questions. The second projectproposes to find obstructions on the fundamental group ofpositively curved manifolds in the presence of continuoussymmetry; other than the classical Synge theorem, there are noknown obstructions. The third project is in the area of geometricgroup theory. In collaboration with N. Brady, M. Bridson andM. Forester we constructed many new examples of first and secondorder Dehn functions by constructing the so called snowflakegroups. We hope to pursue further questions about Dehn functionsfor other classes of finitely presented groups (like CAT(0)groups, higher Dehn functions etc.)Most of us have an intuitive understanding of the termcurvature. Tabletops and desktops are flat while basketballs andsaddles are curved. My research concerns the study of objects inhigher dimensions that admit non-negative curvature. This fallsunder the umbrella of differential geometry which is the languageEinstein used to express the general theory of relativity, ourbest theoretical description of gravity and its effects on theuniverse. Intuitively a positively curved object has the propertythat all triangles drawn on it are fatter than triangles drawn ona tabletop. Similarly, negative curvature corresponds to thin orskinny triangles. So (the surface of) a basketball has positivecurvature while a saddle has negative curvature where the ridersits. In higher dimensions, matters being much less visuallyapparent, one uses equations and sophisticated geometricaltechniques to study the curvature of manifolds which are, roughlyspeaking, objects with no sharp edges. One of the great mysteriesin differential geometry is the dearth of examples ofnon-negatively curved manifolds, and not many structure theoremseither. My work deals with trying to understand the structure ofmanifolds in the presence of certain constraints likenon-negative curvature or symmetry.