Group Actions and Curvature

摘要奖：DMS-0513981首席研究员：Krishnan Shanka非负弯曲黎曼流形的研究是一个具有许多公开问题的丰富学科。国际和平研究所提出了这一领域的两个研究项目。第一个项目是与R。Spatzier是最近与R.Spatzier和B的工作的继续。Wilking；我们证明了沿每条测地线具有上曲率界1和球面Jacobi场的流形一定局部等距于一个紧致的秩为1的对称空间。这引出了更多有趣的问题。第二个方案提出在具有连续对称性的正曲流形的基本群上寻找障碍；除了经典的Synge定理外，没有已知的障碍。第三个项目是几何群论领域。书名/作者声明：[by]M.Forester通过构造所谓的雪花群构造了一阶和二阶Dehn函数的许多新的例子。我们希望对其他有限表示群(如CAT(0)群、高阶Dehn函数等)的Dehn函数有进一步的探讨。我们大多数人对曲率这一术语都有一个直观的理解。桌面和桌面是平的，而篮球和鞍子是弧形的。我的研究涉及允许非负曲率的更高维度的对象的研究。这属于微分几何的范畴，这是爱因斯坦用来表达广义相对论的语言，也是我们对引力及其对宇宙影响的最佳理论描述。直观地说，一个正曲线物体具有这样的特性，即画在它上面的所有三角形都比画在桌面上的三角形胖。同样，负曲率对应于薄的或薄的三角形。所以篮球的表面是正曲率的，而骑手坐的地方是负曲率的。在更高的维度上，事情不那么明显，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，流形是没有锐边的对象。微分几何中最大的谜团之一是缺乏非负曲线流形的例子，也没有太多的结构定理。我的工作是试图在存在某些约束的情况下理解流形的结构，比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.