Group Actions and Curvature

群动作和曲率

• 批准号: 0513981
• 负责人: Krishnan Shankar
• 金额: $ 10.8万
• 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: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0513981&HistoricalAwards=false
• 关键词: Group; Actions; Curvature

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.

摘要：非负弯曲黎曼流形的研究是一个具有许多开放问题的丰富学科。PI在这一领域提出了两个研究项目。第一个与r合作的项目。斯帕齐尔是R.斯帕齐尔和b .斯帕齐尔最近研究的延续。Wilking;我们证明了具有上曲率bound1和沿每个测地线的球面雅可比场的流形必须局部等距于紧致的秩一对称空间。这引出了更多有趣的问题。第二个方案提出在连续对称条件下寻找正弯曲流形基本群上的障碍；除了经典的辛格定理，还有未知的障碍。第三个项目是在几何群理论领域。与N.布雷迪、M.布莱德森和M.布莱德森合作。在此之前，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。我们希望对其他有限表示群（如CAT(0)群，更高的Dehn函数等）的Dehn函数进行进一步的研究，我们大多数人对项曲率有直观的理解。桌面和桌面是平的，而篮球和马鞍是弯曲的。我的研究涉及对具有非负曲率的高维物体的研究。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佳理论描述。直观上，一个正弯曲的物体具有这样的属性，即在其上绘制的所有三角形都比在桌面上绘制的三角形更胖。类似地，负曲率对应于细三角形或细三角形。所以篮球的表面曲率是正的，而鞍座的表面曲率是负的。在更高的维度中，事情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，流形是没有锋利边缘的物体。微分几何中最大的谜团之一是缺乏非负弯曲流形的例子，也没有多少结构定理。我的工作是试图理解流形在某些约束条件下的结构，如非负曲率或对称性。