Curvature, group actions and geometric flows

曲率、群作用和几何流

• 批准号: 1506148
• 负责人: Wolfgang Ziller
• 金额: $ 43.15万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1506148&HistoricalAwards=false
• 关键词: Curvature; group; actions; geometric; flows

Manifolds with positive sectional curvature can be characterized by the property that the sum of the three angles in any triangle is larger than 180 degrees, i.e., their geometry is similar to that of the round sphere (the surface of a ball). Global Riemannian geometry can be described as relating local invariants like curvature to global invariants; manifolds with positive or more generally non-negative curvature play an important role in this subject. Another major theme in mathematics is that of symmetry. The principal investigator has made many contributions by combining both concepts, classifying positively curved manifolds with large symmetry groups and finding new examples, one of the most difficult parts of the subject. The principal investigator also studies such objects as submanifolds of Euclidean space, a classical subject since the seminal work of Gauss.This research project studies non-negative and positive curvature in several contexts. The work aims to find new examples of manifolds with positive curvature and to find obstructions to non-negative curvature by using concave functions arising from Jacobi fields. A frequent theme is the geometry of group actions whose orbits have small codimension, especially cohomogeneity one actions. The PI studies submanifolds of Euclidean space with nonnegative curvature and their rigidity properties. In recent years the Ricci flow has found many important applications in geometry. The PI plans to study further applications of such flows or possible modifications to the existence problem in positive curvature. The PI also studies relationships of positive or non-negative curvature with the concept of geometric formality and that of polar actions. In another project the PI studies initial value problems and existence of Einstein metrics on cohomogeneity one manifolds.

具有正截面曲率的流形的特征可以在于任何三角形中的三个角之和大于180度的性质，即，它们的几何形状类似于圆球（球的表面）的几何形状。整体黎曼几何可以描述为将局部不变量（如曲率）与整体不变量相关联;具有正曲率或更一般的非负曲率的流形在这一主题中扮演着重要角色。 数学中的另一个重要主题是对称性。主要研究者作出了许多贡献，结合这两个概念，分类积极弯曲流形与大型对称群，并找到新的例子，最困难的部分之一的主题。 首席研究员还研究欧几里得空间的子流形等对象，这是自高斯开创性工作以来的经典学科。该研究项目研究多种背景下的非负和正曲率。 这项工作的目的是找到新的例子，流形的正曲率和找到障碍，非负曲率通过使用凹函数产生的雅可比场。一个常见的主题是几何群行动的轨道有小余维，特别是余齐性一行动。 PI研究欧氏空间中具有非负曲率的子流形及其刚性性质。近年来，Ricci流在几何学中有许多重要的应用. PI计划进一步研究这种流动的应用或对正曲率存在问题的可能修改。 PI还研究了正曲率或非负曲率与几何形式和极作用概念的关系。在另一个项目中，PI研究了初值问题和上齐性流形上的爱因斯坦度量的存在性。