Curvature, group actions and geometric flows

具有正截面曲率的流形可以用任意三角形的三个角之和大于180度的性质来表征，即它们的几何形状类似于圆球（球的表面）。全局黎曼几何可以被描述为将曲率等局部不变量与全局不变量联系起来；具有正曲率或更普遍的非负曲率的流形在这一主题中起着重要作用。数学的另一个主要主题是对称。首席研究员通过结合这两个概念，将正弯曲流形与大对称群分类以及寻找新的例子做出了许多贡献，这是该主题最困难的部分之一。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主题。本课题研究了几种情况下的非负曲率和正曲率。本文旨在寻找具有正曲率的流形的新例子，并利用由雅可比场产生的凹函数寻找非负曲率的障碍。一个经常讨论的主题是轨道余维小的群作用的几何问题，特别是同质群作用。PI研究非负曲率欧几里德空间的子流形及其刚性性质。近年来，里奇流在几何学中得到了许多重要的应用。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正。PI还研究了正曲率或非负曲率与几何形式概念和极性作用概念的关系。在另一个项目中，PI研究了齐次1流形上的初值问题和爱因斯坦度量的存在性。

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.