Curvature, group actions and geometric flows

具有正截面曲率的流形可以用任意三角形中三个角的和大于180度的性质来刻画，即它们的几何类似于圆球(球的表面)的几何。整体黎曼几何可以描述为将像曲率这样的局部不变量与整体不变量联系起来，具有正或更一般的非负曲率的流形在这一课题中扮演着重要的角色。数学的另一个主要主题是对称性。这位首席研究者通过将这两个概念结合起来，对具有大对称群的正曲线流形进行分类，并找到新的例子，这是本课题最困难的部分之一，做出了许多贡献。主要研究者还研究了欧几里得空间子流形等对象，这是自高斯开创性工作以来的经典课题。这项工作的目的是寻找具有正曲率的流形的新例子，并利用雅可比场产生的凹函数来寻找非负曲率的障碍。一个常见的主题是群行动的几何，其轨道具有小的余维，特别是上齐次一行动。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流在几何中得到了许多重要的应用。PI计划进一步研究这种流的应用或对正曲率存在问题的可能修改。PI还研究了正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研究初值问题和同调一流形上的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.