Global Riemannian Geometry

PI提出在具有正截面曲率的Gromoll-Meyer球上构造一个黎曼度量。这是一个长期存在的开放问题，现在特别相关的Brendle-Schoen表明，没有异国情调的领域承认度量与积极的复杂的截面曲率。作为一个单独的具体项目，PI还提出要了解什么时候准爱因斯坦和梯度孤子度量被迫成为爱因斯坦度量。这些问题可以追溯到爱因斯坦度量的最早的作品，但随着Perel'man对三维中某些梯度孤子的分类而受到新的关注。理解准爱因斯坦度规在广义相对论中也很重要，因为它们是爱因斯坦场方程的解。一般来说，PI希望研究在特定的拓扑对象上可能存在什么类型的几何。像甜甜圈和茶杯这样的两个物体在几何学上非常不同，但在拓扑学上相似。目标是为特定的拓扑对象找到最好的几何图形。数学家、物理学家、计算机科学家、医学家和许多其他人都对这一点感兴趣，因为越来越清楚的是，我们的平面欧几里得图像并不总是正确的，甚至是最好的模型。

The PI proposes to construct a Riemannian metric on the Gromoll-Meyer sphere with positive sectional curvature. This is a long standing open problem and is now especially relevant as Brendle-Schoen have shown that no exotic sphere admits a metric with positive complex sectional curvatures. As a separate specific project the PI also proposes to understand when quasi-Einstein and gradient soliton metrics are forced to be Einstein metrics. These questions go back to the earliest works on Einstein metrics, but have received new attention with Perel'man classification of certain gradient solitons in three dimensions. Understanding quasi-Einstein metrics is also important in general relativity as they occur as solutions to the Einstein field equations.In general terms the PI wishes to investigate what types of geometries are possible on specific topological objects. Two objects such as a doughnut and a tea cup are geometrically very different but topologically similar. The goal is to find the nicest possible geometries for specific topological objects. This is of interest to mathematicians, physicists, computer scientists,medical scientists and many other people as it is becoming increasingly clear that our flat Euclidean picture is not always the correct or even nicest model to use.