Global Riemannian Geometry

全局黎曼几何

• 批准号: 1006677
• 负责人: Peter Petersen
• 金额: $ 15.3万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1006677&HistoricalAwards=false
• 关键词: Global; Riemannian; Geometry

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.

PI 建议在具有正截面曲率的 Gromoll-Meyer 球面上构建黎曼度量。这是一个长期存在的开放性问题，现在尤其重要，因为 Brendle-Schoen 已经证明，没有奇异球体允许具有正复截面曲率的度量。作为一个单独的具体项目，PI 还建议了解何时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研究，但随着佩雷尔曼对三维某些梯度孤子的分类而受到新的关注。理解准爱因斯坦度量在广义相对论中也很重要，因为它们作为爱因斯坦场方程的解而出现。一般而言，PI 希望研究特定拓扑对象上可能出现的几何类型。甜甜圈和茶杯等两个物体在几何上非常不同，但在拓扑上相似。目标是为特定的拓扑对象找到最好的几何形状。这引起了数学家、物理学家、计算机科学家、医学科学家和许多其他人的兴趣，因为越来越清楚的是，我们的平面欧几里得图并不总是正确的，甚至是最好使用的模型。