Mathematical Sciences: Investigating Global Differential Geometry

数学科学：研究全局微分几何

• 批准号: 8905540
• 负责人: Deane Yang
• 金额: $ 8.37万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-05-15 至 1991-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8905540&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Investigating; Global; Differential

The two senior investigators will study the relationship between Lp bounds on curvature and the geometry and topology of a Riemannian manifold. They will work towards a proof that a new topological invariant, if zero, implies an F-structure in the sense of Cheeger-Gromov. The relationship between the isoperimetric constant and the curvature of a manifold will be investigated. The investigators will generalize the Heintze- Karcher estimate on the volume of a geodesic neighborhood. A lower bound on the shortest closed geodesic would be very useful. A computer will be used to compute some nontrivial examples of convergent Riemannian manifolds. The standard sphere has positive Gaussian curvature everywhere. Even when deformed, the average curvature remains positive. Such is not the case for the two-holed torus. Regardless of how one of these is deformed, the average curvature must be negative. In higher dimensions the picture is much more complicated. Other sorts of curvature, such as Ricci curvature, play the central roles. A central problem in differential geometry, which the two investigators will address, is how the averages of these curvatures relates to topological properties such as numbers of holes.

两位高级研究员将研究曲率上的 Lp 界限与黎曼流形的几何和拓扑之间的关系。 他们将致力于证明新的拓扑不变量（如果为零）意味着 Cheeger-Gromov 意义上的 F 结构。 将研究等周常数与流形曲率之间的关系。 研究人员将推广 Heintze-Karcher 对测地线邻域体积的估计。 最短闭合测地线的下界将非常有用。 计算机将用于计算收敛黎曼流形的一些重要示例。 标准球体处处具有正高斯曲率。 即使变形，平均曲率仍保持正值。 两孔环面的情况并非如此。 无论其中之一如何变形，平均曲率都必须为负。 在更高维度中，情况要复杂得多。 其他类型的曲率，例如 Ricci 曲率，发挥着核心作用。 两位研究人员将解决的微分几何的一个中心问题是这些曲率的平均值如何与拓扑特性（例如孔的数量）相关。