Mathematical Sciences: Investigating Global Differential Geometry

两位高级调查员将研究 在曲率的Lp界与 黎曼流形 他们将致力于证明一个新的 拓扑不变量，如果为零，意味着F-结构在 Cheeger-Gromov。 的关系 等周常数和流形的曲率将是 研究了 调查人员将概括海因策- 测地邻域体积的Karcher估计。 一 最短闭测地线的下界是非常有用的。 计算机将被用来计算一些非平凡的例子， 收敛黎曼流形 标准球面具有正的高斯曲率 到处 即使变形，平均曲率仍然存在 肯定的：第 对于双孔环面来说，情况并非如此。 不管其中一个如何变形，平均曲率 必须是负数。 在更高的维度中， 复杂 其他类型的曲率，比如里奇曲率， 发挥中心作用。 微分学的一个中心问题 几何，这两个调查人员将解决，是如何 这些曲率的平均值与拓扑性质有关 例如孔的数量。

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.