Mathematical Sciences: Differential Geometry and Differential Equations

[505175]克罗克提出的研究是在黎曼几何的一般领域。特别地，测地线流动的各种动力学性质将被研究。其他主题包括周期度量的研究：项目的这一部分涉及到Hopf关于总度量猜想的最新解决方案。Hopf猜想指出，没有共轭点的环面上的黎曼度规一定是平的。黎曼几何是目前数学中一个非常活跃的研究领域；长度最小化路径，距离函数的周期行为，曲率及其与空间整体形状的关系是该领域研究的主题之一。熟悉的欧几里得几何是黎曼几何的一个非常特殊的例子，其特征是完全没有曲率。

9505175 Croke The proposed research lies in the general area of Riemannian geometry. In particular, various dynamical properties of geodesic flow are to be investigated. Other topics include the study of periodic metrics: this portion of the project relates to the recent resolution of Hopf's conjecture on toral metrics. Hopf's conjecture states that a Riemannian metric on a torus without conjugate points must be flat. Riemannian geometry is currently a very active field of research in mathematics; length-minimizing paths, periodic behavior of the distance function, curvature and its relation to the global shape of the space are among the topics studied in this field. The familiar Euclidean geometry is a very special case of Riemannian geometry characterized by a total absence of curvature.