Mathematical Sciences: Differential Geometry and Differential Equations

数学科学：微分几何和微分方程

• 批准号: 9505175
• 负责人: Christopher Croke
• 金额: $ 12万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9505175&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Differential; Geometry; Equations

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.

小行星9505175 建议的研究在于黎曼几何的一般领域。特别是，测地线流的各种动力学性质进行了研究。其他议题包括定期度量的研究：这部分的项目涉及到最近决议的霍普夫猜想toral度量。霍普夫猜想指出，在没有共轭点的环面上的黎曼度量一定是平坦的。 黎曼几何目前是数学中一个非常活跃的研究领域;长度最小化路径、距离函数的周期性行为、曲率及其与空间整体形状的关系都是这一领域研究的主题。我们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。