Mathematical Sciences: Geometry of Nonpositively Curved Manifolds of Rank 1 and Related Homogeneous Spaces

数学科学：一阶非正曲流形几何及相关齐次空间

• 批准号: 9625452
• 负责人: Patrick Eberlein
• 金额: $ 7.5万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-05-15 至 2000-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9625452&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Nonpositively; Curved

9625452 Eberlein The proposed research lies in the general area of Riemannian geometry. The investigator plans to study the geometry of complete Riemannian manifolds of nonpositive curvature and rank 1, a condition implied by but weaker than strictly negative sectional curvature. Particular emphasis will be given to manifolds that are compact or are simply connected and homogeneous. More specifically, the investigator will pursue the following three topics: geometry of compact rank 1 manifolds; geometry of simply connected 2-step nilpotent Lie groups with a left invariant Riemannian metric; homogeneous spaces of strictly negative sectional curvature. Riemannian manifolds are an abstraction of curved spaces possessing a distance function. Unlike curves and surfaces in 3-space these are abstract spaces in that they do not necessarily lie in an ambient space. The proposed research has to do with understanding the large scale structure of certain of these spaces, and looking at various examples coming from Lie group theory.

9625452 Eberlein拟议的研究属于一般领域， 黎曼几何 研究人员计划研究 秩为1的非正曲率完备黎曼流形 条件隐含但弱于严格负截面曲率。 特别强调将给予流形是紧凑的或 单连通和齐次。 更具体地说，调查员将探讨以下三个主题： 紧秩1流形的几何;单连通2步的几何 具有左不变黎曼度量的幂零李群;齐次 严格负截面曲率的空间。 黎曼流形是 具有距离函数的弯曲空间的抽象。 不像 三维空间中的曲线和曲面，这些都是抽象空间，因为它们 不一定位于周围空间中。 这项研究必须 随着对这些空间中某些空间的大尺度结构的理解， 以及李群理论中的各种例子。