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

9625452 Eberlein提出的研究属于黎曼几何的一般领域。研究者计划研究非正曲率和秩1的完全黎曼流形的几何，这是一个由严格负截面曲率隐含但弱于负截面曲率的条件。特别强调将给予流形紧或单连通和齐次。更具体地说，研究者将追求以下三个主题：紧秩1流形的几何；具有左不变riemann度规的单连通二步幂零李群的几何严格负截面曲率的齐次空间。黎曼流形是具有距离函数的弯曲空间的抽象。与三维空间中的曲线和曲面不同，它们是抽象空间，因为它们不一定位于环境空间中。拟议的研究与理解某些空间的大规模结构有关，并研究来自李群理论的各种例子。

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.