Mathematical Sciences: Geometry of Compact Nonpositively Curved Manifolds of Rank 1

首席研究员将继续研究有限体积和非正曲率的完全黎曼流形的几何。将特别强调秩1的流形，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。该研究项目将包括研究具有严格负截面曲率的对称空间的几何特征，以及基流形的基本群的代数性质。一个多世纪以来，数学家一直在研究被称为流形的广义曲面。所谓的“黎曼流形”可能具有明确定义的面积或体积。例如，正如我们通常想象的那样，球体和环面具有有限的表面积。首席研究员将研究非正曲率流形。这种流形的一个例子是双孔环面。这方面的研究在20世纪80年代尤其活跃，这在很大程度上要归功于首席研究员早期的开拓性努力。

The principal investigator will continue his study of the geometry of complete Riemannian manifolds of finite volume and nonpositive curvature. Particular emphasis will be given to manifolds of rank 1, a condition implied by but weaker than strictly negative sectional curvature. The research project will include the investigation of both geometric characterizations of symmetric spaces with strictly negative sectional curvature, and algebraic properties of the fundamental groups of the base manifold. Mathematicians have studied generalized surfaces, called manifolds, for more than a century. A so called "Riemannian manifold" may have a well-defined area or volume. For example, as we usually imagine them, spheres and tori have finite surface area. The principal investigator will study manifolds of nonpositive curvature. One example of such a manifold would be the two-holed torus. This line of research has been particularly active in the 1980's due in large part to earlier pioneering efforts of the principal investigator.