Mathematical Sciences: Manifolds of Nonpositive Curvature, Two-point Homogeneous Spaces and Blaschke Manifolds

主要研究者将研究非正曲率流形中的两个问题，以及涉及齐次空间和Blaschke流形的两个问题。更具体地说，这些问题涉及解流形的对称性、Hadamard流形的几何测度论刻画、特征值的不变性以及完整群例外的Blaschke流形。有两个孔的内管的表面就是流形的一个例子，它可能具有非正曲率。这些都有其更高维度的类似物。其中之一的叶状结构是堆叠在一起的低维表面的集合，这些表面填满了流形。完整群是这样的树叶在扫过其周围流形时扭曲和剪切的量度。首席研究员将研究关于特殊类型流形上的完整学的深入问题。

The principal investigator will study two problems in manifolds of nonpositive curvature as well as two problems involving homogeneous spaces and Blaschke manifolds. More specifically these problems concern the symmetry of solvmanifolds, geometric measure theoretic characterizations of Hadamard manifolds, constancy of eigenvalues, and Blaschke manifolds whose holonomy group is exceptional. The surface of a two-holed inner tube is an example of a manifold which may have nonpositive curvature. These have their higher-dimensional analogues. A foliation of one of these is a collection of lower-dimensional surfaces stacked together which fill up the manifold. The holonomy group is a measure of the twisting and shearing of such leaves as they sweep through their ambient manifolds. The principal investigator will study deep questions concerning holonomy on specialized sorts of manifolds.