论文标题
半摩曼锥
Semi-Riemannian cones
论文作者
论文摘要
由于癌变的结果，在完整的riemannian歧管上的riemannian锥体是扁平的，要么具有不可还原的全能代表。对于不确定的锥体来说，这通常是错误的，但是可以使用全能子空间在锥体上诱导的结构来研究圆锥底部的几何形状。本文的目的是双重的：首先，我们将对具有不可约合的全能代表制的半摩恩曼锥的一般结果进行调查，然后作为主要结果，当锥体允许平行矢量字段时，我们将在此情况下得出这些一般陈述的改进版本。我们将证明，如果基本歧管完成，并且圆锥体的纤维和平行矢量场具有相同的因果特征，则锥体是平坦的，否则，基本歧管则承认了一定的全局扭曲产品结构。我们将使用这些结果为Riemannian流形的分类结果提供了新的证明，其中具有假想的杀伤旋转器和Lorentzian歧管，并带有由Baum和Bohle造成的真实杀戮旋转器。
Due to a result by Gallot a Riemannian cone over a complete Riemannian manifold is either flat or has an irreducible holonomy representation. This is false in general for indefinite cones but the structures induced on the cone by holonomy invariant subspaces can be used to study the geometry on the base of the cone. The purpose of this paper is twofold: first we will give a survey of general results about semi-Riemannian cones with non irreducible holonomy representation and then, as the main result, we will derive improved versions of these general statements in the case when the cone admits a parallel vector field. We will show that if the base manifold is complete and the fibre of the cone and the parallel vector field have the same causal character, then the cone is flat, and that otherwise, the base manifold admits a certain global warped product structure. We will use these results to give a new proof of the classification results for Riemannian manifolds with imaginary Killing spinors and Lorentzian manifolds with real Killing spinors which are due to Baum and Bohle.
