arXiv 1503.082672015-03-28math.DGmath-phmath.MPmath.SG

伪黎曼正规齐性空间上常长Killing向量场的分类

Toward a Classification of Killing Vector Fields of Constant Length on Pseudo--Riemannian Normal Homogeneous Spaces

Joseph A. Wolf, Fabio Podestà, Ming Xu

摘要：

本文发展了伪黎曼齐性空间上定长Killing向量场分类的基本工具。这推广了M. Xu和J. A. Wolf将对$(M,ξ)$进行了分类，其中$M = G/H$是黎曼正规齐性空间，$G$是紧致单李群，而$ξ\in \mathfrak{g}$定义了$M$上的非零常长Killing向量场。那里的方法是直接计算。本文利用Ad（G）的矩映射$M \to \mathfrak{g}^*$和旗流形结构$ξ$给出了一个更短、更几何的证明，它不要求紧性，并且在伪黎曼情形下是有效的.在这种情况下，我们将分类问题分为三个部分。第一个很容易解决。第二个涉及$ξ$是椭圆的，而$G$是简单的（但不一定是紧的）;这种情况是我们这里的主要结果。第三，这仍然是开放的，是一个更组合的问题，涉及前两个元素。

英文摘要：

In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo--riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs $(M,ξ)$ where $M = G/H$ is a Riemannian normal homogeneous space, $G$ is a compact simple Lie group, and $ξ\in \mathfrak{g}$ defines a nonzero Killing vector field of constant length on $M$. The method there was direct computation. Here we make use of the moment map $M \to \mathfrak{g}^*$ and the flag manifold structure of Ad(G)$ξ$ to give a shorter, more geometric proof which does not require compactness and which is valid in the pseudo--riemannian setting. In that context we break the classification problem into three parts. The first is easily settled. The second concerns the cases where $ξ$ is elliptic and $G$ is simple (but not necessarily compact); that case is our main result here. The third, which remains open, is a more combinatorial problem involving elements of the first two.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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