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

TL;DR

This paper develops geometric tools to classify Killing vector fields of constant length on pseudo-Riemannian homogeneous spaces, extending previous Riemannian results to a broader, non-compact setting using moment maps and flag manifolds.

Contribution

It introduces a geometric approach using moment maps and flag manifolds to classify such vector fields in pseudo-Riemannian spaces, generalizing prior methods that relied on direct computation.

Findings

Classification for elliptic cases with simple, possibly non-compact groups

A shorter, geometric proof avoiding compactness assumptions

Open problems in the combinatorial classification of vector fields

Abstract

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 $(M,\xi)$ where $M = G/H$ is a Riemannian normal homogeneous space, $G$ is a compact simple Lie group, and $\xi \in \mathfrak{g}$ 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)$\xi$ 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 $\xi$ 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.