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Toward a classification of killing vector fields of constant length on pseudo-riemannian normal homogeneous spaces

  • Author(s): Wolf, JA
  • Podesta, F
  • Xu, M
  • et al.
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,ξ) where M = G/H is a Riemannian normal homogeneous space, G is a compact simple Lie group, and ξ ϵ 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 → 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.

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