Journal of Differential Geometry

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

Joseph A. Wolf, Fabio Podestà, and Ming Xu

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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 $\mathrm{Ad} (G) \xi$ 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.


JAW: Research partially supported by a Simons Foundation grant and by the Dickson Emeriti Professorship at the University of California, Berkeley.

MX: Research supported by NSFC no. 11271216, State Scholarship Fund of CSC (no. 201408120020), Science and Technology Development Fund for Universities and Colleges in Tianjin (no. 20141005), Doctor fund of Tianjin Normal University (no. 52XB1305). Corresponding author.

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J. Differential Geom., Volume 105, Number 3 (2017), 519-532.

Received: 6 April 2015
First available in Project Euclid: 3 March 2017

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Wolf, Joseph A.; Podestà, Fabio; Xu, Ming. Toward a classification of killing vector fields of constant length on pseudo-Riemannian normal homogeneous spaces. J. Differential Geom. 105 (2017), no. 3, 519--532. doi:10.4310/jdg/1488503006.

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