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arXiv · 2008.00543

Geodesic completeness of some Lorentzian simple Lie groups

E. Ebrahimi·S. M. B. Kashani·M. J. Vanaei
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Abstract

In this paper we investigate geodesic completeness of left-invariant Lorentzian metrics on a simple Lie group $G$ when there exists a left-invariant Killing vector field $Z$ on $G$. Among other results, it is proved that if $Z$ is timelike, or $G$ is strongly causal and $Z$ is lightlike, then the metric is complete. We then consider the special complex Lie group $SL_2(\mathbb{C})$ in more details and show that the existence of a lightlike vector field $Z$ on it, implies geodesic completeness. We also consider the existence of a spacelike vector field $Z$ on $SL_2(\mathbb{C})$ and provide an equivalent condition for the metric to be complete. This illustrates the complexity of the situation when $Z$ is spacelike.

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E. Ebrahimi, S. M. B. Kashani, M. J. Vanaei. 2020-08-02. Geodesic completeness of some Lorentzian simple Lie groups. https://arxiv.org/abs/2008.00543

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