Journal of Lie Theory 35 (2025), No. 2, 239--261
Copyright Heldermann Verlag 2025
Geodesic Completeness of some Lorentzian Simple Lie Groups
Esmail Ebrahimi
Dept. of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran
esmail.ebrahimi@modares.ac.ir
Seyed M. B. Kashani
Dept. of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran
kashanim@modares.ac.ir
Mohammad J. Vanaei
Dept. of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran
javad.vanaei@modares.ac.ir
[Abstract-pdf]
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. The situation is considerably elaborate when $Z$ is spacelike, as our study of the special complex Lie group $SL_2(\mathbb{C})$ illustrates. We show that the existence of a lightlike vector field $Z$ on $SL_2(\mathbb{C})$, implies geodesic completeness. When $Z$ is spacelike and orthogonal to $\sqrt{-1}Z$, we characterize complete metrics on $SL_2(\mathbb{C})$.
Keywords: (Semi)simple Lie group, left-invariant metric, Lorentzian metric, Killing vector field, left-invariant vector field, semisimple element, nilpotent element, compact element, strongly causal, dual Euler equation, generalized conical spiral, limit curve, fir
MSC: 53C22, 53C50, 57M50, 17B08, 22E30.
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