Journal of Lie Theory

no. 2
Geodesic Completeness of some Lorentzian Simple Lie Groups
Journal of Lie Theory, Volume 35 (2025) no. 2, pp. 239-261
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 $SL_2(\mathbb{C})$.
DOI: 10.5802/jolt.1383
Classification: 53C22, 53C50, 57M50, 17B08, 22E30
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 
@article{JOLT_2025_35_2_a1,
     author = {E. Ebrahimi and S. M. B. Kashani and M. J. Vanaei},
     title = {Geodesic {Completeness} of some {Lorentzian} {Simple} {Lie} {Groups}},
     journal = {Journal of Lie Theory},
     pages = {239--261},
     year = {2025},
     volume = {35},
     number = {2},
     doi = {10.5802/jolt.1383},
     zbl = {1570.53043},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1383/}
}
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E. Ebrahimi; S. M. B. Kashani; M. J. Vanaei. Geodesic Completeness of some Lorentzian Simple Lie Groups. Journal of Lie Theory, Volume 35 (2025) no. 2, pp. 239-261. doi: 10.5802/jolt.1383

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