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 $SL_2(\mathbb{C})$.
Esmail Ebrahimi
1
;
Seyed M. B. Kashani
1
;
Mohammad J. Vanaei
1
1
Dept. of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran
Esmail Ebrahimi; Seyed M. B. Kashani; Mohammad J. Vanaei. Geodesic Completeness of some Lorentzian Simple Lie Groups. Journal of Lie Theory, Tome 35 (2025) no. 2, pp. 239-261. http://geodesic.mathdoc.fr/item/JOLT_2025_35_2_a1/
@article{JOLT_2025_35_2_a1,
author = {Esmail Ebrahimi and Seyed M. B. Kashani and Mohammad 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},
url = {http://geodesic.mathdoc.fr/item/JOLT_2025_35_2_a1/}
}
TY - JOUR
AU - Esmail Ebrahimi
AU - Seyed M. B. Kashani
AU - Mohammad J. Vanaei
TI - Geodesic Completeness of some Lorentzian Simple Lie Groups
JO - Journal of Lie Theory
PY - 2025
SP - 239
EP - 261
VL - 35
IS - 2
UR - http://geodesic.mathdoc.fr/item/JOLT_2025_35_2_a1/
ID - JOLT_2025_35_2_a1
ER -
%0 Journal Article
%A Esmail Ebrahimi
%A Seyed M. B. Kashani
%A Mohammad J. Vanaei
%T Geodesic Completeness of some Lorentzian Simple Lie Groups
%J Journal of Lie Theory
%D 2025
%P 239-261
%V 35
%N 2
%U http://geodesic.mathdoc.fr/item/JOLT_2025_35_2_a1/
%F JOLT_2025_35_2_a1
