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On the nonexistence of closed timelike geodesics in flat Lorentz 2-step nilmanifolds

Author: Mohammed Guediri
Journal: Trans. Amer. Math. Soc. 355 (2003), 775-786
MSC (2000): Primary 53C22, 53C50; Secondary 53B30
Published electronically: October 1, 2002
MathSciNet review: 1932725
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Abstract: The main purpose of this paper is to prove that there are no closed timelike geodesics in a (compact or noncompact) flat Lorentz 2-step nilmanifold $ N/\Gamma ,$ where $N$ is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and $\Gamma $ a discrete subgroup of $ N $ acting on $N$ by left translations. For this purpose, we shall first show that if $N$ is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric $g,$ then the restriction of $g$ to the center $Z$of $N$ is degenerate. We shall then determine all 2-step nilpotent Lie groups that can admit a flat left-invariant Lorentz metric. We show that they are trivial central extensions of the three-dimensional Heisenberg Lie group $H_{3}$. If $\left( N,g\right) $ is one such group, we prove that no timelike geodesic in $\left( N,g\right) $ can be translated by an element of $N.$ By the way, we rediscover that the Heisenberg Lie group $H_{2k+1}$admits a flat left-invariant Lorentz metric if and only if $k=1.$

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Additional Information

Mohammed Guediri
Affiliation: Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Keywords: 2-step nilpotent Lie groups, left-invariant Lorentz metrics, closed timelike geodesics
Received by editor(s): July 6, 2001
Received by editor(s) in revised form: June 5, 2002
Published electronically: October 1, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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