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

Guediri, Mohammed

doi:10.1090/s0002-9947-02-03114-8

Cited by 26 publications

(20 citation statements)

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“…Let (g, , ) be a flat pseudo-Euclidean 2-step nilpotent Lie algebra. In [7], the author showed that if the metric , is Lorentzian, then g is an extension trivial of H 3 , where H 3 is a 3-dimensional Heisenberg Lie algebra. Let us studies some properties of (g, , ) in other signatures.…”

Section: Flat Pseudo-euclidean 2-step Nilpotent Lie Algebrasmentioning

confidence: 99%

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On flat pseudo-Euclidean nilpotent Lie algebras

Boucetta

Lebzioui

2019

Journal of Algebra

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A flat pseudo-Euclidean Lie algebra is a real Lie algebra with a non degenerate symmetric bilinear form and a left symmetric product whose the commutator is the Lie bracket and such that the left multiplications are skew-symmetric. We show that the center of a flat pseudo-Euclidean nilpotent Lie algebra of signature (2, n − 2) must be degenerate and all flat pseudo-Euclidean nilpotent Lie algebras of signature (2, n − 2) can be obtained by using the double extension process from flat Lorentzian nilpotent Lie algebras. We show also that the center of a flat pseudo-Euclidean 2-step nilpotent Lie algebra is degenerate and all these Lie algebras are obtained by using a sequence of double extension from an abelian Lie algebra. In particular, we determine all flat pseudo-Euclidean 2-step nilpotent Lie algebras of signature (2, n − 2). The paper contains also some examples in low dimension.2. In [2], Aubert and Medina showed that all flat Lorentzian nilpotent Lie algebras are obtained by the double extension process from Euclidean abelian Lie algebras. 3. Guédiri showed in [7] that a flat Lorentzian 2-step nilpotent Lie algebra is a trivial extension of the 3-dimensional Heisenberg Lie algebra H 3 . 4. In [3,4], the authors showed that flat Lorentzian Lie algebras with degenerate center or flat nonunimodular Lorentzian Lie algebras can be obtained by the double extension process from flat Euclidean Lie algebras.The study of flat pseudo-Euclidean Lie algebras of signature other than (0, n) and (1, n − 1) is an open problem. In this paper, we tackle a part of this problem, namely, we study flat pseudo-Euclidean nilpotent Lie algebras of signature (2, n −2) and flat pseudo-Euclidean 2-step nilpotent Lie algebras of any signature. There are our main results: 1. In Theorem 3.1, we show that the center of a flat pseudo-Euclidean nilpotent Lie algebra of signature (2, n − 2) must be degenerate. From this theorem and Theorem 4.1 we deduce that all flat pseudo-Euclidean nilpotent Lie algebra of signature (2, n − 2) are obtained by the double extension process. 2. We give some general properties of flat pseudo-Euclidean 2-step nilpotent Lie algebras and we show that their center is degenerate. we show also that we can construct all this Lie algebras by applying a sequence of double extension starting from a pseudo-Euclidean abelian Lie algebra. 3. We give all 2-step nilpotent Lie algebras which can admit flat pseudo-Euclidean metrics of signature (2, n − 2) (Theorem 6.1 and Theorem 6.2). We will see that a class of 2step nilpotent Lie algebras which can admit a flat pseudo-Euclidean metrics of signature (2, n − 2) is very rich, contrary to the Euclidean and the Lorentzian cases. As example, we show that any 6-dimensional 2-step nilpotent Lie algebra which is not an extension trivial of a 5-dimensional Heisenberg Lie algebra, admits such metric.The paper is organized as follows. In section 2, we give some generalities on flat pseudo-Euclidean Lie algebras. In section 3 and section 4, we study flat pseudo-Euclidean metrics of signa...

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Section: Flat Pseudo-euclidean 2-step Nilpotent Lie Algebrasmentioning

confidence: 99%

“…3. Guédiri showed in [7] that a flat Lorentzian 2-step nilpotent Lie algebra is a trivial extension of the 3-dimensional Heisenberg Lie algebra H 3 . 4.…”

Section: Introductionmentioning

confidence: 99%

On flat pseudo-Euclidean nilpotent Lie algebras

Boucetta

Lebzioui

2019

Journal of Algebra

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“…More recently, Guediri has proved that compact flat spacetimes contain a causal periodic geodesic [14], and that such spacetimes contain a periodic timelike geodesic if and only if the fundamental group of the underlying manifold contains a nontrivial timelike translation [15]. Non existence results for periodic causal geodesics are also available, see [12,16,17].…”

Section: Introductionmentioning

confidence: 99%

Periodic geodesics and geometry of compact Lorentzian manifolds with a Killing vector field

2009

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ABSTRACT. We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is never vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a never vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.

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“…The existence of closed timelike geodesic has been established also by Galloway in [14], where the author proves the existence of a longest closed timelike curve, which is necessarily a geodesic, in each stable free timelike homotopy class. Also non existence results for nonspacelike geodesics are available, see [15,29].…”

Section: Introductionmentioning

confidence: 99%