We call a connected Lie group endowed with a left-invariant Lorentzian flat
metric Lorentzian flat Lie group. In this Note, we determine all Lorentzian
flat Lie groups admitting a timelike left-invariant Killing vector field. We
show that these Lie groups are 2-solvable and unimodular and hence geodesically
complete. Moreover, we show that a Lorentzian flat Lie group $(\mathrm{G},\mu)$
admits a timelike left-invariant Killing vector field if and only if
$\mathrm{G}$ admits a left-invariant Riemannian metric which has the same
Levi-Civita connection of $\mu$. Finally, we give an useful characterization of
left-invariant pseudo-Riemannian flat metrics on Lie groups $\mathrm{G}$
satisfying the property: for any couple of left invariant vector fields $X$ and
$Y$ their Lie bracket $[X,Y]$ is a linear combination of $X$ and $Y$

Lebzioui, Hicham

Extracta Mathematicae

English

arXiv

We call a connected Lie group endowed with a left-invariant Lorentzian at metric Lorentzian at Lie group. In this Note, we determine all Lorentzian at Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. Moreover, we show that a Lorentzian at Lie group (G,μ) admits a timelike left-invariant Killing vector field if and only if G admits a left-invariant Riemannian metric which has the same Levi-Civita connection of μ. Finally, we give an useful characterization of left-invariant pseudo-Riemannian at metrics on Lie groups G satisfying the property: for any couple of left invariant vector fields X and Y their Lie bracket [X, Y ] is a linear combination of X and Y.peerReviewe

Institutional Repository University of Extremadura

E extracta mathematicae Vol. 29, Num. 1 - 2, 159 { 166 (2014)
Lorentzian Flat Lie Groups Admitting a
Timelike Left-Invariant Killing Vector Field
Hicham Lebzioui
Faculte des Sciences de Meknes, Morocco
hlebzioui@gmail.com
Presented by Manuel de Leon Received January 23, 2014
Abstract : We call a connected Lie group endowed with a left-invariant Lorentzian at metric
Lorentzian at Lie group. In this Note, we determine all Lorentzian at Lie groups admitting
a timelike left-invariant Killing vector eld. We show that these Lie groups are 2-solvable
and unimodular and hence geodesically complete. Moreover, we show that a Lorentzian at
Lie group (G; ) admits a timelike left-invariant Killing vector eld if and only if G admits
a left-invariant Riemannian metric which has the same Levi-Civita connection of . Finally,
we give an useful characterization of left-invariant pseudo-Riemannian at metrics on Lie
groups G satisfying the property: for any couple of left invariant vector elds X and Y their
Lie bracket [X;Y ] is a linear combination of X and Y .
Key words: Lie groups, Lie algebras, Flat Lorentzian metric, Killing vector eld.
AMS Subject Class. (2010): Primary 53C50, 16T25; Secondary 53C20, 17B62.
1. Introduction and main results
A connected Lie group G together with a left-invariant pseudo-Riemannian
metric is called pseudo-Riemannian Lie group. When the metric is de-
nite positive or of signature ( ;+; : : : ;+) the group is called Riemannian
or Lorentzian. Let (G; ) be a pseudo-Riemannian Lie group and g its Lie
algebra endowed with the inner product h ; i = (e). The Levi-Civita con-
nection of (G; ) denes a product (u; v) 7! uv on g called Levi-Civita product
given by:
2huv;wi = h[u; v]; wi   h[v; w]; ui+ h[w; u]; vi : (1)
For any u 2 g, we denote by Lu and Ru respectively the left multiplication
and the right multiplication on g given by Lu(v) = uv and Ru(v) = vu. For
any u 2 g, Lu is skew-symmetric with respect to h ; i, and adu = Lu   Ru
where adu : g! g is given by adu(v) = [u; v].
The curvature of (g; h ; i) is given by K(u; v) = L[u;v]   [Lu;Lv]: If K
vanishes identically (G; ) is called pseudo-Riemannian at Lie group and
159
160 h. lebzioui
(g; h ; i) is called pseudo-Riemannian at Lie algebra. This is equivalent
to the fact that g endowed with the Levi-Civita product is a left symmetric
algebra, i.e., for any u; v; w 2 g,
ass(u; v; w) = ass(v; u; w) ;
where ass(u; v; w) = (uv)w   u(vw). This relation is equivalent to
Ruv   Rv  Ru = [Lu;Rv] ; (2)
for any u; v 2 g.
Let S(g) be the Lie subalgebra of g dened by
S(g) =

u 2 g : adu + adu = 0
	
;
where adu is the adjoint of adu with respect to h ; i. It is easy to see that a
left invariant vector eld X on G is a Killing vector eld i X(e) 2 S(g).
It is a well-known result that a pseudo-Riemannian at Lie group is geode-
sically complete if and only if it is unimodular [3]. In [5], J. Milnor character-
ized Riemannian at Lie groups. In [1, 2] there is a more precise version of
Milnor's Theorem: A Lie group is a Riemannian at Lie group if and only if
its Lie algebra g splits as an orthogonal direct sum: g = S(g)  [g; g], where
S(g) and [g; g] are abelian. Moreover, in this case the dimension of [g; g] is
even and the Levi-Civita product is given by:
La =
8<:ada if a 2 S(g) ;0 if a 2 [g; g] : (3)
In the Lorentzian case, the only known Lorentzian at Lie groups are those
nilpotent [3] and those with degenerate center [1]. Their Lie algebras are
determined by the double extension process, and the metric in both cases is
complete.
Recall that a vector eld X on a Lorentzian manifold (M; g) is called
timelike (resp. spacelike, null) if gp(Xp; Xp) < 0 (resp. gp(Xp; Xp) > 0,
gp(Xp; Xp) = 0) for any p 2 M . Many authors in mathematics and physics
are interested in the existence of timelike Killing vector eld in a Lorentzian
manifold. This condition implies interesting information on the structure of
the manifold. In General Relativity, if this condition is satised, then the
Lorentzian manifold is called stationary, and it is known that any compact
stationary manifold is geodesically complete [7].
lorentzian flat lie groups 161
Our rst and main purpose in this paper is to determine Lorentzian at Lie
groups admitting a timelike left-invariant killing vector eld. More precisely,
we will prove the following result.
Theorem 1.1. Let (G; ) be a Lorentzian Lie group. Then it is at and
carries a timelike left invariant killing vector eld if and only if its Lie algebra
g splits as an orthogonal direct sum g = S(g) [g; g], where [g; g] is abelian,
and S(g) is abelian and contains a timelike vector. Moreover, in this case the
dimension of [g; g] is even and the Levi-Civita product is given by (3).
From this Theorem, we deduce the following corollaries:
Corollary 1.2. A Lorentzian at Lie group which admits a timelike left-
invariant killing vector eld is 2-solvable and unimodular and hence geodesi-
cally complete.
Corollary 1.3. Let (G; ) be a Lorentzian at Lie group. The exis-
tence of timelike left-invariant killing vector eld on G is equivalent to the
existence of left-invariant Riemannian metric on G with the same Levi-Civita
connection of .
On the other hand, we dene the class C consisting of those Lie groups G
which are non commutative and their Lie algebras g satisfy: for all x and y in
g, [x; y] is a linear combination of x and y. This class C has been studied by
several authors. J.Milnor in [5] showed that every left-invariant Riemannian
metric on G 2 C has constant negative sectional curvature. K.Nomizu in [6]
showed that every left invariant Lorentzian metric on G 2 C has constant
sectional curvature which can be positive, negative or null.
The following theorem gives an useful characterization of at pseudo-
Riemannian left invariant metric on a Lie group belonging to C.
Theorem 1.4. Let G be a non commutative Lie group that belongs to C,
and  a left-invariant pseudo-Riemannian metric on G. Then  is at if and
only if the restriction of (e) to [g; g] is degenerate.
We deduce the following corollary.
Corollary 1.5. Let G in C. Then G is non unimodular and hence any
left-invariant pseudo-Riemannian metric  on G such that the restriction of
(e) to [g; g] is degenerate is geodesically incomplete.
162 h. lebzioui
2. Proofs of theorems
This section is devoted to the proofs of the results of the last section. The
following proposition appeared rst in [1, Lemma 3.1] will be useful later.
Proposition 2.1. Let (G; ) be a Riemannian or Lorentzian at Lie
group, then:
S(g) = (gg)? =

u 2 g : Ru = 0
	
:
In particular S(g) is abelian.
Proof. Let (G; ) be a pseudo-Riemannian at Lie group and (g; h ; i) its
pseudo-Riemannian at Lie algebra. We have obviously
(gg)? =

u 2 g : Ru = 0
	
:
Let u 2 S(g). By using (1) one can see easily that u:u = 0, and deduce from
(2) that [Ru;Lu] = R
2
u. Since Ru and Lu are skew-symmetric then [Ru;Lu] is
skew-symmetric. But R2u is symmetric, then R
2
u = 0.
If the metric is Riemannian then Ru = 0. If the metric is Lorentzian, then
Im(Ru) is a totally isotropic subspace and hence there exists an isotropic vec-
tor e 2 g and a covector  2 g such that Ru(v) = (v)e for any v 2 g. Choose
a basis fe; e; f1; : : : ; fn 2g of g such that spanfe; eg and spanff1; : : : ; fn 2g are
orthogonal, ff1; : : : ; fn 2g is orthonormal, e is isotropic and he; ei = 1. We
have, for any i = 1; : : : ; n  2,
hRu(e); ei = (e) =  he;Ru(e)i ;
hRu(e); ei = 0 = (e) ;
hRu(fi); ei = (fi) =  hfi;Ru(e)i ;
hence  = 0 and then Ru = 0.
Proof of Theorem 1.1. Let (g; h ; i) be a at Lorentzian Lie algebra.
Suppose that there exists u 2 S(g) = (gg)? such that hu; ui < 0. Then the
restriction of h ; i to u? is denite positive and so for S(g)? = gg  u?. Then
g = S(g) gg and the restriction of h ; i to S(g) is nondegenerate Lorentzian.
Let us show that h = gg is abelian and h = [g; g]. It is clear that h is a
Riemannian at Lie algebra and hence
h = S(h) [h; h] ;
lorentzian flat lie groups 163
where S(h) and [h; h] are abelian. Moreover, there exist 1; : : : ; p 2 S(h)nf0g
and orthonormal basis (e1; : : : ; e2p) of [h; h] such that for all i = 1; : : : ; p and
for all s 2 S(h):
ads(e2i 1) = i(s)e2i ;
ads(e2i) =  i(s)e2i 1 :
On the other hand, for any s 2 S(g), ads which is skew-symmetric leaves
h invariant and so it leaves [h; h] also invariant and hence S(h). Moreover, if
x 2 S(h) satises [x; s] = 0 for any s 2 S(g), then adx is skew-symmetric and
then x 2 S(g) and hence x = 0. So there exists also 1; : : : ; q 2 S(g) n f0g
and orthonormal basis (f1; : : : ; f2q) of S(h) such that for all i = 1; : : : ; q and
for all s 2 S(g)
ads(f2i 1) = i(s)f2i ;
ads(f2i) =  i(s)f2i 1 :
Suppose that h is non abelian. Without loss of generality we can suppose
that [f1; e1] = 1(f1)e2 6= 0. We have also [f1; e2] =  1(f1)e1 6= 0. Moreover,
there exists s 2 S(g) such that [s; f1] = 1(s)f2 6= 0. We have necessarily
[f2; e1] = 1(f2)e2 6= 0. Otherwise, if [f2; e1] = 0 then [f2; e2] = 0 and hence
ad[s;f2](e1) =  1(s)1(f1)e2 = [ads; adf2 ](e1)
= [[s; e1]; f2] =
2pX
i=3
iei :
This is impossible since 1(s)1(f1) 6= 0. On the other hand, we have
0 = [e1; [f1; s]] + [f1; [s; e1]] + [s; [e1; f1]]
=  1(s)[e1; f2] + [f1;
2pX
i=2
iei]  1(f1)[s; e2]
= 1(s)1(f2)e2 + 1e1 +
2pX
i=3
iei :
This is impossible and hence h is abelian.
Now it is obvious that [g; g]  h and [h; h]  [g; g]. On the other hand, for
any i = 1; : : : ; 2q, there exists s 2 S(g) such that
164 h. lebzioui
[s; f2i 1] = i(s)f2i 6= 0 ;
[s; f2i] =  i(s)f2i 1 6= 0 :
This shows that f2i 1; f2i 2 [g; g]. Conversely, if the Lie algebra g of the
Lorentzian Lie group (G; ) splits as an orthogonal direct sum g = S(g) 
[g; g] where S(g) and [g; g] are abelian, and there exists a timelike vector
in S(g) then the Levi-Civita product is given by (3) which implies that (G; )
is a Lorentzian at Lie group which admits a timelike left-invariant Killing
vector eld.
Proof of Corollary 1.3. Let (G; ) be a Lorentzian at Lie group and
(g; h ; i) its Lorentzian at Lie algebra. If (G; ) admits a timelike left-
invariant Killing vector eld, then according to Theorem 1.1, g = S(g) [g; g]
where S(g) is nondegenerate Lorentzian and [g; g] is nondegenerate Euclidean.
Choose any denite positive product h ; is on S(g) and dene h ; i0 on g as
the orthogonal sum of h ; is and the restriction of h ; i to [g; g]. It is easy to
check that the left invariant Riemannian metric on G associated to h ; i0 has
the same Levi-Civita connection as .
Conversely, suppose that G possesses a left-invariant Riemannian metric
0 with the same Levi-Civita connection of . Put h ; i = (e) and h ; i0 =
0(e). Since  and 0 have the same Levi-Civita connection then, according
to Proposition 2.1  and 0 have also the same space of left invariant Killing
vector elds identied to S(g). Then, by applying both [1, Theorem 3.1] and
Theorem 1.1, we get g = S(g)  [g; g] and this splitting is orthogonal with
respect to h ; i and h ; i0. If the restriction of h ; i to S(g) is Lorentzian then
 admits obviously a left invariant timelike Killing vector eld. Suppose now
that the restriction of h ; i to [g; g] is Lorentzian. Then there exists a basis
fe1; : : : ; epg of [g; g] which is orthogonal with respect to h ; i and orthonormal
with respect to h ; i0 and such that, hei; eii > 0 for i = 1; : : : ; p   1 and
hep; epi < 0. Let s 2 S(g). We have h[s; ep]; epi = 0, and
h[s; ep]; eii = h[s; ep]; eii0hei; eii
=  h[s; ei]; epi =  h[s; ei]; epi0hep; epi
= h[s; ep]; eii0hep; epi
for any i = 1; : : : ; p  1. Since hep; epi < 0 and hei; eii > 0 then h[s; ep]; eii = 0
lorentzian flat lie groups 165
and hence [s; ep] = 0 for any s 2 S(g) and hence ep lies in the center of
g and thus ep 2 S(g) which is impossible. This completes the proof of the
corollary.
Proof of Theorem 1.4. Let (G; ) be a pseudo-Riemannian Lie group and
(g; h ; i) its pseudo-Riemannian Lie algebra.
Milnor in [5] has shown that G 2 C if and only if g contains an abelian
ideal U of codimension 1 and an element b =2 U such that [b; x] = x for every
x 2 U . Note that U = [g; g].
If  is at, then since G is non unimodular, thus according to [1, Propo-
sition 3.1], we deduce that the restriction of h ; i to [g; g] is degenerate.
Conversely, if the restriction of h ; i to [g; g] is degenerate, then there exists
e 2 [g; g] such that he; xi = 0 for any x 2 [g; g]. Let y 2 g such that he; yi 6= 0.
We put:
d =
y
hy; ei  
1
2
hy; yi
hy; ei2 e ;
then hd; ei = 1 and hd; di = 0.
The restriction of h ; i to spanfe; dg is non degenerate, thus if we put
B = spanfe; dg? then we have
g = Re B  Rd ;
where [g; g] = Re B.
We have d = b + u0 where  2 R n f0g and u0 2 [g; g]. Then the only
non-vanishing brackets are [d; e] = e, [d; u] = u for any u 2 [g; g]: By using
(1), we deduce that the Levi-Civita product is given by:
Le = 0 ; de = e ; dd =  d ;
du = ue = 0 ; ud =  u ; uu0 = hu; u0ie
for any u; u0 2 B. From these relations we obtain
K(e; d) = K(e; u) = K(d; u) = K(u; u0) = 0 for any u; u0 2 B ;
which implies that the curvature K vanishes identically.
Acknowledgements
I would like to give my sincere thanks to Professor Mohamed Boucetta
for his helpful discussions and encouragements, and Professor Malika Ait
Ben Haddou for her support and guidance.
166 h. lebzioui
References
[1] M. Ait Ben Haddou, M. Boucetta, H. Lebzioui, Left-invariant
Lorentzian at metrics on Lie groups, J. Lie Theory 22 (2012), 269 { 289.
[2] A. Bahayou, M. Boucetta, Metacurvature of Riemannian Poisson-Lie
groups, J. Lie Theory 19 (2009), 439 { 462.
[3] A. Aubert, A. Medina, Groupes de Lie pseudo-riemanniens plats, Tohoku
Math. J. (2) 55 (2003), 487 { 506.
[4] A. Aubert, \ Structures Anes et Pseudo-metriques Invariantes a Gauche
sur des Groupes de Lie ", Thesis, Univesite Montpellier II, 1996.
[5] J. Milnor, Curvature of left invariant metrics on Lie Groups, Advances in
Math. 21 (1976), 293 { 329.
[6] K. Nomizu, Left-invariant lorentz metrics on Lie groups, Osaka J. Math. 16
(1979), 143 { 150.
[7] M. Sanchez, Lorentzian manifolds admitting a Killing vector eld, Nonlinear
Anal. 30 (1997), 643 { 654.
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Brace Jovanovich, Publishers], New York, 1983.


Lorentzian flat lie groups admitting a timelike left-invariant killing vector field

Abstract: We call a connected Lie group endowed with a left-invariant Lorentzian flat metric Lorentzian flat Lie group. In this Note, we determine all Lorentzian flat Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. Moreover, we show that a Lorentzian flat Lie group (G, µ) admits a timelike left-invariant Killing vector field if and only if G admits a left-invariant Riemannian metric which has the same Levi-Civita connection of µ. Finally, we give an useful characterization of left-invariant pseudo-Riemannian flat metrics on Lie groups G satisfying the property: for any couple of left invariant vector fields X and Y their Lie bracket [X, Y ] is a linear combination of X and Y 

Hicham Lebzioui

CiteSeerX

Lorentzian Flat Lie Groups Admitting a Timelike Left-Invariant Killing Vector Field

Dehesa. Repositorio Institucional de la Universidad de Extremadura

E extracta mathematicae Vol. 29, Núm. 1 - 2, 159 – 166 (2014)Lorentzian Flat Lie Groups Admitting aTimelike Left-Invariant Killing Vector FieldHicham LebziouiFaculté des Sciences de Meknès, Moroccohlebzioui@gmail.comPresented by Manuel de León Received January 23, 2014Abstract : We call a connected Lie group endowed with a left-invariant Lorentzian flat metricLorentzian flat Lie group. In this Note, we determine all Lorentzian flat Lie groups admittinga timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvableand unimodular and hence geodesically complete. Moreover, we show that a Lorentzian flatLie group (G, µ) admits a timelike left-invariant Killing vector field if and only if G admitsa left-invariant Riemannian metric which has the same Levi-Civita connection of µ. Finally,we give an useful characterization of left-invariant pseudo-Riemannian flat metrics on Liegroups G satisfying the property: for any couple of left invariant vector fields X and Y theirLie bracket [X,Y ] is a linear combination of X and Y .Key words: Lie groups, Lie algebras, Flat Lorentzian metric, Killing vector field.AMS Subject Class. (2010): Primary 53C50, 16T25; Secondary 53C20, 17B62.1. Introduction and main resultsA connected Lie group G together with a left-invariant pseudo-Riemannianmetric is called pseudo-Riemannian Lie group. When the metric is defi-nite positive or of signature (−,+, . . . ,+) the group is called Riemannianor Lorentzian. Let (G, µ) be a pseudo-Riemannian Lie group and g its Liealgebra endowed with the inner product ⟨ , ⟩ = µ(e). The Levi-Civita con-nection of (G, µ) defines a product (u, v) 7→ uv on g called Levi-Civita productgiven by:2⟨uv,w⟩ = ⟨[u, v], w⟩ − ⟨[v, w], u⟩+ ⟨[w, u], v⟩ . (1)For any u ∈ g, we denote by Lu and Ru respectively the left multiplicationand the right multiplication on g given by Lu(v) = uv and Ru(v) = vu. Forany u ∈ g, Lu is skew-symmetric with respect to ⟨ , ⟩, and adu = Lu − Ruwhere adu : g → g is given by adu(v) = [u, v].The curvature of (g, ⟨ , ⟩) is given by K(u, v) = L[u,v] − [Lu,Lv]. If Kvanishes identically (G, µ) is called pseudo-Riemannian flat Lie group and159160 h. lebzioui(g, ⟨ , ⟩) is called pseudo-Riemannian flat Lie algebra. This is equivalentto the fact that g endowed with the Levi-Civita product is a left symmetricalgebra, i.e., for any u, v, w ∈ g,ass(u, v, w) = ass(v, u, w) ,where ass(u, v, w) = (uv)w − u(vw). This relation is equivalent toRuv − Rv ◦ Ru = [Lu,Rv] , (2)for any u, v ∈ g.Let S(g) be the Lie subalgebra of g defined byS(g) ={u ∈ g : adu + ad∗u = 0},where ad∗u is the adjoint of adu with respect to ⟨ , ⟩. It is easy to see that aleft invariant vector field X on G is a Killing vector field iff X(e) ∈ S(g).It is a well-known result that a pseudo-Riemannian flat Lie group is geode-sically complete if and only if it is unimodular [3]. In [5], J. Milnor character-ized Riemannian flat Lie groups. In [1, 2] there is a more precise version ofMilnor’s Theorem: A Lie group is a Riemannian flat Lie group if and only ifits Lie algebra g splits as an orthogonal direct sum: g = S(g) ⊕ [g, g], whereS(g) and [g, g] are abelian. Moreover, in this case the dimension of [g, g] iseven and the Levi-Civita product is given by:La =ada if a ∈ S(g) ,0 if a ∈ [g, g] . (3)In the Lorentzian case, the only known Lorentzian flat Lie groups are thosenilpotent [3] and those with degenerate center [1]. Their Lie algebras aredetermined by the double extension process, and the metric in both cases iscomplete.Recall that a vector field X on a Lorentzian manifold (M, g) is calledtimelike (resp. spacelike, null) if gp(Xp, Xp) < 0 (resp. gp(Xp, Xp) > 0,gp(Xp, Xp) = 0) for any p ∈ M . Many authors in mathematics and physicsare interested in the existence of timelike Killing vector field in a Lorentzianmanifold. This condition implies interesting information on the structure ofthe manifold. In General Relativity, if this condition is satisfied, then theLorentzian manifold is called stationary, and it is known that any compactstationary manifold is geodesically complete [7].lorentzian flat lie groups 161Our first and main purpose in this paper is to determine Lorentzian flat Liegroups admitting a timelike left-invariant killing vector field. More precisely,we will prove the following result.Theorem 1.1. Let (G, µ) be a Lorentzian Lie group. Then it is flat andcarries a timelike left invariant killing vector field if and only if its Lie algebrag splits as an orthogonal direct sum g = S(g)⊕ [g, g], where [g, g] is abelian,and S(g) is abelian and contains a timelike vector. Moreover, in this case thedimension of [g, g] is even and the Levi-Civita product is given by (3).From this Theorem, we deduce the following corollaries:Corollary 1.2. A Lorentzian flat Lie group which admits a timelike left-invariant killing vector field is 2-solvable and unimodular and hence geodesi-cally complete.Corollary 1.3. Let (G, µ) be a Lorentzian flat Lie group. The exis-tence of timelike left-invariant killing vector field on G is equivalent to theexistence of left-invariant Riemannian metric on G with the same Levi-Civitaconnection of µ.On the other hand, we define the class C consisting of those Lie groups Gwhich are non commutative and their Lie algebras g satisfy: for all x and y ing, [x, y] is a linear combination of x and y. This class C has been studied byseveral authors. J.Milnor in [5] showed that every left-invariant Riemannianmetric on G ∈ C has constant negative sectional curvature. K.Nomizu in [6]showed that every left invariant Lorentzian metric on G ∈ C has constantsectional curvature which can be positive, negative or null.The following theorem gives an useful characterization of flat pseudo-Riemannian left invariant metric on a Lie group belonging to C.Theorem 1.4. Let G be a non commutative Lie group that belongs to C,and µ a left-invariant pseudo-Riemannian metric on G. Then µ is flat if andonly if the restriction of µ(e) to [g, g] is degenerate.We deduce the following corollary.Corollary 1.5. Let G in C. Then G is non unimodular and hence anyleft-invariant pseudo-Riemannian metric µ on G such that the restriction ofµ(e) to [g, g] is degenerate is geodesically incomplete.162 h. lebzioui2. Proofs of theoremsThis section is devoted to the proofs of the results of the last section. Thefollowing proposition appeared first in [1, Lemma 3.1] will be useful later.Proposition 2.1. Let (G, µ) be a Riemannian or Lorentzian flat Liegroup, then:S(g) = (gg)⊥ ={u ∈ g : Ru = 0}.In particular S(g) is abelian.Proof. Let (G, µ) be a pseudo-Riemannian flat Lie group and (g, ⟨ , ⟩) itspseudo-Riemannian flat Lie algebra. We have obviously(gg)⊥ ={u ∈ g : Ru = 0}.Let u ∈ S(g). By using (1) one can see easily that u.u = 0, and deduce from(2) that [Ru,Lu] = R2u. Since Ru and Lu are skew-symmetric then [Ru,Lu] isskew-symmetric. But R2u is symmetric, then R2u = 0.If the metric is Riemannian then Ru = 0. If the metric is Lorentzian, thenIm(Ru) is a totally isotropic subspace and hence there exists an isotropic vec-tor e ∈ g and a covector α ∈ g∗ such that Ru(v) = α(v)e for any v ∈ g. Choosea basis {e, ē, f1, . . . , fn−2} of g such that span{e, ē} and span{f1, . . . , fn−2} areorthogonal, {f1, . . . , fn−2} is orthonormal, ē is isotropic and ⟨e, ē⟩ = 1. Wehave, for any i = 1, . . . , n− 2,⟨Ru(e), ē⟩ = α(e) = −⟨e,Ru(ē)⟩ ,⟨Ru(ē), ē⟩ = 0 = α(ē) ,⟨Ru(fi), ē⟩ = α(fi) = −⟨fi,Ru(ē)⟩ ,hence α = 0 and then Ru = 0.Proof of Theorem 1.1. Let (g, ⟨ , ⟩) be a flat Lorentzian Lie algebra.Suppose that there exists u ∈ S(g) = (gg)⊥ such that ⟨u, u⟩ < 0. Then therestriction of ⟨ , ⟩ to u⊥ is definite positive and so for S(g)⊥ = gg ⊂ u⊥. Theng = S(g)⊕ gg and the restriction of ⟨ , ⟩ to S(g) is nondegenerate Lorentzian.Let us show that h = gg is abelian and h = [g, g]. It is clear that h is aRiemannian flat Lie algebra and henceh = S(h)⊕ [h, h] ,lorentzian flat lie groups 163where S(h) and [h, h] are abelian. Moreover, there exist λ1, . . . , λp ∈ S(h)∗\{0}and orthonormal basis (e1, . . . , e2p) of [h, h] such that for all i = 1, . . . , p andfor all s ∈ S(h):ads(e2i−1) = λi(s)e2i ,ads(e2i) = −λi(s)e2i−1 .On the other hand, for any s ∈ S(g), ads which is skew-symmetric leavesh invariant and so it leaves [h, h] also invariant and hence S(h). Moreover, ifx ∈ S(h) satisfies [x, s] = 0 for any s ∈ S(g), then adx is skew-symmetric andthen x ∈ S(g) and hence x = 0. So there exists also µ1, . . . , µq ∈ S(g)∗ \ {0}and orthonormal basis (f1, . . . , f2q) of S(h) such that for all i = 1, . . . , q andfor all s ∈ S(g)ads(f2i−1) = µi(s)f2i ,ads(f2i) = −µi(s)f2i−1 .Suppose that h is non abelian. Without loss of generality we can supposethat [f1, e1] = λ1(f1)e2 ̸= 0. We have also [f1, e2] = −λ1(f1)e1 ̸= 0. Moreover,there exists s ∈ S(g) such that [s, f1] = µ1(s)f2 ̸= 0. We have necessarily[f2, e1] = λ1(f2)e2 ̸= 0. Otherwise, if [f2, e1] = 0 then [f2, e2] = 0 and hencead[s,f2](e1) = −µ1(s)λ1(f1)e2 = [ads, adf2 ](e1)= [[s, e1], f2] =2p∑i=3αiei .This is impossible since µ1(s)λ1(f1) ̸= 0. On the other hand, we have0 = [e1, [f1, s]] + [f1, [s, e1]] + [s, [e1, f1]]= −µ1(s)[e1, f2] + [f1,2p∑i=2βiei]− λ1(f1)[s, e2]= µ1(s)λ1(f2)e2 + γ1e1 +2p∑i=3γiei .This is impossible and hence h is abelian.Now it is obvious that [g, g] ⊂ h and [h, h] ⊂ [g, g]. On the other hand, forany i = 1, . . . , 2q, there exists s ∈ S(g) such that164 h. lebzioui[s, f2i−1] = µi(s)f2i ̸= 0 ,[s, f2i] = −µi(s)f2i−1 ̸= 0 .This shows that f2i−1, f2i ∈ [g, g]. Conversely, if the Lie algebra g of theLorentzian Lie group (G, µ) splits as an orthogonal direct sum g = S(g) ⊕[g, g] where S(g) and [g, g] are abelian, and there exists a timelike vectorin S(g) then the Levi-Civita product is given by (3) which implies that (G, µ)is a Lorentzian flat Lie group which admits a timelike left-invariant Killingvector field.Proof of Corollary 1.3. Let (G, µ) be a Lorentzian flat Lie group and(g, ⟨ , ⟩) its Lorentzian flat Lie algebra. If (G, µ) admits a timelike left-invariant Killing vector field, then according to Theorem 1.1, g = S(g)⊕ [g, g]where S(g) is nondegenerate Lorentzian and [g, g] is nondegenerate Euclidean.Choose any definite positive product ⟨ , ⟩s on S(g) and define ⟨ , ⟩′ on g asthe orthogonal sum of ⟨ , ⟩s and the restriction of ⟨ , ⟩ to [g, g]. It is easy tocheck that the left invariant Riemannian metric on G associated to ⟨ , ⟩′ hasthe same Levi-Civita connection as µ.Conversely, suppose that G possesses a left-invariant Riemannian metricµ′ with the same Levi-Civita connection of µ. Put ⟨ , ⟩ = µ(e) and ⟨ , ⟩′ =µ′(e). Since µ and µ′ have the same Levi-Civita connection then, accordingto Proposition 2.1 µ and µ′ have also the same space of left invariant Killingvector fields identified to S(g). Then, by applying both [1, Theorem 3.1] andTheorem 1.1, we get g = S(g) ⊕ [g, g] and this splitting is orthogonal withrespect to ⟨ , ⟩ and ⟨ , ⟩′. If the restriction of ⟨ , ⟩ to S(g) is Lorentzian thenµ admits obviously a left invariant timelike Killing vector field. Suppose nowthat the restriction of ⟨ , ⟩ to [g, g] is Lorentzian. Then there exists a basis{e1, . . . , ep} of [g, g] which is orthogonal with respect to ⟨ , ⟩ and orthonormalwith respect to ⟨ , ⟩′ and such that, ⟨ei, ei⟩ > 0 for i = 1, . . . , p − 1 and⟨ep, ep⟩ < 0. Let s ∈ S(g). We have ⟨[s, ep], ep⟩ = 0, and⟨[s, ep], ei⟩ = ⟨[s, ep], ei⟩′⟨ei, ei⟩= −⟨[s, ei], ep⟩ = −⟨[s, ei], ep⟩′⟨ep, ep⟩= ⟨[s, ep], ei⟩′⟨ep, ep⟩for any i = 1, . . . , p− 1. Since ⟨ep, ep⟩ < 0 and ⟨ei, ei⟩ > 0 then ⟨[s, ep], ei⟩ = 0lorentzian flat lie groups 165and hence [s, ep] = 0 for any s ∈ S(g) and hence ep lies in the center ofg and thus ep ∈ S(g) which is impossible. This completes the proof of thecorollary.Proof of Theorem 1.4. Let (G, µ) be a pseudo-Riemannian Lie group and(g, ⟨ , ⟩) its pseudo-Riemannian Lie algebra.Milnor in [5] has shown that G ∈ C if and only if g contains an abelianideal U of codimension 1 and an element b /∈ U such that [b, x] = x for everyx ∈ U . Note that U = [g, g].If µ is flat, then since G is non unimodular, thus according to [1, Propo-sition 3.1], we deduce that the restriction of ⟨ , ⟩ to [g, g] is degenerate.Conversely, if the restriction of ⟨ , ⟩ to [g, g] is degenerate, then there existse ∈ [g, g] such that ⟨e, x⟩ = 0 for any x ∈ [g, g]. Let y ∈ g such that ⟨e, y⟩ ̸= 0.We put:d =y⟨y, e⟩− 12⟨y, y⟩⟨y, e⟩2e ,then ⟨d, e⟩ = 1 and ⟨d, d⟩ = 0.The restriction of ⟨ , ⟩ to span{e, d} is non degenerate, thus if we putB = span{e, d}⊥ then we haveg = Re⊕ B ⊕ Rd ,where [g, g] = Re⊕ B.We have d = αb + u0 where α ∈ R \ {0} and u0 ∈ [g, g]. Then the onlynon-vanishing brackets are [d, e] = αe, [d, u] = αu for any u ∈ [g, g]. By using(1), we deduce that the Levi-Civita product is given by:Le = 0 , de = αe , dd = −αd ,du = ue = 0 , ud = −αu , uu′ = α⟨u, u′⟩efor any u, u′ ∈ B. From these relations we obtainK(e, d) = K(e, u) = K(d, u) = K(u, u′) = 0 for any u, u′ ∈ B ,which implies that the curvature K vanishes identically.AcknowledgementsI would like to give my sincere thanks to Professor Mohamed Boucettafor his helpful discussions and encouragements, and Professor Malika AitBen Haddou for her support and guidance.166 h. lebziouiReferences[1] M. Ait Ben Haddou, M. Boucetta, H. Lebzioui, Left-invariantLorentzian flat metrics on Lie groups, J. Lie Theory 22 (2012), 269 – 289.[2] A. Bahayou, M. Boucetta, Metacurvature of Riemannian Poisson-Liegroups, J. Lie Theory 19 (2009), 439 – 462.[3] A. Aubert, A. Medina, Groupes de Lie pseudo-riemanniens plats, TohokuMath. J. (2) 55 (2003), 487 – 506.[4] A. Aubert, “ Structures Affines et Pseudo-métriques Invariantes à Gauchesur des Groupes de Lie ”, Thesis, Univesité Montpellier II, 1996.[5] J. Milnor, Curvature of left invariant metrics on Lie Groups, Advances inMath. 21 (1976), 293 – 329.[6] K. Nomizu, Left-invariant lorentz metrics on Lie groups, Osaka J. Math. 16(1979), 143 – 150.[7] M. Sánchez, Lorentzian manifolds admitting a Killing vector field, NonlinearAnal. 30 (1997), 643 – 654.[8] B. O’Neill, “ Semi-Riemannian Geometry ”, Academic Press, Inc. [HarcourtBrace Jovanovich, Publishers], New York, 1983.

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Lorentzian Flat Lie Groups Admitting a Timelike Left-Invariant Killing Vector Field

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