Einstein Lorentzian solvable unimodular Lie groups

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arXiv:2210.15717v1 [math.DG] 27 Oct 2022

Oumaima Tibssirte

aUniversit´e Priv´ee de Marrakech

Km 13, Route d’Amizmiz 42312 - Marrakech - Maroc

e-mail: o.tibssirte@upm.ac.ma

Abstract

The goal of this paper is to show that many key results found in the study of Einstein Lorentzian

nilpotent Lie algebras can still hold in the more general settings of unimodular Lie algebras and

(completely) solvable Lie algebras.

Keywords: Einstein Lorentzian manifolds, Nilpotent Lie groups, Nilpotent Lie algebras,

Unimodular Lie algebras, Solvable Lie algebras, Double extension process

2000 MSC: 53C50,

2000 MSC: 53D15,

2000 MSC: 53B25

1. Introduction

Left-invariant pseudo-Riemannian Einstein metrics on nilpotent Lie groups is a research area that

has known a signiﬁcant progress, especially in the last decade. While the celebrated theorem of

Milnor (see [16]) has among its consequences that in the Riemannian setting no such metrics

could exist, except when the underlying Lie group is abelian, the indeﬁnite case is an entirely

diﬀerent story with a handful of examples available in the literature (see [19]) and a system-

atic study of the Lorentzian case (see [6], [7] and [10]). Although, for the time being, there is

still no decisive theorem that settles the debate concerning the exact conditions for such metrics

to exist, the abundance of results on this particular subject is more apparent now than ever before.

In contrast, there is little to no information when it comes to the investigation of left-invariant Ein-

stein pseudo-Riemannian, indeﬁnite metrics, on general Lie groups, and a line of study that could

potentially lead to a similar development as in the nilpotent case is yet to be found, furethermore

it is not known whether the results obtained for nilpotent Lie groups could hold under weaker

conditions or admit a useful generalization, even solvable Lie groups which represent in some

sense a natural extension of the nilpotent scene, remain a question mark.

The purpose of this paper is therefore to initiate such an inspection by expanding the context of

the main theorems provided by the study of Einstein left-invariant Lorentzian metrics on nilpo-

tent Lie groups to more general situations with an emphasis on the solvable case, and at the same

time to shed light on the limitations of these results. The main reference for this paper is [6], and

an analogy will be drawn between the diﬀerent situations throughout the paper.

Paper Outline. In section 2, we introduce some general preliminaries on pseudo-Euclidean

Preprint submitted to Elsevier October 31, 2022

vector spaces and Lie algebras as well as the notations that will be used in the remainder of the

paper, the proof of all the listed preliminary results can be found in either [6] or [7]. We also

recall the deﬁnition of a solvable Lie algebra and we give a brief account on the Killing form of a

completely solvable lie algebra which shall be useful in the proofs of the main theorems. We start

section 2 with a generalization of Theorem 7 in [19] to the case of an arbitrary Einstein pseudo-

Euclidean, unimodular Lie algebra (see Proposition 3.1), the proof is based on Lemma 3.1 and is

similar in spirit to the one found in [6, Proposition 3.7]. Next we generalize the Lorentzian ver-

sion of [6, Proposition 3.1] to the unimodular solvable setting and we observe that the proof can

be further generalized to include any arbitray Einstein pseudo-Euclidean Lie algebra with non-

degenerate center (see Proposition 3.2). We dedicate section 4 to establish a variant of the double

extension process introduced by Medina-Revoy in [15] (also studied in the nilpotent setting in

[6]), we then give necessary and suﬃcient conditions on the parameters of double extensions that

results in Einstein Lorentzian Lie algebras (see Proposition 4.1 and 4.2), we then prove Theorems

4.1 and 4.2 which are broader versions of Theorem [6, Theorem 4.1] and shows that essentially

the same result holds more generally for completely solvable unimodular Lie algebras.

2. Preliminaries

Apseudo-Euclidean vector space is a real vector space of ﬁnite dimension nendowed with a

nondegenerate symmetric inner product of signature (q,n−q)=(−...−,+...+). When the

signature is (0,n) (resp. (1,n−1)) the space is called Euclidean (resp. Lorentzian).

Let (V,h,i) be a pseudo-Euclidean vector space of signature (q,n−q). A vector u∈Vis called

spacelike if hu,ui>0, timelike if hu,ui<0 and isotropic if hu,ui=0. A family (u1,...,us)

of vectors in Vis called orthogonal if, for i,j=1,...,sand i,j,hui,uji=0. An orthonor-

mal basis of Vis an orthogonal basis (e1,...,en) such that hei,eii=±1. A pseudo-Euclidean

basis of Vis a basis (e1,¯e2,...,eq,¯eq,f1,..., fn−2q) for which the non vanishing products are

h¯ei,eii=hfj,fji=1, i∈ {1,...,q}and j∈ {1,...,n−2q}. When Vis Lorentzian, we call such a

basis Lorentzian. Pseudo-Euclidean basis always exist.

For any endomorphism F:V−→ V, we denote by F∗:V−→ Vits adjoint with respect to h,i.

We shall make use of the following Lemmas whose proofs can be found in [6] :

Lemma 2.1. Let (V,h,i)be a Lorentzian vector space, e an isotropic vector and A a skew-

symmetric endomorphism. Then hAe,Aei ≥ 0. Moreover, hAe,Aei=0if and only if Ae =αe

with α∈R.

Lemma 2.2. Let (V,h,i)be a Lorentzian vector space, e an isotropic vector and A a skew-

symmetric endomorphism such that A(e)=0. Then:

1. tr(A2)≤0,

2. tr(A2)=0if and only if for any x ∈e⊥, A(x)=λ(x)e and in this case tr(A◦B)=0for any

skew-symmetric endomorphism satisfying B(e)=0.

A Lie group Gtogether with a left invariant pseudo-Riemannian metric gis called a pseudo-

Riemannian Lie group. The metric gdeﬁnes a symmetric nondegenerate inner product h,ion

the Lie algebra g=TeGof G, and conversely, any nondegenerate symmetric inner product on g

gives rise to an unique left invariant pseudo-Riemannian metric on G.

We will refer to a Lie algebra endowed with a nondegenerate symmetric inner product as a

pseudo-Euclidean Lie algebra.

Levi-Civita connection of (G,g) deﬁnes a product L : g×g−→ gcalled Levi-Civita product

given by Koszul’s formula

2hLuv,wi=h[u,v],wi+h[w,u],vi+h[w,v],ui.(1)

For any u,v∈g, Lu:g−→ gis skew-symmetric and [u,v]=Luv−Lvu. The curvature on gis

given by

K(u,v)=L[u,v]−[Lu,Lv].

The Ricci curvature ric : g×g−→ Rand its Ricci operator Ric : g−→ gare deﬁned by

hRic(u),vi=ric(u,v)=tr (w−→ K(u,w)v). A pseudo-Euclidean Lie algebra is called ﬂat (resp.

Ricci-ﬂat) if K=0 (resp. ric =0). It is called Einstein if there exists a constant λ∈Rsuch that

Ric =λIdg. For any u∈g, put Ru=Lu−adu. It is well-known that ric is given by

ric(u,v)=−tr(Ru◦Rv)−1

2(hadHu,vi+hadHv,ui),(2)

where His the vector given by hH,ui=tr(adu). Let us transform this formula to get some

useful formulas. To do so, we consider ad : g−→ End(g) and J:g−→ so(g,h,i) the

adjoint representation and the endomorphism given by Ju(v)=ad∗

vu. It is clear that Juis skew-

symmetric, ker ad =Z(g) and ker J=[g,g]⊥. One can deduce easily from (1) that

Ru=−1

2adu+ad∗

u−1

2Ju.

Thus

ric(u,v)=−1

2tr(adu◦adv)−1

2tr(adu◦ad∗

v)−1

4tr(Ju◦Jv)−1

2hadHu,vi − 1

2hadHv,ui.(3)

Deﬁne now the auto-adjoint endomorphisms b

B,J1and J2by

Bu,vi=tr(adu◦adv),hJ1u,vi=tr(adu◦ad∗

v),hJ2u,vi=−tr(Ju◦Jv)=tr(Ju◦J∗

v).

Thus (3) is equivalent to

Ric =−1

2b

B+J1+1

4J2−1

2(adH+ad∗

H).(4)

When gis a unimodular Lie algebra then H=0 and hence (4) becomes

Ric =−1

2(b

B+J1)+1

4J2,(5)

Since we will deal with nilpotent Lie algebras, only J1and J2will be relevant so we are going

to express them in an useful way. This is based on the notion of structure endomorphisms we

introduce now.

Let (e1,...,ep) be a basis of g. Then, for any u,v∈g, the Lie bracket can be written

[u,v]=

i=1

hSiu,viei,(6)

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摘要：

arXiv:2210.15717v1[math.DG]27Oct2022EinsteinLorentziansolvableunimodularLiegroupsOumaimaTibssirteaUniversit´ePriv´eedeMarrakechKm13,Routed’Amizmiz42312-Marrakech-Maroce-mail:o.tibssirte@upm.ac.maAbstractThegoalofthispaperistoshowthatmanykeyresultsfoundinthestudyofEinsteinLorentziannilpotentLiealgebr...

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