Structure of locally conformally flat manifolds satisfying some weakly-Einstein conditions

Jou nal o Geome y and Physics 186 (2023) 104754 Con en s lis s a ailable a ScienceDi ec Jou nal o Geome y and Physics jou nal homepage: www.else ie .com/loca e/geomphys S uc u e o locally con o mally fla mani olds sa is ying some weakly-Eins ein condi ions Rod igo Ma iño-Villa Facul y o Teache T aining, Uni e si y o San iago de Compos ela, 27002 Lugo, Spain a i c l e i n o a b s a c A icle his o y: Recei ed 9 July 2022 Accep ed 20 Janua y 2023 A ailable online 26 Janua y 2023 Keywo ds: C i ical me ic Eins ein and weakly-Eins ein me ics Locally con o mally fla Two-loop eno maliza ion flow I is gi en a comple e s udy o locally con o mally fla me ics sa is ying some weakly- Eins ein condi ions. I is shown ha hey a e ei he a p oduc Mn(c) ×Mn(−c)o a wa ped p oduc R × Rn−1 o some specific wa ping unc ion. Mo eo e , some condi ions on locally con o mally fla fixed poin s o he RG2 flow a e poin ed ou . ©2023 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license (h p://c ea i ecommons .o g /licenses /by /4 .0/). 1. In oduc ion Le (M, g)a n-dimensional Riemannian mani old and Ri s cu a u e enso gi en by R(X, Y) =[∇ X, ∇Y] −∇ [X,Y]. A mani old is called locally con o mally fla i o e e y poin in M, i exis s a neighbo hood o he poin such ha a fla space can be assigned o i ia a con o mal change. Mo eo e , i is a known ac ha a mani old is locally con o mally fla i and only i i s Weyl enso is anishing. In ha case, he cu a u e is de e mined by he Ricci enso , gi en by ρ(X, Y) := {Z→ R(Z, X)Y}. Thus, he cu a u e enso can be w i en as R(X,Y)Z=− τ (n−2)(n−1){g(Y,Z)X−g(X,Z)Y} +1 (n−2){ρ(Y,Z)X−ρ(X,Z)Y+g(Y,Z)QX−g(X,Z)QY},(1.1) whe e Qdeno e he Ricci ope a o , ρ(X, Y) =g(QX, Y)and τ= ρis he scala cu a u e. The e o e, he s udy o he di e en e ms coming om he cu a u e is a much simple ask. Besides, Be ge , in [2], showed he ollowing uni e sal iden i y in dimension ou . ˇ R−R2 4g+τρ−τ 4g−2ˇ ρ−ρ2 4g−2R[ρ]−ρ2 4g=0,(1.2) whe e ˇ Rij =Riabc Rabc j, ˇ ρij =ρiaρa jand R[ρ]ij =Riabjρab. Now, in he ligh o his iden i y, i we assume ha he me ic is Eins ein (i.e., ρ=τ ng), hen, all he b acke s in (1.2) anish, so he Eins ein condi ion o a me ic au oma ically sa isfies ha he o he h ee enso s a e a mul iple o he me ic. So now a na u al ques ion ha a ises is he con e se: I any o hese E-mail add ess: od ig[email p o ec ed]. h ps://doi.o g/10.1016/j.geomphys.2023.104754 0393-0440/©2023 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license (h p:// c ea i ecommons .o g /licenses /by /4 .0/). R. Ma iño-Villa Jou nal o Geome y and Physics 186 (2023) 104754 h ee enso s is a mul iple o he me ic, does a me ic ulfill he Eins ein condi ion? The e a e some coun e examples. In [6]i is shown a p oduc mani old o wo su aces wi h opposi e cu a u e M2(c) ×M2(−c), which sa isfies ha e e yone o he h ee p esen ed enso s a e a mul iple o he me ic whe eas i is no Eins ein. So i is a legi ima e ask looking o examples ha sa isfies hese condi ions bu he Eins ein one. We define he ollowing classes. Defini ion 1. A non–Eins ein Riemannian mani old is called: •ˇ R-Eins ein i ˇ R=||R||2 ng. •ˇ ρ-Eins ein i ˇ ρ=||ρ||2 ng. •R[ρ]-Eins ein i R[ρ] =||ρ||2 ng. Mo eo e , i a Riemannian mani old (M, g)( espec i ely, a me ic) sa isfies any o hese h ee condi ions, hen we will say ha he mani old ( he me ic) sa isfies a weakly-Eins ein condi ion. No ice ha we use a sligh ly di e en defini ion o weakly-Eins ein me ics. In [1] and [6], o example, he au ho s define hese condi ions as wha we call ˇ R-Eins ein. Eins ein me ics a e a main opic in di e en ial geome y and hey appea as c i ical me ics o he Hilbe unc ional, g→ τd olg, es ic ed o olume one me ics. Weakly-Eins ein me ics also appea s na u ally in he s udy o c i ical me ics o some specific unc ional, o ins ance, i we ake he unc ional F ,s:g−→ F ,s(g)= M {||ρ||2+ τ2+s||R||2}d g and compu e i s g adien [4], ∇F ,s=−(1+4s)ρ+(1+2 +2s)Hess(τ)−1+4 2τg −2 τρ−τ 4g−2sˇ R−R2 4g+4sˇ ρ−ρ2 4g −2(1+2s)R[ρ]−ρ2 4g, i in ol es all he enso s men ioned in he defini ion o he weakly-Eins ein classes. Mo eo e , ˇ R-Eins ein condi ion has seem o ecei e a en ion in o he fields In [1], A ias-Ma co and Kowalski classified ˇ R-Eins ein ou dimensional Lie g oups, and ollowing his wo k, a classifica- ion on he same field was gi en in [9], comple ing all he casuis ic. In [5], Chen s udy ˇ R-Eins ein almos con ac mani olds and in [3], he au ho s s udy his enso in he con ex o compac mani olds wi h bounda y. On he o he hand, ˇ R enso appea s in o he fields, such as in he s udy o he wo-loop eno maliza ion flow. The wo-loop eno maliza ion flow (o RG2flow) appea s as a pe u ba ion o he Ricci flow (see [11–13]) and i is gi en by ∂ ∂ g =RG[g],(1.3) whe e RG[g] =−2ρ−α 2ˇ Rand αis a posi i e coupling cons an . One can s udy genuine fixed poin s o (1.3), i.e., me ics sa is ying ρ+α 4ˇ R=0. In dimension wo he condi ion educes o cons an nega i e cu a u e. In dimension h ee, hey we e s udied by Gim e, Guen he and Isenbe g in [11], whe e hey showed solu ions wi h Ricci cu a u es Qρ=−2 diag[1 α, α, 0]o Qρ=−2 diag[2 α, α, 1 α]. Eins ein me ics a e genuine fixed poin s o his flow in dimension ou since i he Ricci enso is a mul iple o he me ic, he ˇ R enso is as well. In [10], i is gi en a classifica ion o genuine fixed poin s in ou dimen- sional Lie g oups. This flow has also been applied in he s udy o black holes me ics, analyzing how hey e ol ed along i and o he s udy o en opy, which has been s a ed as mono onic along his same flow. The eason o use his in such cases is ha he singula i ies appea ing in he s udy o o he flows disappea in RG2, being his a be e app oxima ion o highe cu a u e e ec s [14,15]. The main aim o his wo k is classi ying hese condi ions, bo h weakly-Eins ein and fixed poin s, in he field o locally con o mally fla mani olds. The ˇ R-Eins ein condi ion has been al eady s udied in [8], whe e he me ics sa is ying i we e classified as a p oduc Mn(c) ×Mn(−c)o a wa ped p oduc I× N(c)o a eal in e al and a mani old o cons an sec ional cu a u e cwi h some specific eal unc ion sol ing he di e en ial equa ion ( )2+ ( ) ( ) −c=0. The e o e, 2 R. Ma iño-Villa Jou nal o Geome y and Physics 186 (2023) 104754 we will ocus on he o he wo le , he ˇ ρ-Eins ein and he R[ρ]-Eins ein condi ions along sec ions 2and 3, comple ing he s udy and gi ing he whole classifica ion o weakly-Eins ein locally con o mally fla Riemannian mani olds and finding new examples o his so o mani olds, o which he e is a lack o hem along all he li e a u e. Du ing sec ion 4, we s udy fixed poin s o he RG2 flow, ob aining an algeb aic condi ion. Finally, in sec ion 5, we a e going o s udy he condi ion R[ρ]-Eins ein in a pa icula casuis ic in o de o y o gi e some ligh on i as i seems o be he one ha emains wi h no new examples. 2. R[ρ]-Eins ein condi ion R[ρ]-Eins ein condi ions seems o be oo much igid and hese fields p o ides none new examples o i . The esul o i s calcula ion is b ie ed in he ollowing s a emen . Theo em 2. A locally con o mally fla Riemannian mani old is R[ρ]-Eins ein i and only i M=Mn1(c) ×Mn2(−c)wi h n1=n2. P oo . Fi s o all, we a e going o compu e he R[ρ](1, 1)- enso , which is gi en by R[ρ](X, Y) =g(QR[ρ](X), Y). Using (1.1), a s aigh o wa d calcula ion shows ha QR[ρ]=− 2 (n−2)Q2+nτ (n−1)(n−2)Q+1 (n−2)||ρ||2−τ2 (n−1)(n−2)Id. Since we wan o see when his enso is a mul iply o he iden i y, we need ha QR[ρ]−||ρ||2 nId =0, o equi alen ly, −2 (n−2)Q2+nτ (n−1)(n−2)Q+2 n(n−2)||ρ||2−τ2 (n−1)(n−2)Id =0.(2.4) This equa ion needs o be sa isfied by e e y eigen alue o he enso , and since i is a quad a ic, hen we ha e wo a mos , bu i we ha e jus one, hen he mani old would be Eins ein, so assume ha we ha e eigen alues o he Ricci ope a o λand μwi h mul iplici ies mand n −m, espec i ely. Mo eo e , using he Vie a’s Fo mulae [7], we ob ain ha λ+μ=nτ 2(n−1).(2.5) Thus, as τ=mλ +(n −m)μ, bo h eigen alues a e ela ed by μ=2(n−1)−mn n(n−m)−2(n−1)λ. (2.6) Nex , on he one hand, le S=1 n−2(ρ−τ 2(n−1)g)be he Schou en enso , and on he o he hand, le us in oduce he ollowing echnical esul . Lemma 3. [16]Le Tbe a Codazzi enso . Le γbe an eigen unc ion o Twi h eigenspace Vγ. I dim Vγ≥2, hen ∇γis o hogonal o Vγ. Mo eo e , i Thas exac ly wo di e en eigen unc ions γand δwi h dim Vγ≤dim Vδ, hen (i) Mis locally a p oduc i dim Vγ≥2. (ii) Mis locally a wa ped p oduc wi h one-dimensional i and only i (ii.a) dim Vγ=1, (ii.b) he eigen unc ion δis no cons an and ∇γis o hogonal o Vδ. I is well known ha Sis Codazzi i he mani old is Locally con o mally fla and om (2.6)one can ob ain, h ough a s anda d calcula ion, ha he Schou en enso has wo di e en eigen alues (call hem ¯ λand ¯ μ), and hus, we can apply he lemma. I dim V¯ λ≥2, hen Mis a locally a p oduc by asse ion (i)and due o locally con o mally fla ness i can be ei he R ×N(c)o Mn1(c) ×Mn2(−c). The fi s case implies ha one o he eigen alues is ze o and, as hey a e a mul iple o each o he , hen bo h a e anishing, so Mis fla . Rega ding he second case, one can easily see ha a p oduc mani old Mn1(c1) ×Mn2(c2)is R[ρ]-Eins ein i and only i c2 1(n1−1)2=c2 2(n2−1)2, and since in his case c1=−c2, hen n1=n2. 3 R. Ma iño-Villa Jou nal o Geome y and Physics 186 (2023) 104754 I dim V¯ λ=1, hen dim V¯ μ=n −1, and so ∇¯ μis o hogonal o V¯ μ, bu , as ¯ λis a mul iple o ¯ μ, hen ∇¯ λis also o hogonal o V¯ μ. Besides, ¯ μcanno be cons an . O he wise, ¯ λwould be cons an as well, which would imply ha λand μwould be cons an . Hence, we would ha e a locally con o mally fla mani old wi h cons an Ricci cu a u es, which is cu a u e homogeneous, and by [18], i would be locally symme ic. Then Mwould spli as a p oduc o he o m R ×N(c), whose ac o s co espond o he Ricci cu a u es, so λwould be anishing and so μ, and he e o e, Mwould be fla . Thus, applying he p e ious lemma, we ha e a wa ped p oduc and due o locally con o mally fla ness, he fibe has o be o cons an sec ional cu a u e. Now, we wan o de e mine he wa ping unc ion. In o de o do ha , we use he ollowing esul . Lemma 4 ([17]). Le B × Fbe a wa ped p oduc wi h dim F=d. Le X, Y∈TpBand V, W∈TpF. Then •ρ(X, Y) =ρB(X, Y) −d Hess( )(X, Y). •ρ(X, V) =0. •ρ(V, W) =ρF(V, W) − +(d−1)g(g ad ,g ad ) 2g(V, W). In ou cu en si ua ion, we a e in a wa ped p oduc R × N(c), so he Ricci ope a o is w i en by Q(∂ )=−(n−1)  ∂ ,(2.7) Q(X)=(n−2)c 2−(n−2) 2 2−  X. Since he Ricci eigen alues a e ela ed by λ =(n −1)μ, we ob ain he di e en ial equa ion 2−c=0, which only ha e a sui able solu ion i c>0, and in ha case, i is linea , wha gi es Eins ein me ics. The e o e we canno ha e R[ρ]-Eins ein wa ped p oduc s in his field, which comple es he p oo .  3. ˇ ρ-Eins ein condi ion In sha p con as wi h he p e ious case, we can ge new examples o his me ics. We s a e he ollowing. Theo em 5. A locally con o mally fla Riemannian mani old is ˇ ρ-Eins ein i and only i M=Mn1(c) ×Mn2(−c)wi h n1=n2o a wa ped p oduc R × Rn−1wi h ( )=2(n−1)(a +b) nn 2(n−1) , wi h a, b ∈Rand ∈−b a,+∞. P oo . We p oceed in he same way. This ime, he equa ion desi e equa ion is Q2−||ρ||2 nId =0.(3.8) Consequen ly, we ha e wo eigen alues again and due o Vie a’s o mulae hey a e ela ed by μ =−λ. We shall use Lemma 3again. The e o e, i dim Vλ≥2 hen we ha e a p oduc Mn1(c) ×Mn2(−c)and he condi ion o a p oduc o his kind o be ˇ ρ-Eins ein is ha c2 1(n1−1)2=c2 2(n2−1)2, so n1=n2. I dim Vλ=1, hen we ha e a wa ped p oduc R × N(c), and as we know ha μ =−λ, using ((2.7)), we ob ain n  +(n−2) 2−(n−2)c=0. Now, aking he de i a i e o his equa ion one ob ains n  +(3n−4)   =0. 4 R. Ma iño-Villa Jou nal o Geome y and Physics 186 (2023) 104754 Since and  canno be ze o (o he wise he me ic is fla ), hen one can di ide by hese ac o s and hus (4−3n) n  =   . Nex , in eg a e bo h pa s o he equa ions and ge (4−3n) nln =ln  +K. Taking he exponen ial, he equa ions become (4−3n) n=eK , and now mul iply bo h sides by 2 , 2e−K  (4−3n) n=2  . Call he cons an pa ¯ K. We ha e s anda d in eg als on bo h pa s, so we ge ¯ Kn 4−2n 4−2n n= 2. Finally, isola ing , we ge =˜ K 2−n n, which solu ion is ( )=⎛ ⎝ 2(n−1)˜ K +a n⎞ ⎠ n 2(n−1) , whe e a ∈R. The e o e, we ob ain a solu ion o he second equa ion, which was he de i a i e o he one we go in fi s place. Now, i some unc ion is a solu ion o he fi s equa ions, i is a solu ion o i s de i a i e, and as we know he solu ions o his las one, he solu ion o he o iginal equa ions need o be o his o m. So i we pu his in he o iginal equa ions, we ge ha i is a solu ion o i i and only i c=0. The e o e, we a e in a wa ped p oduc o he o m R × Rn−1and we ha e no o he possibili y he e.  Rema k 6. No ice ha using hese echniques on he wa ped p oduc s, we shall gi e a simple p oo o he classifica ion o he ˇ R-Eins ein case gi en in [8]. F om he e, we ha e ha he ela ion be ween bo h eigen alues was gi en by μ=− 2m+(n−1)(n−4) 2(n−m)+(n−1)(n−4)λ, and i m =1, hen μ=− 2+(n−1)(n−4) 2(n−1)+(n−1)(n−4)λ. Using now (2.7), we ob ain he di e en ial equa ion 2+  −c=0, which is he one ha gi es ˇ R-Eins ein me ics. 4. Locally con o mally fla fixed poin s o he RG2-flow In his sec ion we classi y fixed poin s in he con ex o locally con o mally fla mani olds. Theo em 7. Le (M, g)be a n-dimensional locally con o mally fla fixed poin o he wo-loop eno maliza ion g oup flow wi h cou- pling cons an α. Then (1) I n = 4, hen (M, g)is homo he ic o a p oduc Mn1 1(c) ×Mn2 2(−c)wi h n1=n2o o a wa ped p oduc R × N(c)wi h non i ial wa ping unc ion sa is ying α(n−2)((n−6)n+6) 2−c+α((n−4)(n−2)n−4)  +2(n−2)2 2=0. 5 R. Ma iño-Villa Jou nal o Geome y and Physics 186 (2023) 104754 (2) I n =4, hen R2=ρ2=τ2 3. P oo . Le us ecall, on he one hand, ha fixed poin s o he RG2flow is gi en by a me ic ulfilling ρ+α 4ˇ R=0. On he o he hand, one can see om [8] ha Qˇ Rope a o is gi en by Qˇ R=2 (n−2)2(n−4)Q2+2τ (n−1)Q+(n−1)ρ2−τ2 (n−1)Id. Combining hese wo iden i ies, one can ge ha a me ic in his field is a fixed poin i Q+α 4Qˇ R=0, which is Q+α 2(n−2)2(n−4)Q2+2τ (n−1)Q+(n−1)ρ2−τ2 (n−1)Id=0, and hen, α(n−4) 2(n−2)Q2+ατ +(n−1)(n−2)2 (n−1)(n−2)2Q+α((n−1)ρ2−τ2) 2(n−2)2(n−1)Id =0.(4.9) Now we ha e wo di e en possibili ies depending on he dimension. I n = 4, hen we ha e a quad a ic equa ion on he Ricci ope a o , so we ha e wo Ricci eigen alues ela ed by λ+μ=−2(ατ +(n−1)(n−2)2) α(n−1)(n−2)(n−4). Thus, we ha e wo eigen alues, one a mul iple o he o he , and as he Schou en enso is Codazzi and i has also wo eigen alues, one a mul iple o he o he , hen we ha e ei he a wa ped p oduc R × N(c), wi h a non i ial eal wa ping unc ion and N(c)an (n −1)-dimensional Riemannian mani old o cons an cu a u e o a Riemannian p oduc Mn1 1(c) ×Mn2 2(−c), such ha n1=n2. In o de o de e mine he unc ion , assuming ha λhas mul iplici y one, hen bo h a e ela ed by λ+μ=2α(λ +μ(n−1)) +(n−1)(n−2)2 α(n−4)(n−2)(n−1), and using he o mulas om (2.7) o he Ricci ope a o , we ge ha mus sa is y he di e en ial equa ion α(n−2)((n−6)n+6) 2−c+α((n−4)(n−2)n−4)  +2(n−2)2 2=0 I n =4, hen equa ion (4.9) becomes 12(ατ +12)Q+α(3ρ2−τ2)Id =0. Since his is a lineal equa ion, his only can ha e one solu ion, and hen, he Ricci ope a o has only one eigen alue, so he me ic is Eins ein as long as he equa ion is no iden ically ze o. In o de o ha e ha , we need ha α=−12 τand ρ =τ2 3. Mo eo e , aking aces in ρ+α 4ˇ R=0, one can ob ain ha α=−4τR−2, hen R2=τ2 3, and hence R2=ρ2. No ice ha αcanno be anishing since, in ha case, he Ricci enso is as well.  5. A no e on R[ρ]-Eins ein condi ion Du ing hese sec ions we a e no going o assume ha he mani old is locally con o mally fla . Since his condi ion seems o be he mos igid one, we may hink o he ways o y o ob ain examples. We may hink o an easie casuis ic in o de o ge some sui able algeb aic condi ion. Fo ha , assume ha he Ricci ope a o has wo eigen alues, one simple, i.e., Qρ=diag[λ, μ, ..., μ]. In his si ua ion, as he ρij a e all anishing whene e i = j, hen he enso educes o R[ρ]ij = aRaijaρaa, ha ing a much simple casuis ic. Now we a e compu ing R[ρ]11. R[ρ]11 =R2112ρ22 +R3113ρ33 +···+Rn11nρnn =μ(R2112 +R3113 +···+Rn11n)=μρ11 =μλ Now, ecall ha ρ2=λ2+(n −1)μ2, so one o he equa ions ha needs o be sa isfies, in o de o ha e a mul iply o he iden i y, is λ2+(n−1)μ2−nλμ=0, 6 R. Ma iño-Villa Jou nal o Geome y and Physics 186 (2023) 104754 which, a e doing some simplifica ions, is equi alen o (λ −μ)(λ −(n−1)μ)=0. The e o e, since i λ = μ, we ob ain ha λ =(n −1)μ, which makes he componen R[ρ]11 =(n −1)μ2. The condi ion o R[ρ]being a mul iple o he me ic mus ulfill ha R[ρ]ii =R[ρ]jj =(n −1)μ2and R[ρ]ij =0, o all i, j ∈{1, ..., n}, i = j. Because o ha , we a e s udying he es o he componen s in h ee s eps. Fi s ly we a e analyzing R[ρ]1α, wi h α>1. R[ρ]1α=R21α2ρ22 +R31α3ρ33 +···+Rn1αnρnn =μ(R21α2+R31α3+···+Rn1αn)=μρ1α=0. Using he same compu a ions, we ob ain ha R[ρ]αα =μ((n −2)R1αα1+μ), wi h α>1, and R[ρ]αβ=μ(n −2)R1αβ1, wi h α= β, 1 <α<β. The e o e, om hese wo componen s, we ob ain he equa ions μ((n−2)R1αα1+μ)−μ2(n−1)=0 μ(n−2)R1αβ1=0 On he one hand, as μcanno be ze o, we ge ha R1αα1=μ, and since ραα =μ=R1αα1+R2αα2+···+Rnααn, we ge he condi ion n  i=2 Riααi=0. On he o he hand, we au oma ically ge ha R1αβ1=0. Thus, we ha e he ollowing esul . Lemma 8. Le (M, g)be an n-dimensional Riemannian mani old wi h wo di e en eigen alues o he Ricci ope a o , one o hem simple. Then, (M, g)is R[ρ]-Eins ein i and only i (i) Qρ=diag[(n −1)μ, μ, ..., μ]. (ii) i>1Riααi=0, wi h 1 <α≤n. (iii) R1αβ1=0 o all α, βsuch ha 1 <α<β≤n. Rema k 9. The sui able cu a u e enso could ha e di e en special condi ions depending on he dimension we a e wo king on. Fo ins ance, i n =4, hen ρ23 =0=R1231 +R4234 ρ24 =0=R1241 +R3243 ρ34 =0=R1341 +R2342. Using (iii) om he Lemma, shows ha R4234 =R3243 =R2342 =0. Mo eo e , (ii)gi e us he sys em o equa ions R3223 +R4224 =0 R2332 +R4334 =0 R2442 +R3443 =0, which only possible solu ion, due o he symme ies o he cu a u e enso , is ha e e y Rαββα, wi h 1 <α<β≤4is anishing. Howe e , i n =5, his las sys em becomes R3223 +R4224 +R5225 =0 R2332 +R4334 +R5335 =0 R2442 +R3443 +R5445 =0 R2552 +R3553 +R4554 =0, which does no imply ha e e y e m is anishing. 7 R. Ma iño-Villa Jou nal o Geome y and Physics 186 (2023) 104754 Rema k 10. In he con ex o locally con o mally fla me ics, (i)is sa isfied, bu his does no imply ha Mis locally con o mally fla . Fo example, i n =4, he e m W(e1, e2)o he Weyl enso is gi en by W(e1,e2)=⎛ ⎜ ⎜ ⎝ 00 0 0 00−R1223 −R1224 0R1223 0−R1234 0R1224 R1234 0 ⎞ ⎟ ⎟ ⎠ . In locally con o mally fla me ics wi h his kind o Ricci ope a o , he cu a u e componen s whe e we ha e h ee di e en indices a e ze o, whe eas in his case we do no need any special condi ion o hese componen s. Decla a ion o compe ing in e es The au ho s decla e ha hey ha e no known compe ing financial in e es s o pe sonal ela ionships ha could ha e appea ed o influence he wo k epo ed in his pape . Da a a ailabili y No da a was used o he esea ch desc ibed in he a icle. Re e ences [1] T. A ias-Ma co, O. Kowalski, Classifica ion o 4-dimensional homogeneous weakly Eins ein mani olds, Czechoslo . Ma h. J. 65 (2015) 21–59. [2] M. Be ge , Quelques o mules de a ia ion pou une s uc u e iemannienne, Ann. Sci. Éc. No m. Supé . 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