Jou nal o Geome y and Physics 186 (2023) 104754
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Jou nal o Geome y and Physics
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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,
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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.
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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.
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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,
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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.
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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.
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