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

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Journal of Geometry and Physics

journalhomepage:www.elsevier.com/locate/geomphys

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

Rodrigo Mariño-Villar

FacultyofTeacherTraining,UniversityofSantiagodeCompostela,27002Lugo,Spain

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received9July2022 Accepted20January2023 Availableonline26January2023

Keywords:

Criticalmetric

Einsteinandweakly-Einsteinmetrics Locallyconformallyflat

Two-looprenormalizationflow

It is givenacomplete study oflocallyconformallyflat metrics satisfying someweakly- Einsteinconditions.ItisshownthattheyareeitheraproductMⁿ(c)×^Mn(−^c)orawarped product R×^f Rⁿ⁻¹ for somespecific warping function. Moreover, someconditions on locallyconformallyflatfixedpointsfortheRG2flowarepointedout.

©^{2023TheAuthor(s).PublishedbyElsevierB.V.Thisisanopenaccessarticleunderthe} CCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Let

(

M

,

g

)

an-dimensional Riemannian manifold and R its curvaturetensor givenby R

(

X

,

Y

) = [∇

X

, ∇

Y

] − ∇

[X,Y]. A manifold is calledlocally conformally flat ifforevery point in M, it exists a neighborhood ofthe point such that a flat space can be assigned to itvia a conformal change.Moreover, it is a known fact that a manifold islocally conformally flat if andonly if its Weyl tensor is vanishing. In that case, the curvature is determined by the Ricci tensor, given by

ρ (

X

,

Y

) :=

^tr

{

^Z

→

^R

(

Z

,

X

)

Y

}

^{.Thus,thecurvaturetensorcanbewrittenas} 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

)

Q X

−

^g

(

X

,

Z

)

Q Y

},

^(1.1)

where Q denote theRiccioperator,

ρ (

X

,

Y

) =

^g

(

Q X

,

Y

)

and

τ =

^tr

ρ

isthescalar curvature.Therefore,thestudy of the differenttermscomingfromthecurvatureisamuchsimplertask.

Besides,Berger,in[2],showedthefollowinguniversalidentityindimensionfour.



R

ˇ − 

^R



² 4 g



+ τ ^ ρ ₋ τ

4g

 −

²

 ˇ ρ ₋ ^ ρ 

²

4 g



−

²



R

[ ρ _{] −} ^ ρ 

² 4 g



=

⁰

,

(1.2)

where R

ˇ

i j

=

^RiabcR^abc_j ,

ρ ˇ

i j

= ρ

ia

ρ

^a_j andR

[ ρ _]

i j

=

^Riabj

ρ

^ab.Now,inthelight ofthisidentity,ifwe assumethatthemetricis Einstein(i.e.,

ρ = τ

ng),then,allthebracketsin(1.2) vanish,sotheEinsteinconditionforametricautomaticallysatisfiesthat theother threetensors areamultipleofthemetric.Sonowanaturalquestion thatarisesistheconverse:Ifanyofthese

E-mailaddress:[email protected].

https://doi.org/10.1016/j.geomphys.2023.104754

0393-0440/©^{2023TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://}

creativecommons.org/licenses/by/4.0/).

threetensors isamultipleofthemetric,doesametricfulfill theEinsteincondition?Thereare somecounterexamples. In [6] itisshownaproductmanifold oftwosurfaceswithoppositecurvature M²

(

c

) ×

^M2

(−

^c

)

,whichsatisfiesthateveryone ofthethreepresentedtensorsareamultipleofthemetricwhereas itisnotEinstein.Soitisalegitimatetasklookingfor examplesthatsatisfiestheseconditionsbuttheEinsteinone.Wedefinethefollowingclasses.

Definition1.Anon–EinsteinRiemannianmanifoldiscalled:

• ˇ

^R-EinsteinifR

ˇ = ||

^R

||

² n g.

• ˇ ρ

-Einsteinif

ρ ˇ = || ρ ||

² n g.

•

^R

[ ρ ]

^{-Einsteinif R}

[ ρ ] = || ρ ||

² n g.

Moreover,ifaRiemannianmanifold

(

M

,

g

)

(respectively,ametric)satisfiesanyofthesethreeconditions,thenwewillsay thatthemanifold(themetric)satisfiesaweakly-Einsteincondition.

Notice that we use aslightly differentdefinitionfor weakly-Einsteinmetrics. In [1] and [6],for example,the authors definetheseconditions aswhatwecallR-Einstein.

ˇ

Einsteinmetricsare amaintopicindifferentialgeometryandtheyappearascriticalmetricsfortheHilbertfunctional, g

→ 

τ

dvolg, restricted to volume one metrics. Weakly-Einstein metrics also appears naturally in the study of critical metricsforsomespecificfunctional,forinstance,ifwetakethefunctional

F

t,s

:

g

−→ F

t,s

(

g

) =



M

{|| ρ ||

²

+

t

τ

²

+

s

||

R

||

²

}

dvg

andcomputeitsgradient[4],

∇ F

t,s

= − (

¹

+

^4s

) ρ _{+ (}

1

+

^2t

+

^2s

)

Hess

( τ ) −

¹

+

^4t 2

 τ

g

−

^2t

τ

 ρ − τ

4g

 −

^2s



R

ˇ − 

^R



² 4 g

 +

^4s

 ˇ ρ −  ρ 

²

4 g



−

2

(

1

+

2s

)



R

[ ρ ] −  ρ _

²

4 g

 ,

it involves allthe tensors mentioned inthe definitionofthe weakly-Einstein classes.Moreover, R-Einstein

ˇ

condition has seemtoreceiveattentioninotherfields

In[1],Arias-MarcoandKowalskiclassifiedR-Einstein

ˇ

fourdimensionalLiegroups,andfollowingthiswork,aclassifica- tiononthesamefieldwasgivenin[9],completingallthecasuistic.In[5],Chenstudy R-Einstein

ˇ

almostcontactmanifolds andin[3],theauthorsstudythistensorinthecontextofcompactmanifoldswithboundary.

On the other hand, R tensor

ˇ

appears in other fields,such asin thestudy ofthe two-loop renormalization flow. The two-looprenormalizationflow(or R G2 flow)appearsasaperturbationoftheRicciflow(see[11–13])anditisgivenby

∂

∂

t

g_t

=

^RG

[

^g

],

^(1.3)

where R G

[

^g

] = −

²

ρ −

^α₂R and

ˇ α

isapositivecouplingconstant.

Onecanstudygenuinefixedpointsof(1.3),i.e., metricssatisfying

ρ +

^α₄R

ˇ =

^0.

In dimension two the condition reduces to constant negative curvature. In dimension three, they were studied by Gimre, Guenther and Isenberg in [11], where they showed solutions with Ricci curvatures Q ρ

= −

^2diag

[

_α¹

, α ,

0

]

^or Q ρ

= −

^2diag

[

²_α

, α ,

_α¹

]

^{. Einstein metrics are genuine fixed points ofthisflow in dimension foursince ifthe Ricci tensor} isamultipleofthemetric,the R tensor

ˇ

isaswell.In[10],itisgivenaclassificationofgenuinefixedpointsinfourdimen- sionalLiegroups. Thisflowhasalsobeenappliedinthestudyofblackholesmetrics,analyzinghowtheyevolvedalongit andforthestudyofentropy,whichhasbeenstatedasmonotonicalongthissameflow.Thereasontousethisinsuchcases isthatthesingularitiesappearinginthestudyofotherflowsdisappearinRG2,beingthisabetterapproximationtohigher curvatureeffects[14,15].

The mainaimofthiswork isclassifyingtheseconditions,bothweakly-Einstein andfixed points,inthefield oflocally conformally flatmanifolds.The R-Einstein

ˇ

condition hasbeenalreadystudiedin[8], wherethemetricssatisfyingitwere classified as a product Mⁿ

(

c

) ×

^Mn

( −

^c

)

or a warped product

I ×

f N

(

c

)

of a real interval and a manifold of constant sectionalcurvaturec withsomespecificrealfunctionsolvingthedifferentialequation f^

(

t

)

²

+

^f

(

t

)

f^

(

t

) −

^c

=

^0.Therefore,

we will focuson theother two left, the

ρ ˇ

-Einstein andthe R

[ ρ _]

-Einstein conditionsalong sections 2and3,completing thestudy andgivingthewholeclassificationofweakly-EinsteinlocallyconformallyflatRiemannianmanifoldsandfinding newexamplesofthissortofmanifolds,ofwhichthereisalackofthemalongalltheliterature.Duringsection4,westudy fixed points forthe RG2flow, obtaining an algebraic condition. Finally,insection 5, we aregoing to study thecondition R

[ ρ ]

^{-Einsteininaparticularcasuisticinordertotrytogivesome lightonitasitseemstobetheonethatremains with} nonewexamples.

2. R[

ρ

_]-Einsteincondition

R

[ ρ _]

-Einsteinconditionsseemstobetoomuchrigidandthesefields providesnonenewexamples forit.Theresultof itscalculationisbriefedinthefollowingstatement.

Theorem2.AlocallyconformallyflatRiemannianmanifoldisR

[ ρ _]

-EinsteinifandonlyifM

=

^Mn1

(

c

) ×

^Mn2

( −

^c

)

withn1

=

ⁿ2.

Proof. Firstofall, weare goingto computethe R

[ ρ ] (

¹

,

1

)

-tensor,whichisgivenby R

[ ρ ](

^X

,

Y

) =

^g

(

Q_R[ρ]

(

X

),

Y

)

.Using (1.1),astraightforwardcalculationshowsthat

QR[ρ]

= −

²

(

n

−

²

)

^Q

2

+

ⁿ

τ

(

n

−

¹

)(

n

−

²

)

^Q

+



1

(

n

−

²

) || ρ _||

²

₋ τ

²

(

n

−

¹

)(

n

−

²

)

Id

.

Sincewewanttoseewhenthistensorisamultiplyoftheidentity,weneedthat Q_R[ρ]

− || ρ ||

²

n Id

=

⁰

,

orequivalently,

−

²

(

n

−

2

)

^Q

2

+

ⁿ

τ

(

n

−

1

)(

n

−

2

)

^Q

+



2

n

(

n

−

2

) || ρ _||

²

₋ τ

²

(

n

−

1

)(

n

−

2

)

Id

=

⁰

.

(2.4)

This equation needs tobe satisfiedby every eigenvalue ofthe tensor, and since it is a quadratic,then we havetwo atmost, butifwe have just one,then the manifold wouldbe Einstein,so assume that wehave eigenvaluesof theRicci operator

λ

and

μ

withmultiplicitiesm andn

−

^m,respectively.Moreover,usingtheVieta’sFormulae[7],weobtainthat

λ + μ ₌

ⁿ

τ

2

(

n

−

¹

) .

(2.5)

Thus,as

τ ₌

m

λ + (

ⁿ

−

^m

) μ

,botheigenvaluesarerelatedby

μ ₌

²

⁽

ⁿ

⁻

¹

^{) −}

^mn

n

(

n

−

^m

) −

²

(

n

−

¹

) λ.

(2.6)

Next, on the one hand, let S

=

_n−¹₂

( ρ −

_2(n^τ₋₁₎^g

)

be the Schouten tensor, and on the other hand, let us introduce the followingtechnicalresult.

Lemma3.[16] LetT beaCodazzitensor.Let

γ

beaneigenfunctionofT witheigenspaceV γ .Ifdim V γ

≥

^2,then

∇ γ

isorthogonal toV γ .Moreover,ifT hasexactlytwodifferenteigenfunctions

γ

and

δ

withdim V γ

≤

^{dim V}δ,then

(i) M islocallyaproductifdim V γ

≥

^2.

(ii) M islocallyawarpedproductwithone-dimensionalifandonlyif (ii.a) dim V γ

=

^1,

(ii.b) theeigenfunction

δ

isnotconstantand

∇ γ

isorthogonaltoV_δ.

It iswellknownthat S isCodazziifthemanifold isLocallyconformally flatandfrom(2.6) one canobtain,througha standard calculation,thattheSchoutentensorhastwo differenteigenvalues(call them

¯λ

and

μ ¯

),andthus,we canapply thelemma.

If dim V_¯λ

≥

^{2, then M is a locallya product by assertion}

(

i

)

and dueto locally conformally flatness it can be either

R ×

^N

(

c

)

or Mⁿ¹

(

c

) ×

^Mn2

( −

^c

)

. The firstcaseimpliesthat one ofthe eigenvalues iszero and, asthey are amultiple of each other,then both are vanishing,so M is flat. Regardingthe second case, one caneasily see that aproduct manifold Mⁿ¹

(

c1

) ×

^Mn2

(

c2

)

is R

[ ρ _]

-Einsteinifandonlyif

c²₁

(

n₁

−

¹

)

²

=

^c22

(

n₂

−

¹

)

²

,

andsinceinthiscasec1

= −

^c2,thenn1

=

ⁿ2.

If dim V_¯λ

=

^{1, then dim V}μ¯

=

ⁿ

−

^{1, andso}

∇

μ is¯ orthogonal to Vμ,_¯ but, as

¯λ

is a multiple of

μ ¯

, then

∇

¯λ is also orthogonal toVμ._¯ Besides,

μ ¯

cannotbe constant.Otherwise,

¯λ

wouldbeconstant aswell, whichwouldimplythat

λ

and

μ

wouldbe constant.Hence, we wouldhavea locallyconformally flatmanifold withconstant Riccicurvatures, whichis curvaturehomogeneous,andby[18],itwouldbelocallysymmetric.ThenM wouldsplitasaproductoftheform

R ×

^N

(

c

)

, whosefactorscorrespondto theRiccicurvatures,so

λ

wouldbevanishingandso

μ

,andtherefore,M wouldbeflat.Thus, applying theprevious lemma,we havea warpedproduct and dueto locallyconformally flatness, the fiberhas tobe of constantsectionalcurvature.

Now,wewanttodeterminethewarpingfunction.Inordertodothat,weusethefollowingresult.

Lemma4([17]).LetB

×

f F beawarpedproductwithdim F

=

^d.LetX

,

Y

∈

^TpB andV

,

W

∈

^TpF .Then

• ρ (

X

,

Y

) = ρ

^B

(

X

,

Y

) −

^d

fHess

(

f

)(

X

,

Y

)

.

• ρ (

X

,

V

) =

^0.

• ρ (

V

,

W

) = ρ

^F

(

V

,

W

) −

 

f

f

+ (

^d

−

¹

)

^g

(

grad f

,

grad f

)

f²

g

(

V

,

W

)

.

Inourcurrentsituation,weareinawarpedproduct

R ×

fN

(

c

)

,sotheRiccioperatoriswrittenby Q

(∂

t

) = −(

ⁿ

−

¹

)

^f



f

∂

t

,

(2.7)

Q

(

X

) =



(

n

−

²

)

^c

f²

− (

ⁿ

−

²

)

^f

² f²

−

^f

f



X

.

SincetheRiccieigenvaluesarerelatedby

λ = (

ⁿ

−

¹

) μ

,weobtainthedifferentialequation

f^2

−

^c

=

⁰

,

whichonlyhaveasuitablesolutionifc

>

0,andinthatcase,itislinear,whatgivesEinsteinmetrics.Thereforewecannot haveR

[ ρ _]

-Einsteinwarpedproductsinthisfield,whichcompletestheproof.



3.

ρ

ˇ-Einsteincondition

Insharpcontrastwiththepreviouscase,wecangetnewexamplesforthismetrics.Westatethefollowing.

Theorem5.AlocallyconformallyflatRiemannianmanifoldis

ρ ˇ

-EinsteinifandonlyifM

=

^Mn1

(

c

) ×

^Mn2

(−

^c

)

withn1

=

ⁿ2ora warpedproduct

R ×

f

R

ⁿ⁻¹with

f

(

t

) =



2

(

n

−

¹

) (

at

+

^b

)

n



ⁿ

2(n−1)

,

witha

,

b

∈ R

^andt

∈ 

−b a

, +∞ 

.

Proof. Weproceedinthesameway.Thistime,theequationdesireequationis Q²

− || ρ ||

²

n Id

=

⁰

.

(3.8)

Consequently, we have two eigenvalues againanddue to Vieta’s formulae they are relatedby

μ = −λ

^{.We shall use} Lemma3again.Therefore,ifdim V_λ

≥

^{2 thenwe haveaproduct Mn1}

(

c

) ×

^Mn2

(−

^c

)

andthecondition toaproductofthis kindtobe

ρ ˇ

-Einsteinisthat

c²₁

(

n₁

−

¹

)

²

=

^c2₂

(

n₂

−

¹

)

²

,

son₁

=

ⁿ2.

Ifdim V_λ

=

^{1,thenwehaveawarpedproduct}

R ×

fN

(

c

)

,andasweknowthat

μ = −λ

^{,using((2.7)),weobtain} nf f^

+ (

ⁿ

−

²

)

f^2

− (

ⁿ

−

²

)

c

=

⁰

.

Now,takingthederivativeofthisequationoneobtains nf f^

+ (

³ⁿ

−

⁴

)

f^f^

=

⁰

.

Since f and f^ cannotbezero(otherwisethemetricisflat),thenonecandividebythesefactorsandthus

(

4

−

³ⁿ

)

n

f^ f

=

^f

f^

.

Next,integratebothpartsoftheequationsandget

(

4

−

3n

)

n ln f

=

^{ln f}

+

^K

.

Takingtheexponential,theequationsbecome

f^(4−n3n)

=

^eKf

,

andnowmultiplybothsidesby2 f^,

2e^−Kf^f^(4−n3n)

=

^{2 ff}

.

Calltheconstantpart K .

¯

Wehavestandardintegralsonbothparts,soweget K

¯

n

4

−

2nf⁴⁻ⁿ²ⁿ

=

f^2

.

Finally,isolating f^,weget

f^

= ˜

^{K f2−nn}

,

whichsolutionis

f

(

t

) =

⎛

⎝

²

(

n

−

¹

)



K t

˜ +

^a



n

⎞

⎠

n 2(n−1)

,

where a

∈ R

^{. Therefore, we obtain a solution forthe second equation, which was the derivative of the one we got in} first place.Now, ifsome function isa solutionfor thefirst equations,it isa solution forits derivative, andaswe know the solutions forthislast one,the solutionof theoriginal equationsneed tobe of thisform. Soifwe put this f in the originalequations,wegetthatitisasolutionforitifandonlyifc

=

^{0.Therefore,weareinawarpedproductoftheform}

R ×

f

R

ⁿ⁻¹andwehavenootherpossibilityhere.



Remark6.Noticethatusingthesetechniquesonthewarpedproducts,weshallgiveasimplerprooffortheclassificationof the R-Einstein

ˇ

casegivenin[8].Fromthere,wehavethattherelationbetweenbotheigenvalueswasgivenby

μ = −

^2m

+ (

ⁿ

−

¹

)(

n

−

⁴

)

2

(

n

−

m

) + (

n

−

1

)(

n

−

4

) λ,

andifm

=

^1,then

μ _{= −}

²

^{+ (}

ⁿ

⁻

¹

⁾⁽

ⁿ

⁻

⁴

⁾

2

(

n

−

¹

) + (

ⁿ

−

¹

)(

n

−

⁴

) λ.

Usingnow(2.7),weobtainthedifferentialequation

f^2

+

^{f f}

−

^c

=

⁰

,

whichistheonethatgivesR-Einstein

ˇ

metrics.

4. LocallyconformallyflatfixedpointsoftheRG2-flow

Inthissectionweclassifyfixedpointsinthecontextoflocallyconformallyflatmanifolds.

Theorem7.Let

(

M

,

g

)

bean-dimensionallocallyconformallyflatfixedpointforthetwo-looprenormalizationgroupflowwithcou- plingconstant

α

.Then

(1) Ifn

=

^4,then

(

M

,

g

)

ishomothetictoaproductMⁿ₁¹

(

c

) ×

^Mn₂²

(−

^c

)

withn₁

=

ⁿ2ortoawarpedproduct

R ×

f N

(

c

)

withnon trivialwarpingfunctionsatisfying

α (

n

−

²

)((

n

−

⁶

)

n

+

⁶

)



f^2

−

^c



+ α ((

n

−

⁴

)(

n

−

²

)

n

−

⁴

)

f f^

+

²

(

n

−

²

)

²f²

=

⁰

.

(2) Ifn

=

^4,then



^R



²

=  ρ _

²

₌

^τ₃².

Proof. Letusrecall,ontheonehand,thatfixed pointsforthe R G2 flowisgivenbyametricfulfilling

ρ +

^α₄R

ˇ =

^0.Onthe otherhand,onecanseefrom[8] that Q_Rˇ operatorisgivenby

Q_Rˇ

=

²

(

n

−

²

)

²



(

n

−

⁴

)

Q²

+

²

τ

(

n

−

¹

)

^Q

+ (

n

−

¹

)  ρ _

²

₋ τ

²

(

n

−

¹

)

^Id

.

Combiningthesetwoidentities,onecangetthatametricinthisfieldisafixedpointif Q

+

^α₄^Q_Rˇ

=

^0,whichis

Q

+ α

2

(

n

−

²

)

²



(

n

−

⁴

)

Q²

+

²

τ

(

n

−

¹

)

^Q

+ (

n

−

¹

) ρ 

²

− τ

²

(

n

−

¹

)

^Id

=

⁰

,

andthen,

α (

n

−

4

)

2

(

n

−

2

)

^Q

2

+ ατ + (

n

−

1

)(

n

−

2

)

²

(

n

−

1

)(

n

−

2

)

^{2 Q}

+ α ((

n

−

1

) ρ 

²

− τ

²

)

2

(

n

−

2

)

²

(

n

−

1

)

^Id

=

0

.

(4.9)

Now wehavetwo differentpossibilitiesdepending onthe dimension.Ifn

=

^{4,then wehavea quadraticequationon the} Riccioperator,sowehavetwoRiccieigenvaluesrelatedby

λ + μ = −

²

( ατ _{+ (}

n

−

¹

)(

n

−

²

)

²

) α (

n

−

1

)(

n

−

2

)(

n

−

4

) .

Thus, we have two eigenvalues, one a multiple ofthe other,and asthe Schouten tensoris Codazzi andit hasalso two eigenvalues, one a multiple of the other, then we have either a warped product

R ×

f N

(

c

)

, with f a non trivial real warping functionand N

(

c

)

an

(

n

−

¹

)

-dimensional Riemannian manifold ofconstant curvature ora Riemannian product Mⁿ₁¹

(

c

) ×

^Mn₂²

( −

^c

)

,suchthatn1

=

ⁿ2.Inordertodeterminethefunction f ,assumingthat

λ

hasmultiplicityone,thenboth arerelatedby

λ + μ =

²

 α (λ + μ (

n

−

¹

)) + (

ⁿ

−

¹

)(

n

−

²

)

²



α (

n

−

⁴

)(

n

−

²

)(

n

−

¹

) ,

andusingtheformulasfrom(2.7) fortheRiccioperator,wegetthat f mustsatisfythedifferentialequation

α (

n

−

²

)((

n

−

⁶

)

n

+

⁶

)



f^2

−

^c



+ α ((

n

−

⁴

)(

n

−

²

)

n

−

⁴

)

f f^

+

²

(

n

−

²

)

²f²

=

⁰

Ifn

=

^{4,thenequation}(4.9) becomes

12

( ατ +

¹²

)

Q

+ α (

3

 ρ 

²

− τ

²

)

Id

=

⁰

.

Sincethisisalinealequation,thisonlycanhaveonesolution,andthen,theRiccioperatorhasonlyoneeigenvalue,sothe metricisEinsteinaslongastheequationisnotidenticallyzero.Inordertohavethat,weneedthat

α = −

¹²_{τ and}

 ρ  =

^τ₃^2. Moreover,takingtracesin

ρ +

^α₄R

ˇ =

^{0,onecanobtainthat}

α = −

⁴

τ 

^R



^−2,then



^R



²

=

^τ₃^2,andhence



^R



²

=  ρ 

^2.Notice that

α

cannotbevanishingsince,inthatcase,theRiccitensorisaswell.



5. AnoteonR[

ρ

_]-Einsteincondition

During these sections we are not going to assume that the manifold is locallyconformally flat. Since this condition seemstobethemostrigidone,wemaythinkotherwaystotrytoobtainexamples.Wemaythinkofaneasiercasuisticin ordertogetsomesuitablealgebraiccondition.Forthat,assumethattheRiccioperatorhastwoeigenvalues,onesimple,i.e., Q ρ

=

^diag

[λ, μ , . . . , μ ]

^{.Inthissituation,asthe}

ρ

i j areall vanishingwhenever i

=

^{j,thenthe tensorreducesto R}

[ ρ ]

i j

=



aRai ja

ρ

aa,havingamuchsimplercasuistic.

NowwearecomputingR

[ ρ ]

11.

R

[ ρ ]

11

=

R2112

ρ

22

+

R3113

ρ

33

+ · · · +

Rn11n

ρ

nn

= μ (

R2112

+

^R3113

+ · · · +

^Rn11n

) = μρ

11

= μ λ

Now,recallthat

 ρ _

²

_{= λ}

²

_{+ (}

n

−

¹

) μ

²,sooneoftheequationsthatneedstobesatisfies,inordertohaveamultiplyofthe identity,is

λ

²

+ (

ⁿ

−

¹

) μ

²

−

ⁿ

λ μ =

⁰

,

which,afterdoingsomesimplifications,isequivalentto

(λ − μ )(λ − (

ⁿ

−

¹

) μ ) =

⁰

.

Therefore,sinceif

λ = μ

,weobtainthat

λ = (

ⁿ

−

¹

) μ

,whichmakesthecomponentR

[ ρ ]

11

= (

ⁿ

−

¹

) μ

².

TheconditionofR

[ ρ _]

beingamultipleofthemetricmustfulfillthat R

[ ρ _]

ii

=

^R

[ ρ _]

j j

= (

ⁿ

−

¹

) μ

² andR

[ ρ _]

i j

=

^0,forall i

,

j

∈ {

¹

, . . . ,

n

}

^,i

=

^{j.Becauseofthat,wearestudyingtherestofthecomponentsinthreesteps.}

Firstlyweareanalyzing R

[ ρ ]

1α ,with

α >

1.

R

[ ρ _]

1α

=

^R21α2

ρ

22

+

^R31α3

ρ

33

+ · · · +

^Rn1αn

ρ

nn

= μ (

R21α2

+

^R31α3

+ · · · +

^Rn1αn

) = μρ

1α

=

⁰

.

Usingthesamecomputations,weobtainthatR

[ ρ _]

_αα

₌ μ ((

n

−

²

)

R1αα1

+ μ )

,with

α >

1,andR

[ ρ _]

_αβ

₌ μ (

n

−

²

)

R1αβ1, with

α _{= β}

,1

< α < β

.Therefore,fromthesetwocomponents,weobtaintheequations

μ ((

n

−

²

)

R_1αα1

+ μ ) − μ

²

(

n

−

¹

) =

⁰

μ (

n

−

²

)

R_1αβ1

=

⁰

Ontheonehand,as

μ

cannotbezero,wegetthatR₁αα1

= μ

,andsince

ρ

_αα

= μ =

R1αα1

+

R2αα2

+ · · · +

Rnααn

,

wegetthecondition



n i=2

Riααi

=

0

.

Ontheotherhand,weautomaticallygetthat R₁αβ1

=

^{0.Thus,wehavethefollowingresult.}

Lemma8.Let

(

M

,

g

)

beann-dimensionalRiemannianmanifoldwithtwodifferenteigenvaluesfortheRiccioperator,oneofthem simple.Then,

(

M

,

g

)

isR

[ ρ _]

-Einsteinifandonlyif

(i) Q ρ

=

^diag

[(

ⁿ

−

¹

) μ , μ , . . . , μ ]

^. (ii)



i>1Riααi

=

^0,with1

< α _≤

n.

(iii) R1αβ1

=

^{0 forall}

α

,

β

suchthat1

< α < β ≤

^n.

Remark9.Thesuitablecurvaturetensorcouldhavedifferentspecialconditionsdependingonthedimensionweareworking on.Forinstance,ifn

=

^4,then

ρ

23

=

⁰

=

^R1231

+

^R4234

ρ

24

=

⁰

=

^R1241

+

^R3243

ρ

34

=

⁰

=

^R1341

+

^R2342

.

Using

(

iii

)

fromtheLemma,showsthat R₄₂₃₄

=

^R3243

=

^R2342

=

^0.Moreover,

(

ii

)

giveusthesystemofequations

R3223

+

^R4224

=

⁰ R2332

+

^R4334

=

⁰ R2442

+

^R3443

=

⁰

,

which onlypossible solution,dueto thesymmetriesof thecurvaturetensor, isthat every Rαββα ,with1

< α < β ≤

^{4 is} vanishing.

However,ifn

=

^{5,thislastsystembecomes}

R3223

+

^R4224

+

^R5225

=

⁰ R2332

+

^R4334

+

^R5335

=

⁰ R2442

+

^R3443

+

^R5445

=

⁰ R2552

+

^R3553

+

^R4554

=

⁰

,

whichdoesnotimplythateverytermisvanishing.

Remark10.In the context of locally conformally flat metrics,

(

i

)

is satisfied, but this doesnot imply that M is locally conformallyflat.Forexample,ifn

=

^4,thetermW

(

e₁

,

e₂

)

oftheWeyltensorisgivenby

W

(

e₁

,

e₂

) =

⎛

⎜ ⎜

⎝

0 0 0 0

0 0

−

^R1223

−

^R1224

0 R₁₂₂₃ 0

−

^R1234

0 R₁₂₂₄ R₁₂₃₄ 0

⎞

⎟ ⎟

⎠ .

In locally conformally flat metrics with this kind of Ricci operator, the curvature components where we have three differentindicesarezero,whereasinthiscasewedonotneedanyspecialconditionforthesecomponents.

Declarationofcompetinginterest

The authors declare that they haveno known competingfinancial interests or personal relationships that could have appearedtoinfluencetheworkreportedinthispaper.

Dataavailability

Nodatawasusedfortheresearchdescribedinthearticle.

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