A characterization of Dirac morphisms

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

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aACHARACTERIZATIONOFDIRACMORPHISMSE.LOUBEAUANDR.SLOBODEANUAbstract.RelatingtheDiracoperatorsonthetotalspaceandonthebasemanifoldofahorizontallyconformalsubmersion,wecharacterizeDiracmorphisms,i.e.mapswhichpullback(local)harmonicspinor eldsonto(local)harmonicspinor elds.1.IntroductionIntroducedbyJacobi[10]in1848,harmonicmorphismsaremapswhichpullbacklocalharmonicfunctionsontoharmonicfunctionsand,morerecently,theywerecharacterizedbyFuglede[6]andIshihara[9]ashorizontallyweaklyconformalharmonicmaps.Theirdualnatureofanalyticalandgeometricalobjectshasledtoarichtheory(cf.[2])whichhasencouragedthestudyofvariousothermorphisms,thatismapspreservinggermsofcertaindi erentialoperators.ThecentralroleoftheDiracoperatorindi erentialgeometryandmathematicalphysicscalledforthisapproachtobeappliedtoharmonicspinors.Unlikepreviouscases,the rsthurdleistomakesenseofanotionofpull-backofspinorsbyamap.Thisrequirestheidenti cationofthespinorbundlesinvolved,necessarilyrestrictingourinvestigationtohorizontallyconformalmapsbetweenRiemannianmanifolds(cf.Sec-

tion2).CombiningachainrulefortheDiracoperatorandalocalexistencelemma,weshowthatahorizontallyconformalsubmersionbetweenspinmanifoldsisaDiracmorphismifandonlyifitshori-zontaldistributionisintegrableandthemeancurvatureofthe bres

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

2E.LOUBEAUANDR.SLOBODEANU

isrelatedtothedilationfactor,inamannerreminiscentofthefun-damentalequationforharmonicmorphisms.WeconcludewithsomesimpleexamplesbetweenEuclideanspacesandexplicitourresultsintheset-upof[12],whichinspiredinitiallyourconstruction.

2.Pull-backofaspinor

Let(Mm,g)beaspinRiemannianmanifold,thetwo-sheetedcov-eringSpin(m) ρ→SO(m)inducesadoublecoverχ:PSpin(m)M →PSO(m)MofthebundleofpositivelyorientedorthonormalframesbytheprincipalSpin(m)-bundleoverM,suchthatχ(s·g)=χ(s)·ρ(g), s∈PSpin(m)M,g∈Spin(m).TheassociatedbundleCl(M)=PSO(m)M×clmClmistheCli ordbundle,whereClmistheCli ordalgebraandclmtherepresentationofSO(m)intoAut(Cl(Rm)),andthespinorbun-dleisSM=PSpin(m)M×γSm,withγthespinorialrepresentationofSpin(m)ontheCli ordmoduleS[m/2]

m=C2(cf.[11]).

Aspinor eldisa(smooth)sectionofSM,Ψ:U M →SM,Ψ(x)=[sx,ψ(x)],wheresx∈PSpin(m)Misaspinorialframeatx∈Mandψ:U →Sm,theequivalenceclassbeingde nedby

[s,ψ]=[s·g 1,γ(g)ψ],

forallg∈Spin(m).Thecovariantderivative

ejΨ= iss,dψ(ej)+1

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

ACHARACTERIZATIONOFDIRACMORPHISMS3

E′⊕E′′overM,achoiceofspin-structureonanytwoofthemuniquelydeterminesaspinstructureonthethird([11]).De nition1.Asmoothmapπ:(Mm,g)→(Nn,h)betweenRie-mannianmanifoldsisahorizontallyconformalmapif,atanypointx∈M,dπxmapsthehorizontalspaceHx=(kerdπx)⊥conformallyontoTπ(x)N,i.e.dπxissurjectiveandthereexistsanumberλ(x)=0suchthat 2(πh)x =λ(x)gx .Hx×HxHx×Hx

ThefunctionλisthedilationofπandtheorthogonalcomplementofHxistheverticaldistributionVx=kerdπx.

ThemeancurvaturesofthedistributionsHandVaredenotedµHandµVandIHistheintegrabilitytensorofH.

a=1,...,m nAframe{Va,Xi }iofTMwillbecalledadaptedifVa∈V,a==1,...,n

1,...,m nand{Xi }i=1,...,nisthehorizontalliftbyπofanorthonormalframe{Xi}i=1,...,nonN.

Notethatλ≡1correspondstoRiemanniansubmersions.

Wecallthemapπ:(Mm,g1)→(Nn,h),whereg1=π h+gV,theassociatedRiemanniansubmersionofπ:(Mm,g)→(Nn,h).

Sinceageneralsubmersionπ:(Mm,g)→(Nn,h),betweenspinRiemannianmanifolds,splitsthetangentbundleTMintoH⊕V,ifHadmitsaspinstructure,sodoesVand

(1) Cl(V).Cl(M)=Cl(H)

ThespinstructuresPSpin(n)HandPSpin(m n)VinduceaspinstructurePSpin(m)MbyprolongationoftheprincipalbundlePSpin(n)×Spin(m n)M(cf.[8]).Generalpropertiesofassociatedbundlesofreducedprincipalbundles([8,Theorem3.1])togetherwithadimensioncountyieldthefollowingisomorphismsof(associated)vectorbundles

(2)S+M=SH SV

whenmisevenandnodd,and

(3)

fortheremainingcases.SM=SH SV

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

4E.LOUBEAUANDR.SLOBODEANU

Foranymapπ:M→Nintoaspinmanifold,considerthepull-backspinorbundle

π 1SN={(x,[s,ψ])∈M×SN| S([s,ψ])=π(x)},

where SistheprojectionmapofSN.

IfπisaRiemanniansubmersion,thentheisomorphismπ 1SN=SH,duetotheidenti cationoforthonormalframes,simpli es(2)and(3)into(cf.[4])

(4)S+M=π 1SN SVSM=π 1SN SV.

Remark1.WhenπisaRiemanniansubmersionwithtotallygeodesic bres,Hiscomplete,Nconnectedandthe bresareisometrictoaRiemannianmanifoldF.IfNandFarespinmanifolds,consideronMtheinducedspinstructureand,viatheisomorphismπ 1SN=SH,

(2)and(3)read(see[12])

S+M=π 1SN SF,SM=π 1SN SF.

Remark2.Ifniseven,theCli ordalgebraClnpossessesanirre-duciblecomplexmoduleSnofcomplexdimension2n/2,thecomplexspinormodule.WhenrestrictedtoCl0nthespinormoduledecomposes

+ intoSn=Sn⊕Sn,thesubmodulesofspinorsofpositiveandnegative

chirality,characterizedbytheactionofthevolumeelement,onceanorientationisgiven.Inparticular,thespingroupSpin(n) Cl0nacts

+ onSnandonSn(thespinorrepresentations).

0Ifnisodd,thenClnhastwoinequivalentirreduciblemodulesSnand

1Sn,bothofcomplexdimension2(n 1)/2(againdistinguishedbytheac-tionofthevolumeelement).WhenrestrictedtoCl0nthetwomodules

0becomeequivalentandwesimplywriteSn=Snandthespinorrepre-

sentationγ:Spin0(n)→Aut(Sn)becomesirreducible.

Ifniseven,thenSn+1pullsbacktoSnunderthealgebraisomorphismCln=Cl0n+1.Inotherwords,wecanregardSn+1asthespinorrep-resentationofCln,providedwede netheactionofClnonSn+1byv σ→e0·v·σ.

Similarly,ifnisodd,thentheactionofthevolumeformshowsthat+ 01Sn+1pullsbacktoSnwhileSn+1pullsbacktoSn.

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

ACHARACTERIZATIONOFDIRACMORPHISMS5

Whenπ:(Mn+1,g)→(Nn,h)isaRiemanniansubmersionwithone-dimensional bresintoaspinmanifoldNn,themanifoldMn+1inheritsanaturalspinstructure,cf[13].Moreover,ifniseven,thenSM=π 1SNand,whennisodd,thenS+M=π 1SN.

Theseidenti cationsjustifythefollowingde nition.

De nition2(Riemanniansubmersions).Letπ:(Mm,g)→(Nn,h)beaRiemanniansubmersionbetweenspinmanifoldsandendowtheverticalbundlewiththeinducedspinstructure(ifm=n+1,weconsideronMthenaturalspinstructureinheritedfromN).LetΨ=

[s,ψ]bea(local)spinor eldonN.Sincealocalspinframes={Xi}i=1,...,nonNliftstoanadaptedspinframeonM

s ={X

i,Va}a=1,...,m ni=1,...,n∈PSpin(n)×Z2Spin(m n)M|π 1U,

whereX iisthehorizontalliftofXiand{Va}a=1,...,m nisanorthonor-malframeofV,wede nethepull-backΨ

Ifm 1=n,thesectionΨ =[ s ofΨtobe,ψ π]ofthebundleπ 1SN,identi edwithSM,whenniseven,andwithS+M,whennisodd.

Ifm n≥2,thesectionΨ =[s ,ψ =(ψ π) α]inπ 1SN SV,αbeingasectionofSV,identi edwithS+M,whennisoddandmeven,andwithSMotherwise.

Remark3.NotethatCli ordmultiplicationwiththiskindofspinor eldsisgivenby

X ·ψ =X ·ψ;V·ψ =i

ψ π) V·α,

whenm n≥2,whereX isthehorizontalliftofthevector eldX∈Γ(TN)and

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

6E.LOUBEAUANDR.SLOBODEANU

(cf.[11])

ξγ:SM1→SM

ξγ([s,ψ])=[ξ(s),ψ],

whereξ(s)={Ei=λEi1,Va}ifs={Ei1,Va},thebundleisometryinducedbytheSpin-equivariantmap

ξ:PSpin(n)×Z2Spin(m n)M1→PSpin(n)×Z2Spin(m n)M

givenbythenaturalcorrespondencebetweenadaptedorthogonalframeswithrespecttothetwometrics:E1

i1=λ Ei,Va1=Va.

TheCli ordmultiplicationwillbegivenby

Ei·Ψ=ξγ Ei1·Ψ1 ,Va·Ψ=ξγ(Va·Ψ1),

whereΨ=ξγ Ψ1.

De nition3(Horizontallyconformalsubmersions).Letπ:(Mm,g)→(Nn,h)beahorizontallyconformalsubmersionbetweenspinmani-foldsandendowtheverticalbundlewiththeinducedspinstructure.LetΨ=[s,ψ]bea(local)spinor eldonN.Thepull-backofΨisΨ =ξγ Ψ 1,whereΨ 1isthepull-backofΨbytheassociatedRiemanniansubmersionπ:(Mm,g1)→(Nn,h)andξγthebundleisometrybetweenSM1andSM.

3.Diracmorphismswithhighdimensionalfibres

Throughoutthissectionπhas bresofdimensionatleasttwo.

De nition4.Ahorizontallyconformalsubmersionπ:(Mm,g)→(Nn,h)betweenspinmanifoldsiscalledaDiracmorphismifforanylocalharmonicspinorΨde nedonU N,suchthatπ 1(U)= andthereexistsasectionα∈Γ(SV) V-parallelinhorizontaldirectionsandwithDVα n

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

ACHARACTERIZATIONOFDIRACMORPHISMS7

We rstneedsomelemmas.

Lemma1(Chainrule).Letπ:(Mm,g)→(Nn,h)beahorizontallyconformalsubmersionofdilationλ(m n≥2)andψa(local)spinor

isthepull-backofΨbyπ,withrespecttosomesection eldonN.IfΨ

α∈Γ(SV),then

(5)Nψ =λDDMψ14 IH·ψ

+(2µ·α,H

where{Ei}i=1...,nisalocalorthonormalhorizontalframeonMand

H denotes nIH·ψi<j=1Ei·Ej·I(Ei,Ej)·ψ(thestandardactionof

vector-valued2-formsonspinor elds).

Proof.Letπbeahorizontallyconformalsubmersionofdilationλ.Let

a=1,...,m n{Xi}i=1,...,nbeanorthonormalframeon(N,h)and{Va,Xi }i=1,...,n

anorthonormaladaptedframeon(M,g1),whereg1=πh+gV.With

a=1,...,m nrespecttothemetricg,{Va,λXi }iisanorthonormaladapted=1,...,n

frame.Denoteby and 1the(spinorial)connectionscorrespondingtogandg1,andnoteEi1=Xi ,Ei=λXi .

],forthepull-backspinor eldψ =[ AsDMΨs,DMψ

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

8E.LOUBEAUANDR.SLOBODEANU

=DMψ

=i=1n i=1

1n Ei·Ei((ψ π) α) +Ei· Eiψm n a=1 Va· Vaψ(H0)+

n,m n

2

i,j,a=1

+1 Ei·g( EiEj,Va)Ej·Va·ψ(H2)

n,m n

4

i,j,a=1

+1 Va·g( VaEi,Ej)Ei·Ej·ψ(V1)

4

a,b,c=1

Notethat

1m n Va·g( VaVb,Vc)Vb·Vc·ψ(V3).Nψ=ξ(DNψ) Dγ 1 =ξγ(Xi·Xi(ψ π)+g1( Xi Xj,Xk)Xi·Xj·Xk·(ψ π)) α,

Nwhereg1( 1 Xj,Xk)=h( XXj,Xk).Xii

Thecomputationbreaksdowninto vesteps:

Step1:

Nψ (H0)+(H1)+(H3)=λDn 1

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

ACHARACTERIZATIONOFDIRACMORPHISMS9

As

(H0)+(H1)=n i=1Ei·[Ei(ψ π) α+(ψ π) Ei(α)]

+1

4

i,j,k=1

Thelasttermcanberewritten

1n j k 1Xk(λ)δi Xj(λ)δiXi·Xj·Xk·ψ2 .

and,sincegradH(λ)=λ2gradH1(λ),itbecomes

n 1

2 1,gradH1(λ)·ψ

= n 1 λgradH1(λ)·ψ

2

µV·ψ.

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

10E.LOUBEAUANDR.SLOBODEANU

Asg( VaEj,Vb)= g(Ej, VaVb)andVisintegrable

(V2)= 1

m n m n

22Va·( VaVb)H·Vb+Va·[Va,Vb]H·Vb a<b=1a>b=1m n Va·( VaVb)H·Vb 2a<b=1

Step3:

(8) µV·ψ.(V0)+(V3)=( ·ψm n ·ψ1

ψ π) Va·Va(α)+

ψ π) Va· V

Vaα.

Step4:

(9)(H2)=11

AsforStep2,wehave

(H2)=1H µ·ψ.2

2

i<j=1n Ei·Ej·IH(Ei,Ej)·ψn

H I·ψ,4

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

ACHARACTERIZATIONOFDIRACMORPHISMS11

since

(V1)=1

n,m n

4

i,j,a=1

=1 [g([Va,Ei],Ej) g( EiEj,Va)]Va·Ei·Ej·ψ2

Va(λ)(H2). Butg([Va,Ei],Ej)=g([Va,λXi ],λXj)=

m n a=1

44

4H µ·ψ 4

Summingupthese vestepsyieldsthechainrule. IH·ψ. gradV(lnλ)·ψ1 Va(lnλ)Va·ψ11 Ageneralizationof[1,Proposition2.4]tovector-valuedfunctionsyieldslocalexistenceofharmonicspinorswithprescribedvalue.Lemma2(Localexistence).Foranypointp∈Mandψ0∈SpM,thereexistsanopenneighbourhoodUofpandaharmonicspinorψ:U→SMsuchthatψ(p)=ψ0.

Theorem1(Characterizationforhorizontallyconformalsubmersions).Ahorizontallyconformalsubmersionπ:(M,g) →(N,h)betweenspinmanifoldsisaDiracmorphismifandonlyifitshorizontaldistri-butionisintegrableand

(11)(m n)µV+(n 1)gradH(lnλ)=0,

whereµVisthemeancurvatureofthe bres.

Proof.Letπbeahorizontallyconformalsubmersionwithintegrablehorizontaldistributionandsuchthat(11)issatis ed.Inthiscase,the

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

12E.LOUBEAUANDR.SLOBODEANU

Chainrule(5)simpli estoNψ+ =λDDψMn

i=1Ei·(ψ π) VEiα+(2µ·α.H

Letψbealocalharmonicspinorsuchthatthereexistsasectionα∈Γ(SV) V-parallelinhorizontaldirectionsandwithDVα n

2µH·α=0,wehave,accordingto(5)

n 0= 1

4 Xi ·Xj·(ψ π) IH(Xi ,Xj)·α.

i<j=1

PuttingX=(m n)µV+(n 1)gradH(lnλ)andVij=IH(Xi ,Xj),

(12)becomes

0= 1

4

i<j=1

ButsinceVijisvertical,XandXi ·Xj·Vijhavedi erentdegrees,

necessarilyX=0andVij=0.Therefore,ifahorizontallyconformalsubmersionisDiracmorphism,itmustsatisfyEquation(11)andhaveintegrablehorizontaldistribution. atp∈Mcanbeprescribed,theaboveequationAsthevalueofψimpliesn 0= 1Xi ·Xj·Vij.4i<j=1n Xi ·Xj·Vij·ψ.Remark5.(1)NotetheanalogybetweenEquation(11)andthe

fundamentalequationforharmonicmorphismsin[2].

(2)CompareFormula(5)with[4,(4.26)]and[12,1.1.1].

(3)Ifthe bresaretotallygeodesic,theintegrabilityofthehor-

izontaldistributionmakesthesectionα“basictransversallyharmonic”,asintroducedin[7].

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

ACHARACTERIZATIONOFDIRACMORPHISMS13

Corollary1.ARiemanniansubmersionπ:(Mm,g)→(Nn,h)be-tweenspinmanifoldsisaDiracmorphismifandonlyifits bresareminimalanditshorizontaldistributionisintegrable.

RecallthatifπisaRiemanniansubmersionthenµH=0.

Remark6.Ifthedilationfunctionλisaprojectablefunction(i.e.V(λ)=0),theconformalinvarianceoftheDiracoperator([11])allowsacorrespondencebetweenharmonicspinorsofthespacesinvolvedinthecommutativediagrambelow.

(M,λ2π h+gV)1M-(M,π h+gV)

π

?π1N 2h)(N,λ(N,h)

4.Diracmorphismswithone-dimensionalfibres

Inthissectionm=n+1.

De nition5.Ahorizontallyconformalsubmersionπ:(Mn+1,g)→(Nn,h)betweenspinmanifoldsiscalledaDiracmorphismifforanylocalharmonicspinorΨde nedonU N,suchthatπ 1(U)= and

=ξγ Ψ 1isaharmonicspinoronπ 1(U) M,wherethepullbackΨ

1isthepullbackofΨbytheassociatedRiemanniansubmersion.Ψ

Lemma3.(Chainrule).Letπ:(Mn+1,g)→(Nn,h)beahorizon-tallyconformalsubmersionofdilationλandψa(local)spinor eldonN,then

1Nψ =λDDMψ IH·ψ.(13)4

Proof.Take{Xi}i=1,...,nanorthonormalframeon(Nn,h)and{V,Ei=λXi }i=1,...,nanadaptedframeon(Mm,g).LetΨbea(local)spinor

itspullbackbyπ.Theproofissimilartotheproofof eldonNandΨ

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

14E.LOUBEAUANDR.SLOBODEANU

Lemma1,exceptthat(H0)=π),(V0)=0andtheterms(H3),(V3)donotappear. Ei·Ei(ψ

NotethatµH·ψ =i µH 2 ψ.

Theorem2.Ahorizontallyconformalsubmersionπ:(Mn+1,g)→(Nn,h)betweenspinmanifoldsisaDiracmorphismifandonlyifitshorizontaldistributionisintegrableandminimal,and

µV+(n 1)gradH(lnλ)=0,

whereµVisthemeancurvatureofthe bres.

Proof.TheargumentissimilartotheoneofTheorem1,exceptthat

X=µV+(n 1)gradH(lnλ)+nµH.

ObservethatX=0ifandonlyifµV+(n 1)gradH(lnλ)=0andµH=0,astheybelongtoorthogonaldistributions. Corollary2.ARiemanniansubmersionπ:(Mn+1,g)→(Nn,h)betweenspinmanifoldsisaDiracmorphismifandonlyifitshorizontaldistributionisintegrableandthe bresareminimal.

Remark7.SupposethatπisaRiemanniansubmersion.

(1)TheChainRule(13)givesustheformulaof[13](wherethe bresareminimal)

(14)DMΨ =D NΨ 1

4i n

X

j·AX jV·j=1

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

ACHARACTERIZATIONOFDIRACMORPHISMS15

ψ ,inaglobalspinframe(e=1ψ

theDiracoperatoronR2

02=DR=γ(e1) y2 ψ= + y2)onR2,and

representation,theDiracoperatoronRis

DR3 x13, x3},andwiththePauli=iσk x3

x1 i x1

x1= π2 x2= π2 x3= π2

Letπ:R4 →R2beaRiemanniansubmersionandψ:R2 →C2a

ofψis(ψ π) α:spinor eldon(R2, , standard).Thenthepull-backψ = xk 00 x0 x200 x2 x0 x0 x2 x20000 x0 .

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

16E.LOUBEAUANDR.SLOBODEANU

R4 →C4.Withrespectto{e

1,e2,v,w},v,w∈Kerdπ,assumingH

integrableandchoosingaframe{

x3, x1},

=ψ +ψ

ψ = ψα

ψ α

ψ+α

ψ α+++ .

Theconditionsofharmonicityandparallelismmakeα:R2→C2aharmonicspinor eldwithrespecttothevariablesx0,x1(i.e.α+isholomorphicandα anti-holomorphic).Takeaharmonicspinorψon

+R2(i.e.ψ+isaholomorphicfunction),onecandirectlycheckthatψ

isharmonic,foranyα,ifandonlyifπsatis es

π1

x1 π2 x1 π1 x3 π1 x2==0,,.

merelyintroducesadi erentsign.Again,Thesamequestionforψ

thisforcesπtobeharmonic,i.e.its bresareminimal.

Example3(Moroianu’sprojectablespinors).In[12],Moroianucon-sidersaprincipal brebundleπ:(Mm,g) →Nwithcompactstruc-turalgroupG,overacompactspinmanifold(Nn,h),suchthatπisaRiemanniansubmersionwithtotallygeodesic bresandthehorizontaldistributionHisaprincipalconnection.

Sinceitstangentspaceistrivial,Gadmitsacanonicalspinstructure,andaspinor(ψ π) αiscalledprojectableifα:G →Sm nisaconstantfunctionwithrespecttothecanonicalframeofleft-invariantvector elds.TohaveaDM-invariantnotionofprojectablespinor,itisnecessaryandsu cienttosupposeGcommutative([12])

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

ACHARACTERIZATIONOFDIRACMORPHISMS17

LetX bethehorizontalliftofavector eldXonNand,using

[V,X ]=0forV∈V,sinceHisaprincipalconnection,wehave

V

X α=X(α)+1

m

2X ,Vc)Vb·Vc·α

b<c ng( Vb=1

=X (α) 1

m n

4Va·g( VaVb,Vc)Vb·Vc·α

a,b,c=1

m

= n

Va·Va(α)+3

a=1

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.

18E.LOUBEAUANDR.SLOBODEANU

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D´epartementdeMath´ematiques,Universit´edeBretagneOcciden-tale,6,AvenueVictorLeGorgeu,CS93837,29238BrestCedex3,France

E-mailaddress:loubeau@univ-brest.fr

FacultyofPhysics,BucharestUniversity,405Atomis¸tilorStr.,CPMg-11,RO-077125Bucharest,Romania.

E-mailaddress:radualexandru.slobodeanu@g.unibuc.ro

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