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.WhenrestrictedtoCl0nthespinormoduledec …… 此处隐藏：5537字，全部文档内容请下载后查看。喜欢就下载吧 ……