A characterization of Dirac morphisms(2)

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(Loc …… 此处隐藏：4674字，全部文档内容请下载后查看。喜欢就下载吧 ……