A characterization of Dirac morphisms(3)

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

[3]C.B¨ar,P.Gauduchon,A.Moroianu.Generalizedcylindersinsemi-

RiemannianandSpingeometry,Math.Zeit.249(2005),545–580.

[4]J.-M.Bismut,J.Cheeger.η-invariantsandtheiradiabaticlimits,J.Amer.

Math.Soc.2(1989),33–70.

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