A characterization of Dirac morphisms

1、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. 8 2 ay m 5 g d. h t a

2、m 1 v 1 9 5 . 5 8 :0v i x r aacharacterizationofdiracmorphismse.loubeauandr.slobodeanuabstract.relatingthediracoperatorsonthetotalspaceandonthebasemanifoldofahorizontallyconformalsubmersion,wecharacterizediracmorphisms,i.e.mapswhichpullback(local)harmonicspinor eldsonto(local)harmonicspinor elds.1.i

3、ntroductionintroducedbyjacobi10in1848,harmonicmorphismsaremapswhichpullbacklocalharmonicfunctionsontoharmonicfunctionsand,morerecently,theywerecharacterizedbyfuglede6andishihara9ashorizontallyweaklyconformalharmonicmaps.theirdualnatureofanalyticalandgeometricalobjectshasledtoarichtheory(cf.2)whichha

4、sencouragedthestudyofvariousothermorphisms,thatismapspreservinggermsofcertaindi erentialoperators.thecentralroleofthediracoperatorindi erentialgeometryandmathematicalphysicscalledforthisapproachtobeappliedtoharmonicspinors.unlikepreviouscases,the rsthurdleistomakesenseofanotionofpull-backofspinorsby

5、amap.thisrequirestheidenti cationofthespinorbundlesinvolved,necessarilyrestrictingourinvestigationtohorizontallyconformalmapsbetweenriemannianmanifolds(cf.sec- tion2).combiningachainruleforthediracoperatorandalocalexistencelemma,weshowthatahorizontallyconformalsubmersionbetweenspinmanifoldsisadiracm

6、orphismifandonlyifitshori-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

7、) harmonic spinor fields. 2e.loubeauandr.slobodeanu isrelatedtothedilationfactor,inamannerreminiscentofthefun-damentalequationforharmonicmorphisms.weconcludewithsomesimpleexamplesbetweeneuclideanspacesandexplicitourresultsintheset-upof12,whichinspiredinitiallyourconstruction. 2.pull-backofaspinor le

8、t(mm,g)beaspinriemannianmanifold,thetwo-sheetedcov-eringspin(m) so(m)inducesadoublecover:pspin(m)m pso(m)mofthebundleofpositivelyorientedorthonormalframesbytheprincipalspin(m)-bundleoverm,suchthat(sg)=(s)(g), spspin(m)m,gspin(m).theassociatedbundlecl(m)=pso(m)mclmclmisthecli ordbundle,whereclmisthec

9、li ordalgebraandclmtherepresentationofso(m)intoaut(cl(rm),andthespinorbun-dleissm=pspin(m)msm,withthespinorialrepresentationofspin(m)onthecli ordmodulesm/2 m=c2(cf.11). aspinor eldisa(smooth)sectionofsm,:u m sm,(x)=sx,(x),wheresxpspin(m)misaspinorialframeatxmand:u sm,theequivalenceclassbeingde nedby

10、 s,=sg 1,(g), forallgspin(m).thecovariantderivative ej= iss,d(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

11、fields. acharacterizationofdiracmorphisms3 eeoverm,achoiceofspin-structureonanytwoofthemuniquelydeterminesaspinstructureonthethird(11).de nition1.asmoothmap:(mm,g)(nn,h)betweenrie-mannianmanifoldsisahorizontallyconformalmapif,atanypointxm,dxmapsthehorizontalspacehx=(kerdx)conformallyontot(x)n,i.e.dx

12、issurjectiveandthereexistsanumber(x)=0suchthat 2(h)x =(x)gx .hxhxhxhx thefunctionisthedilationofandtheorthogonalcomplementofhxistheverticaldistributionvx=kerdx. themeancurvaturesofthedistributionshandvaredenotedhandvandihistheintegrabilitytensorofh. a=1,.,m naframeva,xi ioftmwillbecalledadaptedifvav

13、,a=1,.,n 1,.,m nandxi i=1,.,nisthehorizontalliftbyofanorthonormalframexii=1,.,nonn. notethat1correspondstoriemanniansubmersions. wecallthemap:(mm,g1)(nn,h),whereg1= h+gv,theassociatedriemanniansubmersionof:(mm,g)(nn,h). sinceageneralsubmersion:(mm,g)(nn,h),betweenspinriemannianmanifolds,splitsthetan

14、gentbundletmintohv,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)togetherwithadimensio

15、ncountyieldthefollowingisomorphismsof(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

16、back (local) harmonic spinor fields onto (local) harmonic spinor fields. 4e.loubeauandr.slobodeanu foranymap:mnintoaspinmanifold,considerthepull-backspinorbundle 1sn=(x,s,)msn| s(s,)=(x), where sistheprojectionmapofsn. ifisariemanniansubmersion,thentheisomorphism 1sn=sh,duetotheidenti cationoforthon

17、ormalframes,simpli es(2)and(3)into(cf.4) (4)s+m= 1sn svsm= 1sn sv. remark1.whenisariemanniansubmersionwithtotallygeodesic bres,hiscomplete,nconnectedandthe bresareisometrictoariemannianmanifoldf.ifnandfarespinmanifolds,consideronmtheinducedspinstructureand,viatheisomorphism 1sn=sh, (2)and(3)read(see

18、12) s+m= 1sn sf,sm= 1sn sf. remark2.ifniseven,thecli ordalgebraclnpossessesanirre-duciblecomplexmodulesnofcomplexdimension2n/2,thecomplexspinormodule.whenrestrictedtocl0nthespinormoduledecomposes + intosn=snsn,thesubmodulesofspinorsofpositiveandnegative chirality,characterizedbytheactionofthevolumee

19、lement,onceanorientationisgiven.inparticular,thespingroupspin(n) cl0nacts + onsnandonsn(thespinorrepresentations). 0ifnisodd,thenclnhastwoinequivalentirreduciblemodulessnand 1sn,bothofcomplexdimension2(n 1)/2(againdistinguishedbytheac-tionofthevolumeelement).whenrestrictedtocl0nthetwomodules 0become

20、equivalentandwesimplywritesn=snandthespinorrepre- sentation:spin0(n)aut(sn)becomesirreducible. ifniseven,thensn+1pullsbacktosnunderthealgebraisomorphismcln=cl0n+1.inotherwords,wecanregardsn+1asthespinorrep-resentationofcln,providedwede netheactionofclnonsn+1byv e0v. similarly,ifnisodd,thentheactiono

21、fthevolumeformshowsthat+ 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

22、fields. acharacterizationofdiracmorphisms5 when:(mn+1,g)(nn,h)isariemanniansubmersionwithone-dimensional bresintoaspinmanifoldnn,themanifoldmn+1inheritsanaturalspinstructure,cf13.moreover,ifniseven,thensm= 1snand,whennisodd,thens+m= 1sn. theseidenti cationsjustifythefollowingde nition. de nition2(ri

23、emanniansubmersions).let:(mm,g)(nn,h)beariemanniansubmersionbetweenspinmanifoldsandendowtheverticalbundlewiththeinducedspinstructure(ifm=n+1,weconsideronmthenaturalspinstructureinheritedfromn).let= s,bea(local)spinor eldonn.sincealocalspinframes=xii=1,.,nonnliftstoanadaptedspinframeonm s =x i,vaa=1,

24、.,m ni=1,.,npspin(n)z2spin(m n)m| 1u, wherex iisthehorizontalliftofxiandvaa=1,.,m nisanorthonor-malframeofv,wede nethepull-back ifm 1=n,thesection = s oftobe, ofthebundle 1sn,identi edwithsm,whenniseven,andwiths+m,whennisodd. ifm n2,thesection =s , =( ) in 1sn sv,beingasectionofsv,identi edwiths+m,w

25、hennisoddandmeven,andwithsmotherwise. remark3.notethatcli ordmultiplicationwiththiskindofspinor eldsisgivenby x =x ;v =i ) v, whenm n2,wherex isthehorizontalliftofthevector eldx(tn)and relating the dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we

26、 characterize dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields. 6e.loubeauandr.slobodeanu (cf.11) :sm1sm (s,)=(s), where(s)=ei=ei1,vaifs=ei1,va,thebundleisometryinducedbythespin-equivariantmap :pspin(n)z2spin(m n)m1pspin(n)z2spin(m n)m giv

27、enbythenaturalcorrespondencebetweenadaptedorthogonalframeswithrespecttothetwometrics:e1 i1= ei,va1=va. thecli ordmultiplicationwillbegivenby ei= ei11 ,va=(va1), where= 1. de nition3(horizontallyconformalsubmersions).let:(mm,g)(nn,h)beahorizontallyconformalsubmersionbetweenspinmani-foldsandendowtheve

28、rticalbundlewiththeinducedspinstructure.let=s,bea(local)spinor eldonn.thepull-backofis = 1,where 1isthepull-backofbytheassociatedriemanniansubmersion:(mm,g1)(nn,h)andthebundleisometrybetweensm1andsm. 3.diracmorphismswithhighdimensionalfibres throughoutthissectionhas bresofdimensionatleasttwo. de nit

29、ion4.ahorizontallyconformalsubmersion:(mm,g)(nn,h)betweenspinmanifoldsiscalledadiracmorphismifforanylocalharmonicspinorde nedonu n,suchthat 1(u)= andthereexistsasection(sv) v-parallelinhorizontaldirectionsandwithdv n relating the dirac operators on the total space and on the base manifold of a horiz

30、ontally 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).let:(mm,g)(nn,h)beahorizontallyconformalsubmersionofdilation(m n2)an

31、da(local)spinor isthepull-backofby,withrespecttosomesection eldonn.if (sv),then (5)n =ddm14 ih +(2,h whereeii=1.,nisalocalorthonormalhorizontalframeonmand h denotes nihij=1eieji(ei,ej)(thestandardactionof vector-valued2-formsonspinor elds). proof.letbeahorizontallyconformalsubmersionofdilation.let a

32、=1,.,m nxii=1,.,nbeanorthonormalframeon(n,h)andva,xi i=1,.,n anorthonormaladaptedframeon(m,g1),whereg1=h+gv.with a=1,.,m nrespecttothemetricg,va,xi iisanorthonormaladapted=1,.,n frame.denoteby and 1the(spinorial)connectionscorrespondingtogandg1,andnoteei1=xi ,ei=xi . ,forthepull-backspinor eld = asd

33、ms,dm 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 =dm =i=1n i=1 1n eiei( ) ) +

34、ei eim n a=1 va va(h0)+ n,m n 2 i,j,a=1 +1 eig( eiej,va)ejva(h2) n,m n 4 i,j,a=1 +1 vag( vaei,ej)eiej(v1) 4 a,b,c=1 notethat 1m n vag( vavb,vc)vbvc(v3).n=(dn) d 1 =(xixi( )+g1( xi xj,xk)xixjxk( ) , nwhereg1( 1 xj,xk)=h( xxj,xk).xii thecomputationbreaksdowninto vesteps: step1: n (h0)+(h1)+(h3)=dn 1 r

35、elating 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=1eiei( )

36、+( ) ei() +1 4 i,j,k=1 thelasttermcanberewritten 1n j k 1xk()i xj()ixixjxk2 . and,sincegradh()=2gradh1(),itbecomes n 1 2 1,gradh1() = n 1 gradh1() 2 v. relating the dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize dirac morphisms, i.

37、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)hvb+vava,vbhvb ab=1ab=1m n va( vavb)hvb 2ab=1 step3: (8) v.(v0)+(v3)=( m n 1 ) vava()+ ) va v va. step4: (9)(h2

38、)=11 asforstep2,wehave (h2)=1h .2 2 ij=1n eiejih(ei,ej)n h i,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 field

39、s. acharacterizationofdiracmorphisms11 since (v1)=1 n,m n 4 i,j,a=1 =1 g(va,ei,ej) g( eiej,va)vaeiej2 va()(h2). butg(va,ei,ej)=g(va,xi ,xj)= m n a=1 44 4h 4 summingupthese vestepsyieldsthechainrule. ih. gradv(ln)1 va(ln)va11 ageneralizationof1,proposition2.4tovector-valuedfunctionsyieldslocalexisten

40、ceofharmonicspinorswithprescribedvalue.lemma2(localexistence).foranypointpmand0spm,thereexistsanopenneighbourhooduofpandaharmonicspinor:usmsuchthat(p)=0. theorem1(characterizationforhorizontallyconformalsubmersions).ahorizontallyconformalsubmersion:(m,g) (n,h)betweenspinmanifoldsisadiracmorphismifan

41、donlyifitshorizontaldistri-butionisintegrableand (11)(m n)v+(n 1)gradh(ln)=0, wherevisthemeancurvatureofthe bres. proof.letbeahorizontallyconformalsubmersionwithintegrablehorizontaldistributionandsuchthat(11)issatis ed.inthiscase,the relating the dirac operators on the total space and on the base ma

42、nifold 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 eston+ =ddmn i=1ei( ) vei+(2.h letbealocalharmonicspinorsuchthatthereexistsasect

43、ion(sv) v-parallelinhorizontaldirectionsandwithdv n 2h=0,wehave,accordingto(5) n 0= 1 4 xi xj( ) ih(xi ,xj). ij=1 puttingx=(m n)v+(n 1)gradh(ln)andvij=ih(xi ,xj), (12)becomes 0= 1 4 ij=1 butsincevijisvertical,xandxi xjvijhavedi erentdegrees, necessarilyx=0andvij=0.therefore,ifahorizontallyconformals

44、ubmersionisdiracmorphism,itmustsatisfyequation(11)andhaveintegrablehorizontaldistribution. atpmcanbeprescribed,theaboveequationasthevalueofimpliesn 0= 1xi xjvij.4ij=1n xi xjvij.remark5.(1)notetheanalogybetweenequation(11)andthe fundamentalequationforharmonicmorphismsin2. (2)compareformula(5)with4,(4

45、.26)and12,1.1.1. (3)ifthe bresaretotallygeodesic,theintegrabilityofthehor- izontaldistributionmakesthesection“basictransversallyharmonic”,asintroducedin7. relating the dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize dirac morphisms,

46、 i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields. acharacterizationofdiracmorphisms13 corollary1.ariemanniansubmersion:(mm,g)(nn,h)be-tweenspinmanifoldsisadiracmorphismifandonlyifits bresareminimalanditshorizontaldistributionisintegrable. recallthatifisar

47、iemanniansubmersionthenh=0. remark6.ifthedilationfunctionisaprojectablefunction(i.e.v()=0),theconformalinvarianceofthediracoperator(11)allowsacorrespondencebetweenharmonicspinorsofthespacesinvolvedinthecommutativediagrambelow. (m,2 h+gv)1m-(m, h+gv) ?1n 2h)(n,(n,h) 4.diracmorphismswithone-dimensiona

48、lfibres inthissectionm=n+1. de nition5.ahorizontallyconformalsubmersion:(mn+1,g)(nn,h)betweenspinmanifoldsiscalledadiracmorphismifforanylocalharmonicspinorde nedonu n,suchthat 1(u)= and = 1isaharmonicspinoron 1(u) m,wherethepullback 1isthepullbackofbytheassociatedriemanniansubmersion. lemma3.(chainr

49、ule).let:(mn+1,g)(nn,h)beahorizon-tallyconformalsubmersionofdilationanda(local)spinor eldonn,then 1n =ddm ih.(13)4 proof.takexii=1,.,nanorthonormalframeon(nn,h)andv,ei=xi i=1,.,nanadaptedframeon(mm,g).letbea(local)spinor itspullbackby.theproofissimilartotheproofof eldonnand relating the dirac operat

50、ors 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)=),(v0)=0andtheterms(h3),(v3)donotapp

51、ear. eiei( notethath =i h 2 . theorem2.ahorizontallyconformalsubmersion:(mn+1,g)(nn,h)betweenspinmanifoldsisadiracmorphismifandonlyifitshorizontaldistributionisintegrableandminimal,and v+(n 1)gradh(ln)=0, wherevisthemeancurvatureofthe bres. proof.theargumentissimilartotheoneoftheorem1,exceptthat x=v

52、+(n 1)gradh(ln)+nh. observethatx=0ifandonlyifv+(n 1)gradh(ln)=0andh=0,astheybelongtoorthogonaldistributions. corollary2.ariemanniansubmersion:(mn+1,g)(nn,h)betweenspinmanifoldsisadiracmorphismifandonlyifitshorizontaldistributionisintegrableandthe bresareminimal. remark7.supposethatisariemanniansubme

53、rsion. (1)thechainrule(13)givesustheformulaof13(wherethe bresareminimal) (14)dm =d n 1 4i n x jax jvj=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 ,inaglobalspinframe(e=1 thediracoperatoronr2 02=dr=(e1) y2 = + y2)onr2,and representation,thediracoperatoronris dr3 x13, x3,andwiththepauli=ik x3 x1 i x1 x1= 2 x2= 2 x3= 2 let:r4 r2