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1、IntroductiontoConformalGeometryLecture2WeylCurvatureTensorCharacterizationofConformalFlatness(Mn;g)smoothRiemannianmanifold,n2Fundamentalobjects:Levi-CivitaconnectionrandRiemanncurvaturetensorRijklNotation:rkvj=vj;k=@kvjivijkCurvaturetensordenedbycommutingcovariantderivatives:vj;klvj;lk

2、=Rijklvi(indexraisedusinggij)Theorem:ThereexistlocalcoordinatesxisothatPg=(dxi)2ifandonlyifR=0.Therearemanyproofs.SeeVol.2ofSpivak'sAComprehensiveIntroductiontoDierentialGeom-etry.InRiemann'sproof,R=0arisesastheinte-grabilityconditionfortheapplicationofFrobenius'Theoremtoanoverdeterminedsys

3、temofpde's.2Nextsupposegivenaconformalclass[g]ofmetricsgbgifgb=e2!g,!2C1(M).Analogousquestion:giveng,underwhatcon-ditionsdothereexistlocalcoordinatessothatPg=e2!(dxi)2?Suchametricissaidtobelo-callyconformallyat.Ifn=2,alwaystrue:existenceofisothermalco-ordinates.Soassumen3.CertainlyR=0suc

4、esforexistenceofxi,!.Butwe'llseeonlyneedapieceofRtovanish.Decom-poseRintopieces:trace-freepartandtracepart.3(V;g)innerproductspace,g2S2VmetricSimpleranalogousdecompositionforS2V.Anys2S2Vhasatracetrgs=gijsij2R.Fact:anys2S2Vcanbeuniquelywrittens=s0+g;2R;s02S2V;trgs0=01trProof:Taketrace

5、:trgs=n,so=gs.nThens10=s(trgs)gworks.nSoS2V=S2VRg,whereS2V=fs:trgs=0g00Backtocurvaturetensors(linearalgebra):nDef:R=R24V:Rijkl=Rjikl=RijlkandRijkl+Riklj+Riljk=0{z}sameasRi[jkl]=0,where6Ri[jkl]=Rijkl+Riklj+RiljkRijlkRikjlRilkj4Havetr:R!S2V,(trR)ik=Ricik=gjlRijkl.Alsotr2R2R,tr2

6、R=S=gikRicikDenition:W=fW2R:trW=0gNeedawaytoembedS2V,!Rbymultiplyingbyg",analogoustoR,!S2Vby!g.Givens,t2S2V,denes?t2Rby:(s?t)ijkl=siktjlsjktilsiltjk+sjltikTheorem:R=WS2V?gProof:GivenR2R,wanttondW,PsothatRijkl=Wijkl+(PikgjlPjkgilPilgjk+Pjlgik)Taketr:Ricik=0+nPikPikPik+Pjjgik=

7、(n2)Pik+PjjgikAgain:S=(n2)Pjj+nPjj=2(n1)Pjj,sojS1RicSPj=andPik=ikgik2(n1)n22(n1)5Proposition:W=f0gifn=3.Proof:Choosebasissothatgij=ij.ConsidercomponentsWijkl.Twoofijklmustbeequal,sayi=k=1.NowW1j1l=0unlessj,l2f2;3g.trW=0=)W1j1l=W2j2lW3j3l.This