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1、HodgeTheoryonDifferentiableManifoldsThisappendixismeanttoremindthereaderofafewbasicdefinitionsandfactsfromdifferentialgeometry,butitcannotreplaceanintroductiontothesub-ject.Weusetheopportunitytointroducetherelatednotationsusedthrough-outthetext.Noproofsaregiven,thematerialisfarfrombeingcompleteandt

2、hereaderisadvisedtogobacktoanyofthestandardtextbooksfordetails.fcDefinitionA.0.1Anm-dimensionalC-manifoldisatopologicalspaceMtogetherwithanopencoveringM=[jUiandhomeomorphisms

3、

4、ucedifferentiablefunctionsonM.ByCMwedenotethesheafofdifferentiablefunctions,i.e.foranyopensubsetUCMthevalueofCMonUisthespaceofdifferentiablelfunctions/:U—>H.,i.e.functionssuchthat/o(p~:M.isdifferentiableforanychart(Ui,

5、manifolds.Inparticular,thereisthestalkCM,Xofthesheafofdifferentiablefunctionsateverypointx€M.ThetangentspaceTXMofMatthepointx(EMcanbedennedasTxM:=DerR(CM,x,K),thevectorspaceofderivationsD:CM,X—?K,i-e.ofM-linearmapssatisfyingD{f?g)=f(x)?D(g)+D(f)?g{x).'E.g.anycurve7:(-e,e)-?Mwith7(0)=xdefinesatangen

6、tvectorD1by-D7(/)=(d(foj)/dt)(O).282AHodgeTheoryonDifferentiableManifoldsAllthetangentspacesTXMgluetothetangentbundleTM={JxeMTXMwhichisanexampleofadifferentiablerealvectorbundleonM.DefinitionA.0.2LetMbeadifferentiablemanifold.AdifferentiablevectorbundleofrankronMconsistsofadifferentiablemanifoldE,a

7、differen-tiablemap?K:E—>M,andthestructureofarealvectorspaceonanyfibreE(x):=1TT~(X),suchthatthereexistsanopencoveringM=(JUianddif-lrfeomorphismsipi'?n~(Ui)—*UixRwithpr^.o^=nandsuchthatforall1x￡Uithemapipi(x)