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ID：40849896
大小：1.34 MB
页数：33页
时间：2019-08-08
《Hodge Theory on Differentiable Manifolds英文》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
1、HodgeTheoryonDifferentiableManifoldsThisappendixismeanttoremindthereaderofafewbasicdefinitionsandfactsfromdifferentialgeometry,butitcannotreplaceanintroductiontothesub-ject.Weusetheopportunitytointroducetherelatednotationsusedthrough-outthetext.Noproofsaregiven,thematerialisfarfrombeingcompleteandt
2、hereaderisadvisedtogobacktoanyofthestandardtextbooksfordetails.fcDefinitionA.0.1Anm-dimensionalC-manifoldisatopologicalspaceMtogetherwithanopencoveringM=[jUiandhomeomorphisms3、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)=xdefinesatangen6、tvectorD1by-D7(/)=(d(foj)/dt)(O).282AHodgeTheoryonDifferentiableManifoldsAllthetangentspacesTXMgluetothetangentbundleTM={JxeMTXMwhichisanexampleofadifferentiablerealvectorbundleonM.DefinitionA.0.2LetMbeadifferentiablemanifold.AdifferentiablevectorbundleofrankronMconsistsofadifferentiablemanifoldE,a7、differen-tiablemap•K:E—>M,andthestructureofarealvectorspaceonanyfibreE(x):=1TT~(X),suchthatthereexistsanopencoveringM=(JUianddif-lrfeomorphismsipi'•n~(Ui)—*UixRwithpr^.o^=nandsuchthatforall1x£Uithemapipi(x)
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)=xdefinesatangen6、tvectorD1by-D7(/)=(d(foj)/dt)(O).282AHodgeTheoryonDifferentiableManifoldsAllthetangentspacesTXMgluetothetangentbundleTM={JxeMTXMwhichisanexampleofadifferentiablerealvectorbundleonM.DefinitionA.0.2LetMbeadifferentiablemanifold.AdifferentiablevectorbundleofrankronMconsistsofadifferentiablemanifoldE,a7、differen-tiablemap•K:E—>M,andthestructureofarealvectorspaceonanyfibreE(x):=1TT~(X),suchthatthereexistsanopencoveringM=(JUianddif-lrfeomorphismsipi'•n~(Ui)—*UixRwithpr^.o^=nandsuchthatforall1x£Uithemapipi(x)
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)=xdefinesatangen6、tvectorD1by-D7(/)=(d(foj)/dt)(O).282AHodgeTheoryonDifferentiableManifoldsAllthetangentspacesTXMgluetothetangentbundleTM={JxeMTXMwhichisanexampleofadifferentiablerealvectorbundleonM.DefinitionA.0.2LetMbeadifferentiablemanifold.AdifferentiablevectorbundleofrankronMconsistsofadifferentiablemanifoldE,a7、differen-tiablemap•K:E—>M,andthestructureofarealvectorspaceonanyfibreE(x):=1TT~(X),suchthatthereexistsanopencoveringM=(JUianddif-lrfeomorphismsipi'•n~(Ui)—*UixRwithpr^.o^=nandsuchthatforall1x£Uithemapipi(x)
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)
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