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1、INTRODUCTIONTOSMOOTHMANIFOLDSbyJohnM.LeeUniversityofWashingtonDepartmentofMathematicsJohnM.LeeIntroductiontoSmoothManifoldsVersion3.0December31,2000ivJohnM.LeeUniversityofWashingtonDepartmentofMathematicsSeattle,WA98195-4350USAlee@math.washington.eduhttp://www.math.washington.edu/~leec2000
2、byJohnM.LeeCorrectionstoIntroductiontoSmoothManifoldsVersion3.0byJohnM.LeeApril18,2001Page4,secondparagraphafterLemma1.1:Omitredundantthe."Page11,Example1.6:Inthethirdlineabovethesecondequation,changeforeachj"toforeachi."Page12,Example1.7,line5:Changemanifold"tosmoothmanifold."Pag
3、e13,Example1.11:Justbeforeandinthedisplayedequation,change'(')−1toji'(')−1(twice).ijPage21,Problem1-3:Changethede nitionofetoe(x)=−(−x).(Thisisstereographicprojectionfromthesouthpole.)Page24,5thlinebelowtheheading:multiples"ismisspelled.Page24,lastparagraphbeforeExercise2.1:Th
4、ereisasubtleproblemwiththede n-itionofsmoothmapsbetweenmanifoldsgivenhere,becausethisde nitiondoesn'tobviouslyimplythatsmoothmapsarecontinuous.Here'showto xit.ReplacethethirdsentenceofthisparagraphbyWesayFisasmoothmapifforanyp2M,thereexistcharts(U;')con-tainingpand(V;)containingF(p)suchth
5、atF(U)VandthecompositemapF'−1issmoothfrom'(U)to(V).Notethatthisde nitionimplies,inparticular,thateverysmoothmapiscontinuous:IfWNisanyopenset,foreachp2F−1(W)wecanchooseacoordinatedomainVWcontainingF(p),andthenthede nitionguaranteestheexistenceofacoordinatedomainUcontainingpsuchthatUF−
6、1(V)F−1(W),whichimpliesthatF−1(W)isopen."Page25,Lemma2.2:ChangethestatementofthislemmatoLetM,NbesmoothmanifoldsandletF:M!Nbeanymap.ShowthatFissmoothifandonlyifitiscontinuousandsatis esthefollowingcondition:Givenanysmoothatlasesf(U;')gandf(V;)gforMandN,respectively,eachmapF'−1issmootho
7、nitsdomainofde nition."Page30,line6:ChangeopologyofMf"totopologyofM."Page31,Example2.10(e), rstline:Changeomplex"toreal."Page36,Exercise2.9:Replacethe rstsentenceoftheexercisebythefollowing:ShowthatacoverfUgofXbyprecompactopensetsislocally niteifandonlyi
