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1、INTRODUCTIONTOSMOOTHMANIFOLDSbyJohnM.LeeUniversityofWashingtonDepartmentofMathematicsJohnM.LeeIntroductiontoSmoothManifoldsVersion3.0December31,2000ivJohnM.LeeUniversityofWashingtonDepartmentofMathematicsSeattle,WA98195-4350USAlee@math.washington.eduhttp://www.math.washington.edu/~leec200
2、0byJohnM.LeePrefaceThisbookisanintroductorygraduate-leveltextbookonthetheoryofsmoothmanifolds,forstudentswhoalreadyhaveasolidacquaintancewithgeneraltopology,thefundamentalgroup,andcoveringspaces,aswellasbasicundergraduatelinearalgebraandrealanalysis.Itisanaturalsequeltomyearlierbookontopo
3、logicalmanifolds[Lee00].Thissubjectisoftencalleddi erentialgeometry."Ihavemostlyavoidedthisterm,however,becauseitappliesmoreproperlytothestudyofsmoothmanifoldsendowedwithsomeextrastructure,suchasaRiemannianmet-ric,asymplecticstructure,aLiegroupstructure,orafoliation,andofthepropertiestha
4、tareinvariantundermapsthatpreservethestructure.Al-thoughIdotreatallofthesesubjectsinthisbook,theyaretreatedmoreasinterestingexamplestowhichtoapplythegeneraltheorythanasobjectsofstudyintheirownright.Astudentwho nishesthisbookshouldbewellpreparedtogoontostudyanyofthesespecializedsubjectsinm
5、uchgreaterdepth.Thebookisorganizedroughlyasfollows.Chapters1through4aremainlyde nitions.Itisthebaneofthissubjectthattherearesomanyde nitionsthatmustbepiledontopofoneanotherbeforeanythingin-terestingcanbesaid,muchlessproved.Ihavetried,nonetheless,tobringinsigni cantapplicationsasearlyandas
6、oftenaspossible.The rstonecomesattheendofChapter4,whereIshowhowtogeneralizetheclassicaltheoryoflineintegralstomanifolds.Thenextthreechapters,5through7,presentthe rstoffourmajorfoundationaltheoremsonwhichallofsmoothmanifoldstheoryrests
7、theinversefunctiontheorem
8、andsomeapplicationsofit:tosu
9、bmanifoldthe-viPrefaceory,embeddingsofsmoothmanifoldsintoEuclideanspaces,approximationofcontinuousmapsbysmoothones,andquotientsofmanifoldsbygroupactions.Thenextfourchapters,8through11,focusontensorsandtensor eldsonmanifolds,andprogressfromRiemannianmetricsthroughdi
