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1、INTRODUCTIONTOSMOOTHMANIFOLDSbyJohnM.LeeUniversityofWashingtonDepartmentofMathematicsJohnM.LeeIntroductiontoSmoothManifoldsVersion3.0December31,2000ivJohnM.LeeUniversityofWashingtonDepartmentofMathematicsSeattle,WA98195-4350USAlee@math.washington.eduhttp://www.math.was
2、hington.edu/~leec2000byJohnM.LeePrefaceThisbookisanintroductorygraduate-leveltextbookonthetheoryofsmoothmanifolds,forstudentswhoalreadyhaveasolidacquaintancewithgeneraltopology,thefundamentalgroup,andcoveringspaces,aswellasbasicundergraduatelinearalgebraandrealanalysis
3、.Itisanaturalsequeltomyearlierbookontopologicalmanifolds[Lee00].Thissubjectisoftencalleddi erentialgeometry."Ihavemostlyavoidedthisterm,however,becauseitappliesmoreproperlytothestudyofsmoothmanifoldsendowedwithsomeextrastructure,suchasaRiemannianmet-ric,asymplecticstr
4、ucture,aLiegroupstructure,orafoliation,andofthepropertiesthatareinvariantundermapsthatpreservethestructure.Al-thoughIdotreatallofthesesubjectsinthisbook,theyaretreatedmoreasinterestingexamplestowhichtoapplythegeneraltheorythanasobjectsofstudyintheirownright.Astudentwho
5、 nishesthisbookshouldbewellpreparedtogoontostudyanyofthesespecializedsubjectsinmuchgreaterdepth.Thebookisorganizedroughlyasfollows.Chapters1through4aremainlyde nitions.Itisthebaneofthissubjectthattherearesomanyde nitionsthatmustbepiledontopofoneanotherbeforeanythingin-
6、terestingcanbesaid,muchlessproved.Ihavetried,nonetheless,tobringinsigni cantapplicationsasearlyandasoftenaspossible.The rstonecomesattheendofChapter4,whereIshowhowtogeneralizetheclassicaltheoryoflineintegralstomanifolds.Thenextthreechapters,5through7,presentthe rstoffo
7、urmajorfoundationaltheoremsonwhichallofsmoothmanifoldstheoryrests
8、theinversefunctiontheorem
9、andsomeapplicationsofit:tosubmanifoldthe-viPrefaceory,embeddingsofsmoothmanifoldsintoEuclideanspaces,approximationofcontinuousmapsbysmoothones,andquotientsofmanifoldsbygroupacti
10、ons.Thenextfourchapters,8through11,focusontensorsandtensor eldsonmanifolds,andprogressfromRiemannianmetricsthroughdi
