Introduction to smooth manifolds PDF
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218 Graduate Texts in Mathematics Graduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: ColinAdams,WilliamsCollege,Williamstown,MA,USA AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada RuthCharney,BrandeisUniversity,Waltham,MA,USA IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA RogerE.Howe,YaleUniversity,NewHaven,CT,USA DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA JillPipher,BrownUniversity,Providence,RI,USA FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA AmieWilkinson,UniversityofChicago,Chicago,IL,USA GraduateTextsinMathematicsbridgethegapbetweenpassivestudyandcreative understanding, offering graduate-level introductions to advanced topics in mathe- matics.Thevolumesarecarefullywrittenasteachingaidsandhighlightcharacter- isticfeaturesofthetheory.Althoughthesebooksarefrequentlyusedastextbooks ingraduatecourses,theyarealsosuitableforindividualstudy. Forfurthervolumes: www.springer.com/series/136 John M. Lee Introduction to Smooth Manifolds Second Edition JohnM.Lee DepartmentofMathematics UniversityofWashington Seattle,WA,USA ISSN0072-5285 ISBN978-1-4419-9981-8 ISBN978-1-4419-9982-5(eBook) DOI10.1007/978-1-4419-9982-5 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2012945172 MathematicsSubjectClassification: 53-01,58-01,57-01 ©SpringerScience+BusinessMediaNewYork2003,2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Manifoldscropupeverywhereinmathematics.Thesegeneralizationsofcurvesand surfaces to arbitrarily many dimensions provide the mathematical context for un- derstanding“space”inallofitsmanifestations.Today,thetoolsofmanifoldtheory are indispensable in most major subfields of pure mathematics, and are becoming increasingly important in such diverse fields as genetics, robotics, econometrics, statistics, computer graphics, biomedical imaging, and, of course, the undisputed leader among consumers (and inspirers) of mathematics—theoretical physics. No longertheprovinceofdifferentialgeometersalone,smoothmanifoldtechnologyis nowabasicskillthatallmathematicsstudentsshouldacquireasearlyaspossible. Overthepastcenturyortwo,mathematicianshavedevelopedawondrouscollec- tionofconceptualmachinesthatenableustopeerevermoredeeplyintotheinvisi- bleworldofgeometryinhigherdimensions.Oncetheiroperationismastered,these powerfulmachinesenableustothinkgeometricallyaboutthe 6-dimensionalsolu- tionsetofapolynomialequationinfourcomplexvariables,orthe10-dimensional manifold of 5 5 orthogonal matrices, as easily as we think about the familiar ! 2-dimensionalsphere in R3. The price we pay for thispower, however,is that the machinesareassembledfromlayeruponlayerofabstractstructure.Startingwiththe familiar raw materials of Euclidean spaces, linear algebra, multivariable calculus, anddifferentialequations,onemustprogressthroughtopologicalspaces,smoothat- lases,tangentbundles,immersedandembeddedsubmanifolds,vectorfields,flows, cotangent bundles, tensors, Riemannian metrics, differential forms, foliations, Lie derivatives,Liegroups,Liealgebras,andmore—justtogettothepointwhereone can even think about studying specialized applications of manifold theory such as comparisontheory,gaugetheory,symplectictopology,orRicciflow. This book is designed as a first-year graduate text on manifold theory, for stu- dentswhoalreadyhaveasolidacquaintancewithundergraduatelinearalgebra,real analysis,andtopology.Ihavetriedtofocusontheportionsofmanifoldtheorythat willbeneededbymostpeoplewhogoontousemanifoldsinmathematicalorsci- entific research. I introduce and use all of the standard tools of the subject, and provemostofitsfundamentaltheorems,whileavoidingunnecessarygeneralization v vi Preface or specialization. I try to keep the approach as concrete as possible, with pictures andintuitivediscussionsofhowoneshouldthinkgeometricallyabouttheabstract concepts,butwithoutshyingawayfromthepowerfultoolsthatmodernmathemat- icshastooffer.Tofitinallofthebasicsandstillmaintainareasonablysanepace, Ihavehadtoomitorbarelytouchonanumberofimportanttopics,suchascomplex manifolds,infinite-dimensionalmanifolds,connections,geodesics,curvature,fiber bundles,sheaves,characteristicclasses,andHodgetheory.Thinkofthemasdessert, tobesavoredaftercompletingthisbookasthemaincourse. Toconveythebook’scompass,itiseasiesttodescribewhereitstartsandwhere itends.Thestartinglineisdrawnjustaftertopology:I assumethatthereaderhas had a rigorous introduction to general topology, including the fundamental group andcoveringspaces.OneconvenientsourceforthismaterialismyIntroductionto TopologicalManifolds[LeeTM],whichIwrotepartlywiththeaimofprovidingthe topologicalbackgroundneededforthisbook.Thereareotherbooksthatcoversim- ilarmaterialwell;IamespeciallyfondofthesecondeditionofMunkres’sTopology [Mun00].Thefinishlineisdrawnjustafterabroadandsolidbackgroundhasbeen established, but before getting into the more specialized aspects of any particular subject.Inparticular,IintroduceRiemannianmetrics,butIdonotgointoconnec- tions,geodesics,orcurvature.TherearemanyRiemanniangeometrybooksforthe interestedstudenttotakeupnext,includingonethatIwrote[LeeRM]withthegoal ofmovingexpedientlyinaone-quartercoursefrombasicsmoothmanifoldtheory tonontrivialgeometrictheoremsaboutcurvatureandtopology.Similarmaterialis covered in the last two chapters of the recent book by Jeffrey Lee (no relation) [LeeJeff09],anddoCarmo[dC92]coversabitmore.Formoreambitiousreaders, IrecommendthebeautifulbooksbyPetersen[Pet06],Sharpe[Sha97],andChavel [Cha06]. This subject is often called “differential geometry.” I have deliberately avoided usingthattermtodescribewhatthisbookisabout,however,becausethetermap- plies more properly to the study of smooth manifolds endowed with some extra structure—such as Lie groups, Riemannian manifolds, symplectic manifolds, vec- tor bundles, foliations—and of their properties that are invariant under structure- preservingmaps.AlthoughIdogiveallofthesegeometricstructurestheirdue(after all,smoothmanifoldtheoryisprettysterilewithoutsomegeometricapplications), Ifeltthatitwasmorehonestnottosuggestthatthebookisprimarilyaboutoneor allofthesegeometries.Instead,itisaboutdevelopingthegeneraltoolsforworking with smooth manifolds, so that the reader can go on to work in whatever field of differentialgeometryoritscousinsheorshefeelsdrawnto. Thereisnocanonicallinearpaththroughthismaterial.Ihavechosenanorder- ing of topics designed to establish a good technical foundation in the first half of thebook,sothatIcandiscussinterestingapplicationsinthesecondhalf.Oncethe firsttwelvechaptershavebeencompleted,thereissomeflexibilityinorderingthe remaining chapters. For example, Chapter 13 (Riemannian Metrics) can be post- poned if desired, although some sections of Chapters 15 and 16 would have to be postponedaswell.Ontheotherhand,Chapters19–21(DistributionsandFoliations, TheExponentialMap,andQuotientManifolds,respectively)couldinprinciplebe Preface vii inserted any time after Chapter 14, and much of the material can be covered even earlier if you are willing to skip over the references to differential forms. And the finalchapter(SymplecticManifolds)wouldmakesenseanytimeafterChapter17, orevenafterChapter14ifyouskipthereferencestodeRhamcohomology. Asyoumighthaveguessedfromthesizeofthebook,andwillquicklyconfirm when you start reading it, my style tends toward more detailed explanations and proofs than one typically finds in graduate textbooks. I realize this is not to every instructor’staste,butinmyexperiencemoststudentsappreciatehavingthedetails spelledoutwhentheyarefirstlearningthesubject.Thedetailedproofsinthebook provide students with useful models of rigor, and can free up class time for dis- cussionofthemeaningsandmotivationsbehindthedefinitionsaswellasthe“big ideas”underlyingsomeofthemoredifficultproofs.Thereareplentyofopportuni- tiesintheexercisesandproblemsforstudentstoprovideargumentsoftheirown. Ishouldsaysomethingaboutmychoicesofconventionsandnotations.Theold joke that “differential geometry is the study of properties that are invariant under change of notation” is funny primarily because it is alarmingly close to the truth. Everygeometerhashisorherfavoritesystemofnotation,andwhilethesystemsare allinsomesenseformallyisomorphic,thetransformationsrequiredtogetfromone toanotherareoftennotatallobvioustostudents.Becauseoneofmycentralgoals is to prepare students to read advanced texts and research articles in differential geometry, I have tried to choose notations and conventionsthat are as close to the mainstreamas Icanmakethemwithoutsacrificingtoomuchinternalconsistency. (One difference between this edition and the previous one is that I have changed a number of my notational conventions to make them more consistent with main- stream mathematicalusage.) When there are multipleconventionsin commonuse (suchasforthewedgeproductortheLaplaceoperator),Iexplainwhatthealterna- tivesareandalertthestudenttobeawareofwhichconventionisinusebyanygiven writer.Strivingfortoomuchconsistencyinthissubjectcanbeamistake,however, and I have eschewed absolute consistency whenever I felt it would get in the way ofeaseofunderstanding.Ihavealsointroducedsomecommonshortcutsatanearly stage, such as the Einstein summationconventionand the systematic confounding ofmapswiththeircoordinaterepresentations,bothofwhichtendtodrivestudents crazyatfirst,butpayoffenormouslyinefficiencylater. Prerequisites Thissubjectdrawsonmostofthetopicsthatarecoveredinatypicalundergraduate mathematicseducation.Theappendices(whichmostreadersshouldread,oratleast skim,first)containacursorysummaryofprerequisitematerialontopology,linear algebra,calculus,anddifferentialequations.Althoughstudentswhohavenotseen thismaterialbeforewillnotlearnitfromreadingtheappendices,Ihopereaderswill appreciate having all of the background material collected in one place. Besides giving me a convenient way to refer to results that I want to assume as known, it also gives the reader a splendid opportunity to brush up on topics that were once (hopefully)understoodbutmayhavefaded. viii Preface Exercisesand Problems This book has a rather large number of exercises and problems for the student to workout.Embeddedinthetextofeachchapterarequestionslabeledas“Exercises.” Theseare(mostly)shortopportunitiestofillingapsinthetext.Someofthemare routineverificationsthatwouldbetedioustowriteoutinfull,butarenotquitetrivial enoughtowarranttossingoffasobvious.Irecommendthatseriousreaderstakethe time at least to stop and convince themselves that they fully understand what is involvedindoingeachexercise,ifnottowriteoutacompletesolution,becauseit willmaketheirreadingofthetextfarmorefruitful. Attheendofeachchapterisacollectionof(mostly)longerandharderquestions labeled as “Problems.” These are the ones from which I select written homework assignmentswhenIteachthismaterial.Manyofthemwilltakehoursforstudentsto workthrough.Onlybydoingasubstantialnumberoftheseproblemscanonehope to absorb this material deeply. I have tried insofar as possible to choose problems that are enlightening in some way and have interesting consequences in their own right.Whentheresultofaproblemisusedinanessentialwayinthetext,thepage whereitisusedisnotedattheendoftheproblemstatement. Ihavedeliberatelynotprovidedwrittensolutionstoanyoftheproblems,either in the back of the book or on the Internet. In my experience, if written solutions to problems are available, even the most conscientious students find it very hard to resist the temptation to look at the solutions as soon as they get stuck. But it is exactlyatthatstageofbeingstuckthatstudentslearnmosteffectively,bystruggling togetunstuckandeventuallyfindingapaththroughthethicket.Readingsomeone else’ssolutiontooearlycangiveoneacomforting,butultimatelymisleading,sense ofunderstanding.Ifyoureallyfeelyouhaverunoutofideas,talkwithaninstructor, afellowstudent,oroneoftheonlinemathematicaldiscussioncommunitiessuchas math.stackexchange.com.Evenifsomeoneelsegivesyouasuggestionthatturnsout tobethekeytogettingunstuck,youwillstilllearnmuchmorefromabsorbingthe suggestion and working out the details on your own than you would from reading someoneelse’spolishedproof. AbouttheSecondEdition Those who are familiar with the first edition of this book will notice first that the topicshavebeensubstantiallyrearranged.ThisisprimarilybecauseIdecideditwas worthwhiletointroducethetwomostimportantanalytictools(theranktheoremand thefundamentaltheoremonflows)muchearlier,sothattheycanbeusedthroughout thebookratherthanbeingrelegatedtolaterchapters. Afewnewtopicshavebeenadded,notablySard’stheorem,sometransversality theorems,aproofthatinfinitesimalLiegroupactionsgenerateglobalgroupactions, amorethoroughstudyoffirst-orderpartialdifferentialequations,abrieftreatment ofdegreetheoryforsmoothmapsbetweencompactmanifolds,andanintroduction tocontactstructures.IhaveconsolidatedtheintroductorytreatmentsofLiegroups, Preface ix Riemannian metrics, and symplectic manifolds in chapters of their own, to make iteasiertoconcentrateonthespecialfeaturesofthosesubjectswhentheyarefirst introduced(althoughLiegroupsandRiemannianmetricsstillappearrepeatedlyin later chapters). In addition, manifolds with boundary are now treated much more systematicallythroughoutthebook. Apart from additions and rearrangement, there are thousands of small changes and also some large ones. Parts of every chapter have been substantially rewritten to improve clarity. Some proofs that seemed too labored in the original have been streamlined,whileothersthatseemedunclearhavebeenexpanded.Ihavemodified someofmynotations,usuallymovingtowardmoreconsistencywithcommonno- tationsintheliterature.Thereisanewnotationindexjustbeforethesubjectindex. There are also some typographical improvements in this edition. Most impor- tantly, mathematical terms are now typeset in bold italics when they are officially defined, to reflect the fact that definitions are just as important as theorems and proofs but fit better into the flow of paragraphs rather than being called out with special headings. The exercises in the text are now indicated more clearly with a specialsymbol(I),andnumberedconsecutivelywiththetheoremstomakethem easier to find. The symbol !, in addition to marking the ends of proofs, now also marks the ends of statements of corollaries that follow so easily that they do not needproofs;andIhaveintroducedthesymbol//tomarktheendsofnumberedex- amples.TheentirebookisnowsetinTimesRoman,supplementedbytheexcellent MathTimeProfessionalII mathematicsfontsfromPersonalTEX,Inc. Acknowledgments Many people have contributed to the development of this book in indispensable ways.IwouldliketomentionTomDuchamp,JimIsenberg,andSteveMitchell,all of whom generously shared their own notes and ideas about teaching this subject; and Gary Sandine, who made lots of helpful suggestions and created more than a third of the illustrations in the book. In addition, I would like to thank the many otherswhohavereadthebookandsenttheircorrectionsandsuggestionstome.(In theInternetage,textbookwritingbecomesevermoreacollaborativeventure.)And mostofall,IoweadebtofgratitudetoJudithArms,whohasimprovedthebookin countlesswayswithherthoughtfulandpenetratingsuggestions. Forthesakeoffuturereaders,Ihopeeachreaderwilltakethetimetokeepnotes of any mistakes or passages that are awkward or unclear, and let me know about themassoonasitisconvenientforyou.Iwillkeepanup-to-datelistofcorrections on my website, whose address is listed below. (Sad experience suggests that there willbeplentyofcorrectionsdespitemybesteffortstorootthemoutinadvance.)If thatsitebecomesunavailableforanyreason,thepublisherwillknowwheretofind me.Happyreading! Seattle,Washington,USA JohnM.Lee www.math.washington.edu/~lee
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