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首页 Fundamentals of Differential Geometry (Serge Lan…

Fundamentals of Differential Geometry (Serge Lang, Springer, 1999).pdf

Fundamentals of Differential Ge…

mathworker
2010-04-11 0人阅读 举报 0 0 0 暂无简介

简介:本文档为《Fundamentals of Differential Geometry (Serge Lang, Springer, 1999)pdf》,可适用于工程科技领域

SergeLangFundamentalsofDifferentialGeometryWithIllustrationslllllllllllllllllllllllulllllllYlllllllllllllllllllllIllllllliylMATgEqSpringerSergeLangDepartmentofMathematicsYaleUniversityNewHaven,CTUSAEditorialBoardSAxlerFWGehringKARibetMathematicsDepartmentMathematicsDepartmentMathematicsDepartmentSanFranciscoStateEastHallUniversityofCaliforniaSanFrancisco,CAAnnArbor,MIUSAUSAUSAUniversityUniversityofMichiganatBerkeleyBerkeley,CAMathematicsSubjectClassification():LibraryofCongressCataloginginPublicationDataLang,Serge,FundamentalsofdifferentialgeometrySergeLangpcm(Graduatetextsinmathematics:)IncludesbibliographicalreferencesandindexISBNX(alkpaper)Geometry,DifferentialITitleSeriesQAL’dcPrintedonacidfreepaperSpringerVerlagNewYork,IncAllrightsreservedThisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermissionofthepublisher(SpringerVerlagNewYork,Inc,FifthAvenue,NewYork,NY,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysisUseinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbiddenTheuseofgeneraldescriptivenames,tradenames,trademarks,etc,inthispublication,eveniftheformerarenotespeciallyidentified,isnottobetakenasasignthatsuchnames,asunderstoodbytheTradeMarksandMerchandiseMarksAct,mayaccordinglybeusedfreelybyanyoneProductioncoordinatedbyBrianHoweandmanagedbyTerryKornakmanufacturingsupervisedbyJeffreyTaubTypesetbyAscoTradeTypesettingLtd,HongKongPrintedandboundbyEdwardsBrothers,Inc,AnnArbor,MIPrintedintheUnitedStatesofAmericaIISBNXSpringerVerlagNewYorkBerlinHeidelbergSPINForewordThepresentbookaimstogiveafairlycomprehensiveaccountofthefundamentalsofdifferentialmanifoldsanddifferentialgeometryThesizeofthebookinfluencedwheretostop,andtherewouldbeenoughmaterialforasecondvolume(thisisnotathreat)Atthemostbasiclevel,thebookgivesanintroductiontothebasicconceptswhichareusedindifferentialtopology,differentialgeometry,anddifferentialequationsIndifferentialtopology,onestudiesforinstancehomotopyclassesofmapsandthepossibilityoffindingsuitabledifferentiablemapsinthem(immersions,embeddings,isomorphisms,etc)Onemayalsousedifferentiablestructuresontopologicalmanifoldstodeterminethetopologicalstructureofthemanifold(forexample,AlaSmaleSm)Indifferentialgeometry,oneputsanadditionalstructureonthedifferentiablemanifold(avectorfield,aspray,aform,aRiemannianmetric,adlib)andstudiespropertiesconnectedespeciallywiththeseobjectsFormally,onemaysaythatonestudiespropertiesinvariantunderthegroupofdifferentiableautomorphismswhichpreservetheadditionalstructureIndifferentialequations,onestudiesvectorfieldsandtheirintegralcurves,singularpoints,stableandunstablemanifolds,etcAcertainnumberofconceptsareessentialforallthree,andaresobasicandelementarythatitisworthwhiletocollectthemtogethersothatmoreadvancedexpositionscanbegivenwithouthavingtostartfromtheverybeginningsThoseinterestedinabriefintroductioncouldrunthroughChapters,,IV,V,VII,andmostofPartIonvolumeforms,Stokes’theorem,andintegrationTheymayalsoassumeallmanifoldsfinitedimensionalChartsandlocalcoordinatesAchartonamanifoldisclassicallyarepresentationofanopensetofthemanifoldinsomeeuclideanspace,FOREWORDviUsingachartdoesnotnecessarilyimplyusingcoordinatesChartswillbeusedsystematicallyItwillbeobservedequallysystematicallythatfinitedimensionalityisherebynotusedItispossibletolaydownatnoextracostthefoundations(andmuchmorebeyond)formanifoldsmodeledonBanachorHilbertspacesratherthanfinitedimensionalspacesInfact,itturnsoutthattheexpositiongainsconsiderablyfromthesystematiceliminationoftheindiscriminateuseoflocalcoordinatesXI,,x,anddxl,,dx,Thesearereplacedbywhattheystandfor,namelyisomorphismsofopensubsetsofthemanifoldonopensubsetsofBanachspaces(localcharts),andalocalanalysisofthesituationwhichismorepowerfulandequallyeasytouseformallyInmostcases,thefinitedimensionalproofextendsatoncetoaninvariantinfinitedimensionalproofFurthermore,instudyingdifferentialforms,oneneedstoknowonlythedefinitionofmultilinearcontinuousmapsAnabuseofmultilinearalgebrainstandardtreatisesarisesfromanunnecessarydoubledualizationandanabusiveuseofthetensorproductIdon’tpropose,ofcourse,todoawaywithlocalcoordinatesTheyareusefulforcomputations,andarealsoespeciallyusefulwhenintegratingdifferentialforms,becausethedxlAAdx,correspondstothedxldx,ofLebesguemeasure,inorientedchartsThusweoftengivethelocalcoordinateformulationforsuchapplicationsMuchoftheliteratureisstillcoveredbylocalcoordinates,andIthereforehopethattheneophytewillthusbehelpedingettingacquaintedwiththeliteratureIalsohopetoconvincetheexpertthatnothingislost,andmuchisgained,byexpressingone’sgeometricthoughtswithouthidingthemunderanirrelevantformalismIamawareofawidespreadapprehensivereactionthemomentsomegeometersorstudentsseethewords“Banachspace”or“Hilbertmanifold”Asapossiblepalliative,IsuggestreadingthematerialassumingfromthestartthatBanachspacemeansfinitedimensionalspaceoverthereals,andHilbertmanifoldorRiemannianmanifoldmeansafinitedimensionalmanifoldwithametric,withthelocalconstantmodelbeingordinaryeuclideanspaceTheseassumptionswillnotmakeanyproofshorterOnemajorfunctionoffindingproofsvalidintheinfinitedimensionalcaseistoprovideproofswhichareespeciallynaturalandsimpleinthefinitedimensionalcaseEvenforthosewhowanttodealonlywithfinitedimensionalmanifolds,IurgethemtoconsidertheproofsgiveninthisbookInmanycases,proofsbasedoncoordinatefreelocalrepresentationsinchartsareclearerthanproofswhicharerepletewiththeclawsofaratherunpleasantpryinginsectsuchasIIndeed,thebilinearmapassociatedwithaspray(whichisthequadraticmapcorrespondingtoasymmetricconnection)satisfiesquiteanicelocalformalisminchartsIthinkthelocalrepresentationofthecurvaturetensorasin~OPositionofChapterIXshowstheefficiencyofthisformalismanditssuperiorityoverFOREWORDviilocalcoordinatesReadersmayalsofinditinstructivetocomparetheproofofPropositionofChapterIXconcerningtherateofgrowthofJacobifieldswithmoreclassicalonesinvolvingcoordinatesasinHe,ppApplicationsinInfiniteDimensionItisprofitabletodealwithinfinitedimensionalmanifolds,modeledonaBanachspaceingeneral,aselfdualBanachspaceforpseudoRiemanniangeometry,andaHilbertspaceforRiemanniangeometryInthestandardpseudoRiemannianandRiemanniantheory,readerswillnotethatthedifferentialtheoryworksintheseinfinitedimensionalcases,withtheHopfRinowtheoremasthesingleexception,butnottheCartanHadamardtheoremanditscorollariesOnlywhenonecomestodealingwithvolumesandintegrationdoesfinitedimensionalityplayamajorroleEvenifviathephysicistswiththeirFeynmanintegrationoneeventuallydevelopsacoherentanalogoustheoryintheinfinitedimensionalcase,therewillstillbesomethingspecialaboutthefinitedimensionalcaseThefailureofHopfRinowintheinfinitedimensionalcaseisduetoaphenomenonofpositivecurvatureThevalidityofCartanHadamardinthecaseofnegativecurvatureisaverysignificantfact,anditisonlyrecentlybeingrealizedasprovidingasettingformajorapplicationsItisageneralphenomenonthatspacesparametrizingcertainstructuresareactuallyinfinitedimensionalCartanHadamardspaces,inmanycontexts,egTeichmullerspaces,spacesofRiemannianmetrics,spacesofKahlermetrics,spacesofconnections,spacesassociatedwithcertainpartialdifferentialequations,adlibCfforinstancetheapplicationtotheKdVequationinScTZ,andthecommentsattheendofChapterXI,$concerningotherapplicationsActually,theuseofinfinitedimensionalmanifoldsinconnectionwithTeichmullerspacesdatesbacksometime,becauseasshownbyBers,thesespacescanbeembeddedassubmanifoldsofacomplexBanachspaceCfGa,viViewingtheseasCartanHadamardmanifoldscomesfromnewerinsightsForfurthercommentsonsomerecentaspectsoftheuseofinfinitedimension,includingreferencestoKlingenberg’sbookKl,seetheintroductiontoChapterXIIIOfcourse,thereareotherolderapplicationsoftheinfinitedimensionalcaseSomeofthemaretothecalculusofvariationsandtophysics,forinstanceasinAbrahamMarsdenAbMItmayalsohappenthatonedoesnotneedformallytheinfinitedimensionalsetting,butthatitisusefultokeepinmindtomotivatethemethodsandapproachtakeninvariousdirectionsForinstance,bythedeviceofusingcurves,onecanreducewhatisapriorianinfinitedimensionalquestiontoordinarycalculusinfinitedimensionalspace,asinthestandardvariationformulasgiveninChapterXI,$VlllFOREWORDSimilarly,theproperdomainforthegeodesicpartofMorsetheoryistheloopspace(orthespaceofcertainpaths),viewedasaninfinitedimensionalmanifold,butasubstantialpartofthetheorycanbedevelopedwithoutformallyintroducingthismanifoldThereductiontothefinitedimensionalcaseisofcourseaveryinterestingaspectofthesituation,fromwhichonecandeducedeepresultsconcerningthefinitedimensionalmanifolditself,butitstopsshortofacompleteanalysisoftheloopspace(CfBootBo,MilnorMi)SeealsothepapersofPalaisPaandSmaleSmInaddition,giventwofinitedimensionalmanifoldsX,YitisfruitfultogivethesetofdifferentiablemapsfromXtoYaninfinitedimensionalmanifoldstructure,aswasstartedbyEellsEe,Ee,Ee,EeS,andEeBysodoing,onetranscendsthepurelyformaltranslationoffinitedimensionalresultsgettingessentiallynewones,whichwouldinturnaffectthefinitedimensionalcaseForotherconnectionswithdifferentialgeometry,seeElFoundationsforthegeometryofmanifoldsofmappingsaregiveninAbraham’snotesofSmale’slecturesAbandPalais’smonographPaFormorerecentapplicationstocriticalpointtheoryandsubmanifoldgeometry,seePaTInthedirectionofdifferentialequations,theextensionofthestableandunstablemanifoldtheoremtotheBanachcase,alreadymentionedasapossibilityinearlierversionsofDiferentiulManifolds,wasprovedquiteelegantlybyIrwinIr,followingtheideaofPughandRobbinfordealingwithlocalflowsusingtheimplicitmappingtheoreminBanachspacesIhaveincludedthePughRobbinproof,butrefertoIrwin’spaperforthestablemanifoldtheoremwhichbelongsattheverybeginningofthetheoryofordinarydifferentialequationsThePughRobbinproofcanalsobeadjustedtoholdforvectorfieldsofclassH*(Sobolevspaces),ofimportanceinpartialdifferentialequations,asshownbyEbinandMarsdenEbMItisastandardremarkthattheCmfunctionsonanopensubsetofaeuclideanspacedonotformaBanachspaceTheyformaFrkchetspace(denumerablymanynormsinsteadofone)Ontheotherhand,theimplicitfunctiontheoremandthelocalexistencetheoremfordifferentialequationsarenottrueinthemoregeneralcaseInordertorecoversimilarresults,amuchmoresophisticatedtheoryisneeded,whichisonlybeginningtobedeveloped(CfNash’spaperonRiemannianmetricsNa,andsubsequentcontributionsofSchwartzScandMoserMo)Inparticular,someadditionalstructuremustbeadded(smoothingoperators)CfalsomyBourbakiseminartalkonthesubjectLaThisgoesbeyondthescopeofthisbook,andpresentsanactivetopicforresearchOntheotherhand,forsomeapplications,onemaycompletetheCmspaceunderasuitableHilbertspacenorm,dealwiththeresultingHilbertFOREWORDixmanifold,andthenuseanappropriateregularitytheoremtoshowthatsolutionsoftheequationunderstudyactuallyareCmIhaveemphasizeddifferentialaspectsofdifferentialmanifoldsratherthantopologicalonesIamespeciallyinterestedinlayingdownbasicmaterialwhichmayleadtovarioustypesofapplicationswhichhavearisensincethesixties,vastlyexpandingtheperspectiveondifferentialgeometryandanalysisForinstance,IexpectthebooksBGVandGitobeonlythefirstofmanytopresenttheaccumulatedvisionfromtheseventiesandeighties,aftertheworkofAtiyah,Bismut,Bott,Gilkey,McKean,Patodi,Singer,andmanyothersNegativeCurvatureMosttextsemphasizepositivecurvatureattheexpenseofnegativecurvatureIhavetriedtoredressthisimbalanceInalgebraicgeometry,itiswellrecognizedthatnegativecurvatureamountsmoreorlessto“generaltype”Forinstance,curvesofgenusarespecial,curvesofgenusaresemispecial,andcurvesofgenusareofgeneraltypeThusIhavedevotedanentirechaptertothefundamentalexampleofaspaceofnegativecurvatureActually,IprefertoworkwiththeRiemanntensorIuse“curvature”simplyasacodewordwhichiseasilyrecognizablebypeopleinthefieldFurthermore,Iincludeacompleteaccountoftheequivalencebetweenseminegativecurvature,themetricincreasingpropertyoftheexponentialmap,andtheBruhatTitssemiparallelogramlawThird,IemphasizetheCartanHadamardfurtherbygivingaversionforthenormalbundleofatotallygeodesicsubmanifoldIamindebtedtoWuforvaluablemathematicalandhistoricalcommentsonthistopicThereareseveralcurrentdirectionswherebyspacesofnegativecurvaturearethefundamentalbuildingblocksofsometheoriesTheyarequotientsofCartanHadamardspacesImyselfgotinterestedindifferentialgeometrybecauseofthejointworkwithJorgenson,whichnaturallyledustosuchspacesfortheconstructionandtheoryofcertainzetafunctionsQuitegenerally,wewereledtoconsiderspaceswhichadmitastratificationsuchthateachstratumisaquotientofaCartanHadamardspace(especiallyasymmetricspace)byadiscretegroupThatsuchstratificationsexistverywidelyisafactnotgenerallytakenintoaccountForinstance,itisatheoremofGrifEthsthatgivenanalgebraicvarietyoverthecomplexnumbers,thereexistsaproperZariskiclosedsubsetwhosecomplementisaquotientofacomplexboundeddomain,sointhisway,everyalgebraicvarietyadmitsastratificationasabove,evenwithconstantnegativecurvatureThurston’sapproachtomanifoldscouldbeviewedfromourperspectivealsoThegeneralproblemthenariseshowzetafunctions,spectralinvariants,homotopyandhomologyinvariants,adXFOREWORDlibbehavewithrespecttostratifications,whetheradditivelyorotherwiseIntheJorgensonLangprogram,weassociateazetafunctiontoeachstratum,andthezetafunctionsoflowerstrataaretheprincipalfudgefactorsinthefunctionalequationofthezetafunctionassociatedtothemainstratumThespectralexpansionoftheheatkernelamountstoathetarelation,andwegetthezetafunctionbytakingtheGausstransformofthethetarelationFromaquitedifferentperspective,certainnatural“moduli”spacesforstructuresonfinitedimensionalmanifoldshaveaverystrongtendencytobeCartanHadamardspaces,forinstancethespaceofRiemannianmetrics,spacesofKahlermetrics,spacesofconnections,etcwhichdeservetobeincorporatedinageneraltheoryInanycase,Ifindtheexclusivehistoricalemphasisatthefoundationallevelonpositivecurvature,spheres,projectivespaces,grassmanians,attheexpenseofquotientsofCartanHadamardspaces,tobemisleadingastothewaymanifoldsarebuiltupTimewilltell,butIdon’tthinkwe’llhavetowaitverylongbeforearadicalchangeofviewpointbecomesprevalentNewHaven,SERGELANGAcknowledgmentsIhavegreatlyprofitedfromseveralsourcesinwritingthisbookThesesourcesincludesomefromthes,andsomemorerecentonesFirst,IoriginallyprofitedfromDieudonnC’sFoundationsofModernAnalysis,whichstartedtoemphasizetheBanachpointofviewSecond,IoriginallyprofitedfromBourbaki’sFasciculederksultutsBouforthefoundationsofdifferentiablemanifoldsThisprovidesagoodguideastowhatshouldbeincludedIhavenotfolloweditentirely,asIhaveomittedsometopicsandaddedothers,butonthewhole,IfounditquiteusefulIhaveputtheemphasisonthedifferentiablepointofview,asdistinguishedfromtheanalyticHowever,tooffsetthisalittle,IincludedtwoanalyticapplicationsofStokes’formula,theCauchytheoreminseveralvariables,andtheresiduetheoremThird,Milnor’snotesMi,Mi,MiprovedinvaluableTheywereofcoursedirectedtowarddifferentialtopology,butofnecessityhadtocoveradhocthefoundationsofdifferentiablemanifolds(or,atleast,partofthem)Inparticular,Ihaveusedhistreatmentoftheoperationsonvectorbundles(Chapter,$)andhiselegantexpositionoftheuniquenessoftubularneighborhoods(ChapterIV,$,andChapterVII,$)Fourth,IamverymuchindebtedtoPalaisforcollaboratingonChapterIV,andgivingmehisexpositionofsprays(ChapterIV,$)Asheshowedme,thesecanbeusedtoconstructtubularneighborhoodsPalaisalsoshowedmehowonecanrecoverspraysandgeodesicsonaRiemannianmanifoldbymakingdirectuseofthecanonicalformandthemetric(ChapterVII,)ThisisaconsiderableimprovementonpastexpositionsInthedirectionofdifferentialgeometry,IfoundBergerGauduchonMazetBGMextremelyvaluable,especiallyinthewaytheyleadtothestudyoftheLaplacianandtheheatequationThisbookhasbeenxixiiACKNOWLEDGMENTSveryinfluential,forinstanceforGHL,whichIhavealsofoundusefulIalsofoundusefulKlingenberg’sbookKl,seeespeciallychapterXIIIIamverythankfultoKarcherandWuforinstructingmeonseveralmatters,includingestimatesforJacobifieldsIhavealsobenefitedfromHelgason’sbookHe,whichcontainssomematerialofinterestindependentlyofLiegroups,concerningtheLaplacianIamespeciallyindebtedtoWu’sinvaluableguidanceindealingwiththetraceofthesecondfundamentalformanditsa

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