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首页 GTM 176-Lee J.M___Riemannian Manifolds.. An Intr…

GTM 176-Lee J.M___Riemannian Manifolds.. An Introduction to Curvature (Springer 1997).pdf

GTM 176-Lee J.M___Riemannian Man…

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简介:本文档为《GTM 176-Lee J.M___Riemannian Manifolds.. An Introduction to Curvature (Springer 1997)pdf》,可适用于人文社科领域,主题内容包含RiemannianManifolds:AnIntroductiontoCurvatureJohnMLeeSpringerPrefaceThisbo符等。

RiemannianManifolds:AnIntroductiontoCurvatureJohnMLeeSpringerPrefaceThisbookisdesignedasatextbookforaonequarteroronesemestergraduatecourseonRiemanniangeometry,forstudentswhoarefamiliarwithtopologicalanddifferentiablemanifoldsItfocusesondevelopinganintimateacquaintancewiththegeometricmeaningofcurvatureInsodoing,itintroducesanddemonstratestheusesofallthemaintechnicaltoolsneededforacarefulstudyofRiemannianmanifoldsIhaveselectedasetoftopicsthatcanreasonablybecoveredintentofifteenweeks,insteadofmakinganyattempttoprovideanencyclopedictreatmentofthesubjectThebookbeginswithacarefultreatmentofthemachineryofmetrics,connections,andgeodesics,withoutwhichonecannotclaimtobedoingRiemanniangeometryItthenintroducestheRiemanncurvaturetensor,andquicklymovesontosubmanifoldtheoryinordertogivethecurvaturetensoraconcretequantitativeinterpretationFromthenon,alleffortsarebenttowardprovingthefourmostfundamentaltheoremsrelatingcurvatureandtopology:theGauss–Bonnettheorem(expressingthetotalcurvatureofasurfaceintermsofitstopologicaltype),theCartan–Hadamardtheorem(restrictingthetopologyofmanifoldsofnonpositivecurvature),Bonnet’stheorem(givinganalogousrestrictionsonmanifoldsofstrictlypositivecurvature),andaspecialcaseoftheCartan–Ambrose–Hickstheorem(characterizingmanifoldsofconstantcurvature)ManyotherresultsandtechniquesmightreasonablyclaimaplaceinanintroductoryRiemanniangeometrycourse,butcouldnotbeincludedduetotimeconstraintsInparticular,IdonottreattheRauchcomparisontheorem,theMorseindextheorem,Toponogov’stheorem,ortheirimportantapplicationssuchasthespheretheorem,excepttomentionsomeofthemviiiPrefaceinpassingandIdonottouchontheLaplace–BeltramioperatororHodgetheory,orindeedanyofthemultitudeofdeepandexcitingapplicationsofpartialdifferentialequationstoRiemanniangeometryTheseimportanttopicsareforother,moreadvancedcoursesThelibrariesalreadycontainawealthofsuperbreferencebooksonRiemanniangeometry,whichtheinterestedreadercanconsultforadeepertreatmentofthetopicsintroducedhere,orcanusetoexplorethemoreesotericaspectsofthesubjectSomeofmyfavoritesaretheelegantintroductiontocomparisontheorybyJeffCheegerandDavidEbinCE(whichhassadlybeenoutofprintforanumberofyears)ManfredodoCarmo’smuchmoreleisurelytreatmentofthesamematerialandmoredCBarrettO’Neill’sbeautifullyintegratedintroductiontopseudoRiemannianandRiemanniangeometryO’NIsaacChavel’smasterfulrecentintroductorytextCha,whichstartswiththefoundationsofthesubjectandquicklytakesthereaderdeepintoresearchterritoryMichaelSpivak’sclassictomeSpi,whichcanbeusedasatextbookifplentyoftimeisavailable,orcanprovideenjoyablebedtimereadingand,ofcourse,the“EncyclopaediaBritannica”ofdifferentialgeometrybooks,FoundationsofDifferentialGeometrybyKobayashiandNomizuKNAttheotherendofthespectrum,FrankMorgan’sdelightfullittlebookMortouchesonmostoftheimportantideasinanintuitiveandinformalwaywithlotsofpicturesIenthusiasticallyrecommenditasapreludetothisbookItisnotmypurposetoreplaceanyoftheseInstead,itismyhopethatthisbookwillfillanicheintheliteraturebypresentingaselectiveintroductiontothemainideasofthesubjectinaneasilyaccessiblewayTheselectionissmallenoughtofitintoasinglecourse,butbroadenough,Ihope,toprovideanynovicewithafirmfoundationfromwhichtopursueresearchordevelopapplicationsinRiemanniangeometryandotherfieldsthatuseitstoolsThisbookiswrittenundertheassumptionthatthestudentalreadyknowsthefundamentalsofthetheoryoftopologicalanddifferentialmanifolds,astreated,forexample,inMas,chapters–andBoo,chapters–Inparticular,thestudentshouldbeconversantwiththefundamentalgroup,coveringspaces,theclassificationofcompactsurfaces,topologicalandsmoothmanifolds,immersionsandsubmersions,vectorfieldsandflows,LiebracketsandLiederivatives,theFrobeniustheorem,tensors,differentialforms,Stokes’stheorem,andelementarypropertiesofLiegroupsOntheotherhand,IdonotassumeanypreviousacquaintancewithRiemannianmetrics,orevenwiththeclassicaltheoryofcurvesandsurfacesinR(Inthissubject,anythingprovedbeforecanbeconsidered“classical”)Althoughatonetimeitmighthavebeenreasonabletoexpectmostmathematicsstudentstohavestudiedsurfacetheoryasundergraduates,fewcurrentNorthAmericanundergraduatemathmajorsseeanydifferenPrefaceixtialgeometryThusthefundamentalsofthegeometryofsurfaces,includingaproofoftheGauss–Bonnettheorem,areworkedoutfromscratchhereThebookbeginswithanonrigorousoverviewofthesubjectinChapter,designedtointroducesomeoftheintuitionsunderlyingthenotionofcurvatureandtolinkthemwithelementarygeometricideasthestudenthasseenbeforeThisisfollowedinChapterbyabriefreviewofsomebackgroundmaterialontensors,manifolds,andvectorbundles,includedbecausethesearethebasictoolsusedthroughoutthebookandbecauseoftentheyarenotcoveredinquiteenoughdetailinelementarycoursesonmanifoldsChapterbeginsthecourseproper,withdefinitionsofRiemannianmetricsandsomeoftheirattendantfloraandfaunaTheendofthechapterdescribestheconstantcurvature“modelspaces”ofRiemanniangeometry,withagreatdealofdetailedcomputationThesemodelsformasortofleitmotifthroughoutthetext,andserveasillustrationsandtestbedsfortheabstracttheoryasitisdevelopedOtherimportantclassesofexamplesaredevelopedintheproblemsattheendsofthechapters,particularlyinvariantmetricsonLiegroupsandRiemanniansubmersionsChapterintroducesconnectionsInordertoisolatetheimportantpropertiesofconnectionsthatareindependentofthemetric,aswellastolaythegroundworkfortheirfurtherstudyinsucharenasastheChern–WeiltheoryofcharacteristicclassesandtheDonaldsonandSeiberg–Wittentheoriesofgaugefields,connectionsaredefinedfirstonarbitraryvectorbundlesThishasthefurtheradvantageofmakingiteasytodefinetheinducedconnectionsontensorbundlesChapterinvestigatesconnectionsinthecontextofRiemannianmanifolds,developingtheRiemannianconnection,itsgeodesics,theexponentialmap,andnormalcoordinatesChaptercontinuesthestudyofgeodesics,focusingontheirdistanceminimizingpropertiesFirst,someelementaryideasfromthecalculusofvariationsareintroducedtoprovethateverydistanceminimizingcurveisageodesicThentheGausslemmaisusedtoprovethe(partial)conversethateverygeodesicislocallyminimizingBecausetheGausslemmaalsogivesaneasyproofthatminimizingcurvesaregeodesics,thecalculusofvariationsmethodsarenotstrictlynecessaryatthispointtheyareincludedtofacilitatetheiruselaterincomparisontheoremsChapterunveilsthefirstfullygeneraldefinitionofcurvatureThecurvaturetensorismotivatedinitiallybythequestionofwhetherallRiemannianmetricsarelocallyequivalent,andbythefailureofparalleltranslationtobepathindependentasanobstructiontolocalequivalenceThisleadsnaturallytoaqualitativeinterpretationofcurvatureastheobstructiontoflatness(localequivalencetoEuclideanspace)Chapterdepartssomewhatfromthetraditionalorderofpresentation,byinvestigatingsubmanifoldtheoryimmediatelyafterintroducingthecurvaturetensor,soastodefinesectionalcurvaturesandgivethecurvatureamorequantitativegeometricinterpretationxPrefaceThelastthreechaptersaredevotedtothemostimportantelementaryglobaltheoremsrelatinggeometrytotopologyChaptergivesasimplemovingframesproofoftheGauss–Bonnettheorem,completewithacarefultreatmentofHopf’srotationangletheorem(theUmlaufsatz)Chapterislargelyofatechnicalnature,coveringJacobifields,conjugatepoints,thesecondvariationformula,andtheindexformforlateruseincomparisontheoremsFinallyinChaptercomesthedenouementproofsofsomeofthe“big”globaltheoremsillustratingthewaysinwhichcurvatureandtopologyaffecteachother:theCartan–Hadamardtheorem,Bonnet’stheorem(anditsgeneralization,Myers’stheorem),andCartan’scharacterizationofmanifoldsofconstantcurvatureThebookcontainsmanyquestionsforthereader,whichdeservespecialmentionTheyfallintotwocategories:“exercises,”whichareintegratedintothetext,and“problems,”groupedattheendofeachchapterBothareessentialtoafullunderstandingofthematerial,buttheyareofsomewhatdifferentcharacterandservedifferentpurposesTheexercisesincludesomebackgroundmaterialthatthestudentshouldhaveseenalreadyinanearliercourse,someproofsthatfillinthegapsfromthetext,somesimplebutilluminatingexamples,andsomeintermediateresultsthatareusedinthetextortheproblemsTheyare,ingeneral,elementary,buttheyarenotoptionalindeed,theyareintegraltothecontinuityofthetextTheyarechosenandtimedsoastogivethereaderopportunitiestopauseandthinkoverthematerialthathasjustbeenintroduced,topracticeworkingwiththedefinitions,andtodevelopskillsthatareusedlaterinthebookIrecommendstronglythatstudentsstopanddoeachexerciseasitoccursinthetextbeforegoinganyfurtherTheproblemsthatconcludethechaptersaregenerallymoredifficultthantheexercises,someofthemconsiderablyso,andshouldbeconsideredacentralpartofthebookbyanystudentwhoisseriousaboutlearningthesubjectTheynotonlyintroducenewmaterialnotcoveredinthebodyofthetext,buttheyalsoprovidethestudentwithindispensablepracticeinusingthetechniquesexplainedinthetext,bothfordoingcomputationsandforprovingtheoremsIfmorethanasemesterisavailable,theinstructormightwanttopresentsomeoftheseproblemsinclassAcknowledgments:IoweanunpayabledebttotheauthorsofthemanyRiemanniangeometrybooksIhaveusedandcherishedovertheyears,especiallytheonesmentionedaboveIhavedonelittlemorethanrearrangetheirideasintoaformthatseemshandyforteachingBeyondthat,IwouldliketothankmyPhDadvisor,RichardMelrose,whomanyyearsagointroducedmetodifferentialgeometryinhiseccentricbutthoroughlyenlighteningwayJudithArms,who,asafellowteacherofRiemanniangeometryattheUniversityofWashington,helpedbrainstormaboutthe“idealcontents”ofthiscourseallmygraduatestudentsattheUniversityPrefacexiofWashingtonwhohavesufferedwithamazinggracethroughtheflawedearlydraftsofthisbook,especiallyJedMihalisin,whogavethemanuscriptameticulousreadingfromauser’sviewpointandcameupwithnumerousvaluablesuggestionsandInaLindemannofSpringerVerlag,whoencouragedmetoturnmylecturenotesintoabookandgavemefreereinindecidingonitsshapeandcontentsAndofcoursemywife,PmWeizenbaum,whocontributedprofessionaleditinghelpaswellasthelovingsupportandencouragementIneedtokeepatthisdayafterdayContentsPrefaceviiWhatIsCurvatureTheEuclideanPlaneSurfacesinSpaceCurvatureinHigherDimensionsReviewofTensors,Manifolds,andVectorBundlesTensorsonaVectorSpaceManifoldsVectorBundlesTensorBundlesandTensorFieldsDefinitionsandExamplesofRiemannianMetricsRiemannianMetricsElementaryConstructionsAssociatedwithRiemannianMetricsGeneralizationsofRiemannianMetricsTheModelSpacesofRiemannianGeometryProblemsConnectionsTheProblemofDifferentiatingVectorFieldsConnectionsVectorFieldsAlongCurvesxivContentsGeodesicsProblemsRiemannianGeodesicsTheRiemannianConnectionTheExponentialMapNormalNeighborhoodsandNormalCoordinatesGeodesicsoftheModelSpacesProblemsGeodesicsandDistanceLengthsandDistancesonRiemannianManifoldsGeodesicsandMinimizingCurvesCompletenessProblemsCurvatureLocalInvariantsFlatManifoldsSymmetriesoftheCurvatureTensorRicciandScalarCurvaturesProblemsRiemannianSubmanifoldsRiemannianSubmanifoldsandtheSecondFundamentalFormHypersurfacesinEuclideanSpaceGeometricInterpretationofCurvatureinHigherDimensionsProblemsTheGauss–BonnetTheoremSomePlaneGeometryTheGauss–BonnetFormulaTheGauss–BonnetTheoremProblemsJacobiFieldsTheJacobiEquationComputationsofJacobiFieldsConjugatePointsTheSecondVariationFormulaGeodesicsDoNotMinimizePastConjugatePointsProblemsCurvatureandTopologySomeComparisonTheoremsManifoldsofNegativeCurvatureContentsxvManifoldsofPositiveCurvatureManifoldsofConstantCurvatureProblemsReferencesIndexWhatIsCurvatureIfyou’vejustcompletedanintroductorycourseondifferentialgeometry,youmightbewonderingwherethegeometrywentInmostpeople’sexperience,geometryisconcernedwithpropertiessuchasdistances,lengths,angles,areas,volumes,andcurvatureTheseconcepts,however,arebarelymentionedintypicalbeginninggraduatecoursesindifferentialgeometryinstead,suchcoursesareconcernedwithsmoothstructures,flows,tensors,anddifferentialformsThepurposeofthisbookistointroducethetheoryofRiemannianmanifolds:thesearesmoothmanifoldsequippedwithRiemannianmetrics(smoothlyvaryingchoicesofinnerproductsontangentspaces),whichallowonetomeasuregeometricquantitiessuchasdistancesandanglesThisisthebranchofmoderndifferentialgeometryinwhich“geometric”ideas,inthefamiliarsenseoftheword,cometotheforeItisthedirectdescendantofEuclid’splaneandsolidgeometry,bywayofGauss’stheoryofcurvedsurfacesinspace,anditisadynamicsubjectofcontemporaryresearchThecentralunifyingthemeincurrentRiemanniangeometryresearchisthenotionofcurvatureanditsrelationtotopologyThisbookisdesignedtohelpyoudevelopboththetoolsandtheintuitionyouwillneedforanindepthexplorationofcurvatureintheRiemanniansettingUnfortunately,asyouwillsoondiscover,anadequatedevelopmentofcurvatureinanarbitrarynumberofdimensionsrequiresagreatdealoftechnicalmachinery,makingiteasytolosesightoftheunderlyinggeometriccontentToputthesubjectinperspective,therefore,let’sbeginbyaskingsomeverybasicquestions:WhatiscurvatureWhataretheimportanttheoremsaboutitWhatIsCurvatureInthischapter,weexploretheseandrelatedquestionsinaninformalway,withoutproofsInthenextchapter,wereviewsomebasicmaterialabouttensors,manifolds,andvectorbundlesthatisusedthroughoutthebookThe“official”treatmentofthesubjectbeginsinChapterTheEuclideanPlaneTogetasenseofthekindsofquestionsRiemanniangeometersaddressandwherethesequestionscamefrom,let’slookbackattheveryrootsofoursubjectThetreatmentofgeometryasamathematicalsubjectbeganwithEuclideanplanegeometry,whichyoustudiedinschoolItselementsarepoints,lines,distances,angles,andareasHereareacoupleoftypicaltheorems:Theorem(SSS)TwoEuclideantrianglesarecongruentifandonlyifthelengthsoftheircorrespondingsidesareequalTheorem(AngleSumTheorem)ThesumoftheinterioranglesofaEuclideantriangleisπAstrivialastheyseem,thesetwotheoremsservetoillustratetwomajortypesofresultsthatpermeatethestudyofgeometryinthisbook,wecallthem“classificationtheorems”and“localglobaltheorems”TheSSS(SideSideSide)theoremisaclassificationtheoremSuchatheoremtellsusthattodeterminewhethertwomathematicalobjectsareequivalent(undersomeappropriateequivalencerelation),weneedonlycompareasmall(oratleastfinite!)numberofcomputableinvariantsInthiscasetheequivalencerelationiscongruenceequivalenceunderthegroupofrigidmotionsoftheplaneandtheinvariantsarethethreesidelengthsTheanglesumtheoremisofadifferentsortItrelatesalocalgeometricproperty(anglemeasure)toaglobalproperty(thatofbeingathreesidedpolygonortriangle)Mostofthetheoremswestudyinthisbookareofthistype,which,forlackofabettername,wecalllocalglobaltheoremsAfterprovingthebasicfactsaboutpointsandlinesandthefiguresconstructeddirectlyfromthem,onecangoontostudyotherfiguresderivedfromthebasicelements,suchascirclesTwotypicalresultsaboutcirclesaregivenbelowthefirstisaclassificationtheorem,whilethesecondisalocalglobaltheorem(Itmaynotbeobviousatthispointwhyweconsiderthesecondtobealocalglobaltheorem,butitwillbecomeclearersoon)Theorem(CircleClassificationTheorem)TwocirclesintheEuclideanplanearecongruentifandonlyi

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