Universitext Editorial Board (North America): S. Axler K.A. Ribet Loring W. Tu An Introduction to Manifolds Loring W. Tu Department of Mathematics Tufts University Medford, MA 02155 [email protected] Editorial Board (North America): S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California at Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA [email protected] [email protected] ISBN-13: 978-0-387-48098-5 e-ISBN-13: 978-0-387-48101-2 Mathematics Classification Code (2000): 58-01, 58Axx, 58A05, 58A10, 58A12 Library of Congress Control Number: 2007932203 © 2008 Springer Science + Business Media, LLC. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identifi ed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 www.springer.com (JLS/MP) DedicatedtothememoryofRaoulBott Preface IthasbeenmorethantwodecadessinceRaoulBottandIpublishedDifferentialForms inAlgebraicTopology. Whilethisbookhasenjoyedacertainsuccess,itdoesassume some familiarity with manifolds and so is not so readily accessible to the average first-yeargraduatestudentinmathematics. Ithasbeenmygoalforquitesometime tobridgethisgapbywritinganelementaryintroductiontomanifoldsassumingonly one semester of abstract algebra and a year of real analysis. Moreover, given the tremendous interaction in the last twenty years between geometry and topology on theonehandandphysicsontheother,myintendedaudienceincludesnotonlybudding mathematiciansandadvancedundergraduates,butalsophysicistswhowantasolid foundationingeometryandtopology. Withsomanyexcellentbooksonmanifoldsonthemarket,anyauthorwhoun- dertakestowriteanotherowestothepublic,ifnottohimself,agoodrationale. First andforemostismydesiretowriteareadablebutrigorousintroductionthatgetsthe reader quickly up to speed, to the point where for example he or she can compute deRhamcohomologyofsimplespaces. Asecondconsiderationstemsfromtheself-imposedabsenceofpoint-settopology intheprerequisites. Mostbookslaboringunderthesameconstraintdefineamanifold as a subset of a Euclidean space. This has the disadvantage of making quotient manifolds, of which a projective space is a prime example, difficult to understand. My solution is to make the first four chapters of the book independent of point-set topologyandtoplacethenecessarypoint-settopologyinanappendix. Whilereading thefirstfourchapters,thestudentshouldatthesametimestudyAppendixAtoacquire thepoint-settopologythatwillbeassumedstartinginChapter5. The book is meant to be read and studied by a novice. It is not meant to be encyclopedic. Therefore,Idiscussonlytheirreducibleminimumofmanifoldtheory whichIthinkeverymathematicianshouldknow. Ihopethatthemodestyofthescope allowsthecentralideastoemergemoreclearly. Inseveralyearsofteaching,Ihave generallybeenabletocovertheentirebookinonesemester. Inordernottointerrupttheflowoftheexposition,certainproofsofamoreroutine orcomputationalnatureareleftasexercises. Otherexercisesarescatteredthroughout theexposition,intheirnaturalcontext. Inadditiontotheexercisesembeddedinthe viii Preface text, there are problems at the end of each chapter. Hints and solutions to selected exercisesandproblemsaregatheredattheendofthebook. Ihavestarredtheproblems forwhichcompletesolutionsareprovided. This book has been conceived as the first volume of a tetralogy on geometry andtopology. ThesecondvolumeisDifferentialFormsinAlgebraicTopologycited above. IhopethatVolume3, DifferentialGeometry: Connections, Curvature, and CharacteristicClasses,willsoonseethelightofday. Volume4,ElementsofEquiv- ariantCohomology,along-runningjointprojectwithRaoulBottbeforehispassing awayin2005,shouldappearinayear. Thisprojecthasbeentenyearsingestation. DuringthistimeIhavebenefitedfrom thesupportandhospitalityofmanyinstitutionsinadditiontomyown;morespecif- ically,IthanktheFrenchMinistèredel’EnseignementSupérieuretdelaRecherche foraseniorfellowship(boursedehautniveau),theInstitutHenriPoincaré,theInstitut deMathématiquesdeJussieu,andtheDepartmentsofMathematicsattheÉcoleNor- maleSupérieure(rued’Ulm), theUniversitéParisVII,andtheUniversitédeLille, forstaysofvariouslength.Allofthemhavecontributedinsomeessentialwaytothe finishedproduct. IoweadebtofgratitudetomycolleaguesFultonGonzalez, ZbigniewNitecki, andMontserratTeixidor-i-Bigas,whotestedthemanuscriptandprovidedmanyuse- fulcommentsandcorrections,tomystudentsCristianGonzalez,ChristopherWatson, andespeciallyAaronW.BrownandJeffreyD.Carlsonfortheirdetailederrataandsug- gestionsforimprovement,toAnnKostantofSpringerandherteamJohnSpiegelman andElizabethLoewforeditingadvice,typesetting,andmanufacturing,respectively, andtoSteveSchnablyandPaulGérardinforyearsofunwaveringmoralsupport. I thankAaronW.BrownalsoforpreparingtheListofSymbolsandtheTEXfilesfor manyofthesolutions. SpecialthanksgotoGeorgeLegerforhisdevotiontoallofmy bookprojectsandforhiscarefulreadingofmanyversionsofthemanuscripts. His encouragement,feedback,andsuggestionshavebeeninvaluabletomeinthisbook as well as in several others. Finally, I want to mention Raoul Bott whose courses on geometry and topology helped to shape my mathematical thinking and whose exemplarylifeisaninspirationtousall. Medford,Massachusetts LoringW.Tu June2007 Contents Preface ......................................................... vii 0 ABriefIntroduction .......................................... 1 PartI EuclideanSpaces 1 SmoothFunctionsonaEuclideanSpace ......................... 5 ∞ 1.1 C VersusAnalyticFunctions ............................... 5 1.2 Taylor’sTheoremwithRemainder ............................ 7 Problems ...................................................... 9 2 TangentVectorsinRnasDerivations............................ 11 2.1 TheDirectionalDerivative................................... 12 2.2 GermsofFunctions......................................... 13 2.3 DerivationsataPoint ....................................... 14 2.4 VectorFields .............................................. 15 2.5 VectorFieldsasDerivations.................................. 17 Problems ...................................................... 18 3 Alternatingk-LinearFunctions................................. 19 3.1 DualSpace................................................ 19 3.2 Permutations .............................................. 20 3.3 MultilinearFunctions ....................................... 22 3.4 PermutationActiononk-LinearFunctions...................... 23 3.5 TheSymmetrizingandAlternatingOperators ................... 24 3.6 TheTensorProduct ......................................... 25 3.7 TheWedgeProduct......................................... 25 3.8 AnticommutativityoftheWedgeProduct....................... 27 3.9 AssociativityoftheWedgeProduct............................ 28 3.10 ABasisfork-Covectors ..................................... 30 Problems ...................................................... 31 x Contents 4 DifferentialFormsonRn ...................................... 33 4.1 Differential1-FormsandtheDifferentialofaFunction ........... 33 4.2 Differentialk-Forms ........................................ 35 4.3 DifferentialFormsasMultilinearFunctionsonVectorFields ...... 36 4.4 TheExteriorDerivative ..................................... 36 4.5 ClosedFormsandExactForms............................... 39 4.6 ApplicationstoVectorCalculus............................... 39 4.7 ConventiononSubscriptsandSuperscripts ..................... 42 Problems ...................................................... 42 PartII Manifolds 5 Manifolds................................................... 47 5.1 TopologicalManifolds ...................................... 47 5.2 CompatibleCharts.......................................... 48 5.3 SmoothManifolds.......................................... 50 5.4 ExamplesofSmoothManifolds .............................. 51 Problems ...................................................... 53 6 SmoothMapsonaManifold ................................... 57 6.1 SmoothFunctionsandMaps ................................. 57 6.2 PartialDerivatives.......................................... 60 6.3 TheInverseFunctionTheorem ............................... 60 Problems ...................................................... 62 7 Quotients ................................................... 63 7.1 TheQuotientTopology...................................... 63 7.2 ContinuityofaMaponaQuotient ............................ 64 7.3 IdentificationofaSubsettoaPoint............................ 65 7.4 ANecessaryConditionforaHausdorffQuotient ................ 65 7.5 OpenEquivalenceRelations ................................. 66 7.6 TheRealProjectiveSpace ................................... 68 ∞ 7.7 TheStandardC AtlasonaRealProjectiveSpace .............. 71 Problems ...................................................... 73 PartIII TheTangentSpace 8 TheTangentSpace ........................................... 77 8.1 TheTangentSpaceataPoint................................. 77 8.2 TheDifferentialofaMap.................................... 78 8.3 TheChainRule ............................................ 79 8.4 BasesfortheTangentSpaceataPoint ......................... 80 8.5 LocalExpressionfortheDifferential .......................... 82 8.6 CurvesinaManifold ....................................... 83 Contents xi 8.7 ComputingtheDifferentialUsingCurves ...................... 85 8.8 Rank,CriticalandRegularPoints ............................. 86 Problems ...................................................... 87 9 Submanifolds................................................ 91 9.1 Submanifolds.............................................. 91 9.2 TheZeroSetofaFunction................................... 94 9.3 TheRegularLevelSetTheorem .............................. 95 9.4 ExamplesofRegularSubmanifolds ........................... 97 Problems ...................................................... 98 10 CategoriesandFunctors ...................................... 101 10.1 Categories ................................................ 101 10.2 Functors .................................................. 102 10.3 DualMaps ................................................ 103 Problems ...................................................... 104 11 TheRankofaSmoothMap.................................... 105 11.1 ConstantRankTheorem..................................... 106 11.2 ImmersionsandSubmersions ................................ 107 11.3 ImagesofSmoothMaps..................................... 109 11.4 SmoothMapsintoaSubmanifold ............................. 113 11.5 TheTangentPlanetoaSurfaceinR3 .......................... 115 Problems ...................................................... 116 12 TheTangentBundle .......................................... 119 12.1 TheTopologyoftheTangentBundle .......................... 119 12.2 TheManifoldStructureontheTangentBundle.................. 121 12.3 VectorBundles............................................. 121 12.4 SmoothSections ........................................... 123 12.5 SmoothFrames ............................................ 125 Problems ...................................................... 126 13 BumpFunctionsandPartitionsofUnity ......................... 127 ∞ 13.1 C BumpFunctions........................................ 127 13.2 PartitionsofUnity.......................................... 131 13.3 ExistenceofaPartitionofUnity .............................. 132 Problems ...................................................... 134 14 VectorFields ................................................ 135 14.1 SmoothnessofaVectorField................................. 135 14.2 IntegralCurves ............................................ 136 14.3 LocalFlows ............................................... 138 14.4 TheLieBracket............................................ 141 14.5 RelatedVectorFields ....................................... 143 14.6 ThePush-ForwardofaVectorField ........................... 144 Problems ...................................................... 144 xii Contents PartIV LieGroupsandLieAlgebras 15 LieGroups.................................................. 149 15.1 ExamplesofLieGroups..................................... 149 15.2 LieSubgroups ............................................. 152 15.3 TheMatrixExponential ..................................... 153 15.4 TheTraceofaMatrix ....................................... 155 15.5 TheDifferentialofdetattheIdentity .......................... 157 Problems ...................................................... 157 16 LieAlgebras................................................. 161 16.1 TangentSpaceattheIdentityofaLieGroup.................... 161 16.2 TheTangentSpacetoSL(n,R)atI ........................... 161 16.3 TheTangentSpacetoO(n)atI .............................. 162 16.4 Left-InvariantVectorFieldsonaLieGroup .................... 163 16.5 TheLieAlgebraofaLieGroup............................... 165 16.6 TheLieBracketongl(n,R).................................. 166 16.7 ThePush-ForwardofaLeft-InvariantVectorField............... 167 16.8 TheDifferentialasaLieAlgebraHomomorphism ............... 168 Problems ...................................................... 170 PartV DifferentialForms 17 Differential1-Forms.......................................... 175 17.1 TheDifferentialofaFunction ................................ 175 17.2 LocalExpressionforaDifferential1-Form ..................... 176 17.3 TheCotangentBundle ...................................... 177 ∞ 17.4 CharacterizationofC 1-Forms.............................. 177 17.5 Pullbackof1-forms......................................... 179 Problems ...................................................... 179 18 Differentialk-Forms.......................................... 181 18.1 LocalExpressionforak-Form ............................... 182 18.2 TheBundlePointofView ................................... 183 ∞ 18.3 C k-Forms .............................................. 183 18.4 Pullbackofk-Forms ........................................ 184 18.5 TheWedgeProduct......................................... 184 18.6 InvariantFormsonaLieGroup............................... 186 Problems ...................................................... 186