IllinoisJournalofMathematicsVolume54,Number2,Summer2010,Pages708 - PDF document

IllinoisJournalofMathematicsVolume54,Number2,Summer2010,Pages708…741S0019-2082AUSTERESUBMANIFOLDSOFDIMENSIONFOUR:EXAMPLESANDMAXIMALTYPESMARIANTYIONELANDTHOMASIVEYAbstract.AusteresubmanifoldsinEuclideanspacewerein-troducedbyHarveyandLawsoninconnectionwiththeirstudyofcalibratedgeometries.Thealgebraicpossibilitiesforsecondfundamentalformsof4-dimensionalausteresubmanifoldswereclassi“edbyBryant,intothreetypeswhichwelabelA,BandC.Inthispaper,weshowthattypeAsubmanifoldscorrespondexactlytorealK¨ahlersubmanifolds,weconstructnewexamplesofsuchsubmanifoldsin,andweobtainclassi“ca-tionresultsonsubmanifoldswithsecondfundamentalformsofmaximaltype.1.IntroductionDe“nitionsandbackground.Recallthatforanimmersedsubmanifoldwithnormalbundle),thesecondfundamentalformII:)isde“nedbyII(X,YX,YaretangentvectorstoistheEuclideanconnectioninistheorthogonalprojectionontothenormalbundle.Thenaustereif,foranynormalvector“eld,theeigenvaluesofthequadraticformX,Y):=II(X,Y)withrespecttothemetricareateachpointsym-metricallyarrangedaroundzeroontherealline(equivalently,allodddegreesymmetricpolynomialsintheseeigenvaluesvanish).When=2,thisjustmeansthatisaminimalsurfacein.However,when2theaustereconditionisstrongerthanminimality,andleadstoahighlyoverdeterminedsystemofPDEsfortheimmersion.Forexample,becauseofnonlinearityof ReceivedJune15,2009;receivedin“nalformDecember11,2009.MathematicsSubjectClassi“cation.Primary53B25.Secondary53B35,53C38,2011UniversityofIllinois 710M.IONELANDT.IVEYInthiscontext,ageneralizedhelicoidistheimageofaparametriza-arepositiveconstantsand.Fortheconstructionofthetwistedcone,see[2].2(Bryant)Letandletbeamaximalausteresubspace-conjugatetooneofthefollowingA,BaresymmetricmatricesmIBisawhereinthelastcaseparametersWewillsaythatanausteresubmanifoldisoftypeA,BorCrespec-tively,ifforeverypointthespace(4)-conjugatetoasubspaceofthecorrespondingmaximalausteresubspacegiveninTheorem2.(InthecaseoftypeC,weallowtheparameterstovaryfrompointtopointinItispossiblefortobeconjugatetoasubspaceofmorethanonemaximalsubspace(e.g.,whenisahypersurface),butwewillassumethatthereisoneparticularmaximalsubspacewhichappliesatallpointsofItiseasytogiveexamplesofaustere4-foldsoftypesAandC.Whenthecodimensioniseven,anyholomorphicsubmanifoldisanaustere4-foldoftypeA.Toseethis,letbethecomplexstructure,andnotethatforanyvector“eldsX,YtangenttoII(andthereforeII(II(X,YDajczerandGromoll[5]de“nedasubmanifoldtobecircularifitcarriesaparallelcomplexstructuresuchthat(3)holds,andtheyobservedthatthisconditionimpliesthatisaustere.Itfollowsfrom(3)thatIIisrepresentedbymatricesinwhenwechooseamovingframesuchthat.Asforaustere4-foldsoftypeC,anotherresultofBryant(see[2],Theorem3.1) AUSTERESUBMANIFOLDSOFDIMENSIONFOUR711impliesthatisageneralizedhelicoidinifandonlyiftheparametersareidenticallyzero.Ontheotherhand,wedonotknowifotheraustere4-foldsoftypeCexist.Ourapproach.Ourgoalsinstudyingausteresubmanifoldsaretoobtainnewexamplesand,wherepossible,toclassifyaustere4-foldsofagiventype.Weemploythemethodofmovingframestogenerateexteriordierentialsystems(EDS)whosesolutionscorrespondtoaustere4-foldsofagiventypeandcodimension.Whensuchsystemsareinvolutive,Cartan…K¨ahlertheory(see[3])givesusameasureofthesizeofthesolutionspace,intheformofwhatinitialdatamaybechosenforasequenceofCauchyproblemsthatdetermineeverypossiblelocalsolution.Studyingthestructureoftheexteriordierentialsystemcanalsoenableustoestablishglobalpropertiesofsolutions(see,e.g.,Proposition5andProposition14below).Onecouldorganizeaclassi“cationschemeforaustere4-foldsinEuclideanspacebytypeandthedimension(assumedconstantoverthesub-However,weexpecttoobtainthestrongestclassi“cationtheoremswhentheaustereconditionisstrongest,thatis,whenisaslargeaspossibleforagiventype.Thus,liketheearlierresultsofBryanton3-folds,theclassi-“cationresultsinthispaperareobtainedassumingthatisconjugatetooneof.(Inthiscase,wesayisoftypeA,B,orC,respectively.)Classifyingaustere4-foldsofofnonmaximaltypewouldinvolveparametrizingthepossiblesubspacesofagivendimensionwithinandanalyzingtheassociatedEDS.Inmanyinstances,themanyadditionalparametersinvolvedmaketheEDSintractable,evenwiththeas-sistanceofcomputeralgebrasystems.Toobtainnewexamples,anoftensuccessfulstrategyistoassumeaddi-tionalconditions.Inthelastpartofthispaper,weobtainnewexamplesofaustere4-foldsofnonmaximaltypeAbyassumingthat=2andthespaceliesonanonprincipalorbitoftheactionofthesymmetrygroupoftheGrassmannianoftwo-dimensionalsubspacesof.OnecanalsocarryoutthisapproachfortypeBwith=2,butthisyieldsnonewexamples.TheapproachisnotfeasiblefortypeCbecauseinthatcasethesymmetrygroupisdiscrete.Outlineandsummaryofresults.InSection2,wede“nethemovingframesandassociatedgeometricstructureswewilluseintherestofthepaper.Becausetheexteriordierentialsystemsweusearetailoredforsubmanifoldsinaspeci“ccodimension,weproveapreliminaryresultinSection2.1totheeectthat,whensatis“escertainalgebraiccriteria,thenequalsthe Byduality,thisisalsotherankofIIasalinearmapintothenormalbundle;hence,wewilloftenrefertoitasthenormalrank 712M.IONELANDT.IVEYeectivecodimension(i.e.,thecodimensionofwithinthesmallesttotallygeodesicsubmanifoldcontainingit).Section3isconcernedwithaustere4-foldsoftypeA.Notethatsuchsub-manifoldscarryawell-de“nedalmostcomplexstructuresatisfying(3).InSection3.1,wederivesucientconditionsontobeparallel,imply-ingthatisK¨ahler.(ThisgivesaconversetoDajczerandGromollsresult.)WeshowthattheK¨ahlernessconditionsapplywheneverhasdimensionatleasttwo.ThemainresultofSection3.2isthedescriptionofthegen-eralityofmaximaltypeAaustere4-folds.Weshowthatsuchsubmanifoldsdependonachoiceof2(1)functionsof2variables,inthesenseoftheCartan…K¨ahlerTheorem.Weconcludethat,generically,theseausteresubmanifoldsoftypeAarenotholomorphicsubmanifolds.Section4isconcernedwithclassifyingaustere4-foldsofmaximaltypesBandC.InSection4.1,weshowthataustere4-foldsofmaximaltypeBdonotexist.InSection4.2,weprovetworesultsaboutmaximaltypeC.First,ifisateachpointconjugatetoa“xedmaximalausteresubspacetheparametersareassumedtobeconstantover)thenmustbeageneralizedhelicoid.Second,evenwithoutrequiringtheparametervaluestobe“xed,weshowthatthereisonlya“nite-dimensionalfamilyofsubmani-foldsofmaximaltypeC.Thisfollowsfromshowingthat,awayfromcertainexceptionalparametervalues,thecharacteristicvarietyoftherelevantEDSisempty;wealsoshowthattheparameterstakevalueintheexceptionallocuson(atmost)thecomplementofanopendensesubsetofInSection5,wegivesomeinterestingexamplesofaustere4-foldsofnon-maximaltype.Asmentionedabove,oneapproachistoassumethatasapointintheGrassmannianoftherelevantmaximalausteresubspace,isnongenericfortheactionofthesymmetrygroup(i.e.,itliesalonganonprin-cipalorbit).InSection5,weusethesymmetrygroupoftonormalize2-dimensionalsubspacesof,andidentifythenongenericsubspaces.WethenclassifythetypeAaustere4-foldsforwhichhasdimensiontwoandisof“xednongenerictype,assumingthattheGaussmapisnondegenerate.(OnecanshowthatiftheGaussmapofaustere4-foldisdegenerate,thenitmusthaverankatmost2.Austeresubmanifoldswithrank2Gaussmapwereclassi“edbyDajczerandFlorit[4].)Thesesubmanifolds,whichalllieinatotallygeodesic,turnouttobeeitherholomorphicsubmanifolds,productsofminimalsurfaces,orelse2-ruledsubmanifolds.Thelatterhavetheprop-ertythattheimageofthemap6),takingpointintothesubspaceofparalleltotherulingthrough,isaholomorphiccurve.Suchcurvesarenotarbitrary,however;wealsoshowhowtheseruledsubmanifoldsmaybeconstructedbyinsteadchoosingageneralholomorphiccurveinWeplantocarryoutafullclassi“cationof2-ruledaustere4-foldsinournextpaper. AUSTERESUBMANIFOLDSOFDIMENSIONFOUR717First,supposewewishtoconstructanausteresubmanifoldsuchthatateachpointisconjugatetoa“xedausteresubspaceofdimensionisoftype).Letsymmetricmatricesbea“xedbasisforthissubspace.Thenanysuchsubmanifoldcanbelocallyequippedwithanadaptedframesuchthat(15)II(holds.Conversely,ifsubmanifoldissuchthatthenitistheimageofasectionofforsomeausteremanifoldoftypeThus,wemayde“neonaPfaanexteriordierentialsystem,thestandardwhoseintegralsubmanifoldscorrespondtoausteremanifoldsofthistype.Wewillalsoneedtoconsiderausteremanifoldsisconjugatetoanausteresubspaceof“xeddimensionbutwhichdependsonparameterswhichareallowedtovaryalong.Supposethatabasisofthissubspaceisgivenbysymmetricmatrices),andtheparametersareallowedtorangeoveranopenset.Thenwemayde“nethestandardsystemwithparameterswhichisanalogoustotheabove,butnowde“nedontheproductGivenanyausteremanifoldofthiskind,wemayconstructanadaptedframealongsuchthatII(forfunctions.Thentheimageofthe“beredproductofthep,e))and))willbeanintegralsubmanifoldof.Conversely,anyintegralsubmanifoldofsatisfyingtheindependenceconditiongives(byprojectingontothe“rstfactorinsectionofwhichisanadaptedframeforanausteremanifoldForlateruse,wecomputethe1-formsof.Wenotethat0modulothe1-formsof,sothattheonlyalgebraicgenerator2-formsareobtainedfromdierentiatingthe1-forms.Using(12)and(13),wemodulo.The2-formsforthestandardsystemwithoutparametersareobtainedreplacingin(16)withaconstant AUSTERESUBMANIFOLDSOFDIMENSIONFOUR719Proof.bea“xedbasisfor.Letbetheeectivecodi-mensionof,notassumedtobethesameasthenormalrankof.Locally,wemayconstructanadaptedframesuchthatII(aresomefunctionsonand16.Thentheadaptedframede“nesalocalsectionsuchthatspanthecotangentspaceofandtheimageofisanintegralofthe1-formsByspecializingthecomputation(16)tothecasewhere,wemodulotheforms,where(17)Sh, ]ij jand[]denotesthecommutator.These2-formsmustvanishunderpullbackvia.Considerthe4-forms:=Usingthefactthat,wecanexpandtheseassSh, ]ij(JT)k+Jim[Sh, ]mjTk j k .Next,write,wheretakesvalueintakesvalueinUsingthefactthatthematricesand[]anticommutewithwithSh,]commuteswith,wehaveeSh, ]ij (Š tJ+J t)jk+[Sh,]ij ( tJ+J t)jk Tk.Itiseasytoverifythat(=0.ComputingtheremainingtermsgivessSh,]ijUjk (k),where(k)denotesthe3-formwhichisthewedgeproductofthesuchthatWrite,where )and ).Suppose.Thenthevanishingofandthefactthat=0,impliesthatthemustsatisfy 722M.IONELANDT.IVEYSincetheseconditionsareconstraintsonhowthenormalvectorsmaybearranged,theyholdonlyonacodimension-twosubmanifoldoftheframe.Letdenotethissubmanifold.WenowapplyCartanstestforinvolutivitytothepullbackofthePfaansystemPropositionthesystemisinvolutivewithCartancharacters=24,=10.Proof.AsintheproofofProposition5,letbetheprojectionofinto.Thenmodwherewede“neNext,letstandforthematrix-valued1-formwhoseentriesare.Onthe361-formsarelinearlyindependent.Because,foreachtakesvalueinthe6-dimensionalspace,itfollowsthatonthereareexactly36linearlyindependentformsamongthe.Dierentiating(20)showsthattwooftheseformspullbacktotobelinearlydependentontheothers;forexample,onecansolveforintermsoftheother.Itfollowsthat,whenpulledbackto,thereare24linearlyindependent1-formsamongand10furtherindependent1-formsamongthe.ThisgivesustheclaimedvaluesfortheCartancharacters.ToapplyCartanstest,weneedtocalculatethe“berdimensionofthespaceof4-dimensionalintegralelementsatpointson.Supposethatanintegralelementisde“nedbyijkijkissymmetricini,j,k,andforany“xedisinthespaceForeach,thespaceofsymmetrictensorssatisfyingtheseconditionsisiso-morphictotheprolongation,whichhasdimension8.Asvaries,weobtaina48-dimensionalspaceofsolutionsijk.However,thecorrespondingintegral4-planesmustbetangenttothesubmanifold.Thisrequirementim-poses4additionallinearlyindependenthomogeneousconditionsontheijksoweconcludethatthe“berdimensionofthespaceofintegralelementstan-genttois44.Sincethisdimensioncoincideswith,thesystemisinvolutive.Wenowstatethefollowingtheorem.Austere-foldsinofmaximaltypeAexistanddependlocallyonachoiceoffunctionsofEachofthemcarriesacomplexstructurewithrespecttowhichthemetricinheritedfromambientspaceisK¨buttheyaregenericallynotcomplexsubmanifolds 724M.IONELANDT.IVEYSuchmovingframes,assectionsof,giveintegralsubmanifoldsofthefollowingPfaansystem:Again,werestricttothesubmanifoldwheretheintegrabilityconditions(20)hold.Wecomputemodulothe1-formsin.Forevery“xedindex,thetableaucomponentgivenbyisisomorphicto,andisinvolutivewithcharacters=2.CombiningthiswiththeresultsofProposition8weconcludethattheEDSisinvolutivewithcharacters=24+4(6)=410+2(6)=2WeconcludethattypeAaustere4-foldsinwithmaximal“rstnormalspace(sothat6)dependonachoiceof2(1)functionsof2variables.Bycontrast,wheniseven,holomorphicsubmanifoldsofrealdimension4dependonfunctionsof2variables.4.MaximaltypesBandC4.1.SubmanifoldsofmaximaltypeB.beanausteresubman-ifoldoftypeofnormalrank.Byhypothesis,thereisamovingframe)suchthattheareorthonormalandtangenttoandineachnormaldirectiontheshapeoperatortakestheformII=Weconsiderthestandardsystemwithparameters,whereisthesemi-orthonormalframebundleofandII(formatricesoftheform(22).(Foreach,theparametersarethescalarandtheentriesof.)TheintegralsubmanifoldsofthisEDScorrespondtoausteresubmanifoldsoftypeB.Asin(16),wecomputethesystem2-formsassSa, ]ij+ abSbij) jmodulothe1-formsoftheideal,where[]denotesthecommutator.Hence,thetableauofthesystemisspannedbythe1-forms AUSTERESUBMANIFOLDSOFDIMENSIONFOUR727Thus,anyintegral4-foldoftheEDSwillalsobeanintegralofthe1-.Letbethedierentialidealresultingfromaddingthese1-formsto.Theexteriorderivativesofthes,modulothe1-formsofarelinearcombinationsofwedgeproductsoftheswitheachother,andwiththes.Thus,isanonlinearPfaansystem.Inparticular,ifwesubstitutethevaluesgivenbyijkintothenew2-forms,andtakeco-ecientswithrespecttothe2-forms,weobtain66quadraticpolynomialsintheijkwhichmustvanishinorderforanintegralelementoftobeanintegralelement.Eliminatingtheijkfromthesepolynomialsyieldsintegrabilitycon-ditionsintermsofthewhichinclude=0.Sincethisisimpossibleforcomponentsofapositivede“nitemetric,weconcludethatthesetofintegral4-planesofsatisfyingtheindependenceconditionisempty.4.2.SubmanifoldsofmaximaltypeC.Webeginbynotingthattheofquadraticformsisinvariantunderconjugationbyadiscretesub-groupof(4)thatsimultaneouslypermutes.Thesepermutationswill,ofcourse,preservetheequationin(1)satis“edbythebutwillnotpreservetheinequalitiesin(1).WenowdiscusssubmanifoldsoftypeCwhose“rstnormalspaceisof=3.ThesesubmanifoldslieinasseeninProposition3.AswasthecasewithsubmanifoldsoftypeBwhosesecondfundamentalformhadmaximalspan,wecanchooseanorthonormalframeforthetangentspaceandabasisforthe“rstnormalspacewithrespecttowhichthesecondfundamentalformisrepresentedbyanybasisforthespacewechoose.Accordingly,letbethebundleofsuchframesonusethebasismatrices0100100000000011000whereweassumethat 730M.IONELANDT.IVEY 2( 21+ 43) 1 2 3,2 3 4 3 1 2 3=1 2( 21+ 43) 3 2 4,1 1 3 4 1 1 2=1 Becausethe2-formsvanishonanyintegral4-plane,thesameistrueforthe4-formsontheright.Then,sincemustrestricttotobealinearcombinationofthe,thesimultaneousvanishingofthe4-formsontherightin(30)impliesthatthislinearcombinationmustbezero.Similarly,onecanusetheotherpiecesofthetableautoshowthattherearethreemore1-formsthatmustvanishontheintegralelements.Inall,theseadditionalformsare Š1 41+ 12,2= 41+ 32=4 wherewenowassumethat=1and1.(Wewillconsiderthecase1below.)Letbethedierentialidealobtainedbyaddingtheabovefour1-formsto.ThisyieldsanonlinearPfaansystem,sincetheexteriorderivativesofthenewaddedformswillcontainlinearcombinationsofwedgesofthe.Computingmodulo,the1-formsofgivesThus,integralelementsexistonlyonthesubmanifoldwhere=0.Werestricttothesubmanifoldwhere=0.Theintegral4-planeswillbede“nedbytheequationsijkwherenowonly15ofthe1-formsarelinearlyindependent,aslinearcom-binationsof.Nowwesubstitutethevaluesin(31)intothenew2-forms4.Foreachofthese,thecoecientswithrespecttoshouldallbezero.Fromtheseconditions,weget12quadraticpolynomialsinthewhichmustvanishonanyintegralsubmanifoldofAGr¨obnerbasiscalculationshowsthatthesepolynomialshavenocommonzero,sothesetof4-integralelementsofisempty.=1or1,theconclusionisthesame.Itturnsoutthatinthiscasethereare7more1-formsthatvanishonanyintegralelementofandwhichhavetobeaddedtotheideal.Amongthe1-formsoftheaugmentedideal 732M.IONELANDT.IVEYThus,weconcludethattheonlypossiblesolutionswithparametersconstantarethoseforwhichalltheseparametersarezero.5.ExamplesInthissection,weexaminesomeinterestingexamplesofausteresubman-ifoldswhosenormalrankisnotmaximal.Inparticular,wedescribesomenongenericaustere4-foldsoftypeAwith=2.Moreprecisely,wenormalizethe2-dimensionalsubspacesofwhichlieonnonprincipalorbitsofthesymmetrygroup,andclassifythecorrespondingaustere4-folds.AsstatedinSection3.1,thesymmetrygroupof,withgivenby(9),anditsactiononThisgroupisisomorphictotheusualgroup(2)of22unitarymatrices,whichactsinasimilarwayonthespaceof22complexmatrices.Infact,wecande“neanisomorphism(2)thatintertwinestheseactions:ifwelet :ŠAŠi (M·S(M)· ).Inwhatfollows,wewillusethisactiontonormalizerealsubspacesofbeasubspaceofrealdimension2,andletS,T.We“rstconsiderthefollowingspecialcases:S,TarelinearlydependentoverInthiscase,wecanuse(2)tosimultaneouslydiagonalize.Usinglinearcombinationswithrealcoecients,wecanarrangethatS,x,yWedistinguishtwosubcases:(a)everymatrixinhasfullrank,sothatx,yarenotbothzero;and(b)thematricesaresingular(i.e.,=0).S,TarelinearlyindependentoverWe“rstnotethattheremustbeasingularmatrixinthecomplexspanof,thatis,(34)det(Wedistinguishseveralsubcases:containsasingularmatrix).Inthiscase,wecanlinearlycombinesothathasrank1.Using(2),wecanarrangethatkerisspannedbyy,0];thenusingthediagonalsubgroup(1)andrealscalefactors,wecanassumethat 734M.IONELANDT.IVEY ,andletbethespanof.Itiseasytocheckthat=0ineachcase,sobytheargumentofProposition3,liesinatotallygeodesic(i)Assumethatisoftype1(a);thenisparametrizedbyx,y.TakinggSandTaregivenby(33)]asbasismatrices,letbethestandardsystemwithparameters,de“nedonminustheorigin.ByProposition6,anyausteremanifoldofthistypewillbeK¨withrespecttothecomplexstructuregivenby(9).Thus,theconnection1-formsmustsatisfyforanyadaptedframethatmakesconjugateto.Therefore,suchadaptedframesgiveintegralsoftheaugmentedsystemTakingexteriorderivativesofthelasttwo1-formsmodulothealgebraicidealgeneratedbyformsinshowsthatintegralsubmanifoldsexistonlyatpointswhereInotherwords,itisnecessarythattheframevectorsbeorthogonalandhavethesamelength.Wepullbackthesystemtothesubmanifoldwheretheseconditionshold.(Pulledbackto,theconnectionformssatisfytheadditionalrelations)On,thesystemisinvolutivewithCartancharacters=4,=2.Toseethatthecorrespondingausteresubmanifoldsareholomorphic,weneedtoendowwiththeappropriatecomplexstructurewhichrestrictsto.Because,equation(2)impliesthatthisambientcomplexstructuremustsatisfy.Thus,ifweletbeamatrixwhosecolumnsarethevectors,thenmustsatisfyFC,C 0 0 istobethestandardcomplexstructureon,thenitmustbegivenbyaconstantmatrix.By(39),thismatrixmustequalFCF.Thus,wehaveonlytoshowthat,foranyintegralof,thismatrixisaconstant.Thestructureequations(12)implythat,whereisthe6matrixofconnectionforms:Usingthis,wecomputethathat,C]FŠ1.Then,itiseasytoverifythat,foranyintegralof,thevaluesoftheconnectionformsimplythatthat,C]=0.Theargumentinthecasethatisconjugatetoaspaceoftype2(c)issimilar,savethatinthatcaseisK¨ahlerwithrespecttothecomplexstructurerepresentedby 736M.IONELANDT.IVEYThus,themap6)hasranktwo.Toseethattheimageisaholomorphiccurve,wemustexaminethecomplexstructureon6).Asin([9],loc.cit.),wethinkof6)asSO(6)SO(4).Dieren-tialformson6)„inparticular,(10)-formsforthecomplexstructure„maybelifteduptothegroupSO(6).Inthiscase,weusealiftingofthetoamap :SO(6)de“nedby(41) : r1=1 r2,= .(Notethechangeinorder,chosensothatthevectorstangenttotherulingatarethe“rsttwocolumnsof ().)Let =betheMaurer…CartanformonSO(6),withcomponents.Thentheformsfor36aresemibasicforthequotientmap:SO(6)6),whichsendstothe2-planespannedbyits“rsttwocolumns.Moreover,thecomplexspanofthe1-formsiswellde“nedonthequotient,andspansthespaceof(10)-formson6).NotealsothatbycomparingtheMaurer…Cartanequation withthede“ningproperties(10)oftheconnectionformsshowsthat(42) for1i,j4,whereisthepermutationthatexchangesindices2and3.Toshowholomorphicityof,wehavetoshowthatthepullbackunder ofthe(10)-formson6)are(10)-formson,thatis,inthespanof.(Notethatitisequivalenttoshowthisforthepullbacksunder fortheforms.)Using(40)and(42),wecomputeNoticethattheholomorphiccurvein6)isnotgeneric;indeed,ifweredeterminedbyspecifyinganarbitraryholomorphiccurveintheGrassmannianastheimage),thenonewouldexpecttheCartancharacterbe6.Instead,aswewillseebelow,is(inpart)determinedbyageneralholomorphiccurveinadierentHermitiansymmetricspace.Fortherestofthissubsection,wewillfocuson2-ruledausteresubmanifolds=2,thelasttypediscussedinTheorem15.Asintheproofofthattheorem,anadaptedframealongsuchasubmanifoldde“nesthemap :SO(6)givenby(41).Nowlet:SO(6)(3)bethe 738M.IONELANDT.IVEYbeidenti“edwith,inthefollowingway.withthestan-dardHermitianmetric,andletwiththeEuclideanmetric.Then,andeachcomplexstructurecorresponds(byas-sociatingtoits+ieigenspace)toatotallyisotropicsubspaceEachsuchsubspaceisoftheformforsomevector.Themapiswell-de“neduptocomplexmultiple,andidenti“esSO(6)(3)with.Moreover,thestandardK¨ahlerformonpullsbackto 1+2 2+3 onSO(6),andSO(6)maybeiden-ti“edwiththeunitaryframebundleof,withconnectionformsgivenbythecomponentsof.Thefollowingresultshowsthattheassociationofwithaholomorphiccurveinissurjectivebutnot1-to-1.LetbeaholomorphiccurveinGivenanonplanarpointthereisanopenneighborhoodcontaininganda-ruledausteremanifoldsuchthatSuchmanifoldsdependonachoiceoffunctionsofProof.TheproofofTheorem15part(iii)showsthatalongthereisframe()suchthatII(II(formatrices0100100000andsomepositivefunctionsr,x,y.Toconstruct,wewillsetupaPfaansystem,similartotheaugmentedstandardsystem,satis“edbytheorthonormalframe.betheorthonormalframebundleof;intermsofthebundleofsemi-orthonormalframesde“nedinSection2.2,isthesubbundleofonwhichhold.Thestructureequationsofarethesameasthosegivenbyequa-tions(10)through(12),butwiththespecialization(46)takenintoaccount,isskew-symmetricandWeadjoinr,x,yasnewvariables,takingvalueinthepositiveoctant,andde“neourPfaansystemtobegeneratedbyAsin(41),wede“neamap :SO(6),whosevalueisthematrixwithcolumns().Wewillnowshowhow,givenanarbitrary Welearnedthisidenti“cationinapaperofAbbena,GarbieroandSalamon[1]. AUSTERESUBMANIFOLDSOFDIMENSIONFOUR739holomorphiccurve,weconstructanintegralmanifoldofimage,under ,isanopenneighborhoodof),acodimension-4submanifoldofSO(6).Webeginbycon-structinganintegralofthe1-forms.On,thecomplexspanofthe1-formsisone-dimensionalateachpoint.Ifisalocalholomorphiccoordinateon,thentherewillbecomplexfunctionssuchthat.Substitutingthisin(44)givesmoddz.bea“berof.Sincesuch“bersareleftcosetsofSO(6),(3)actssimplytransitivelyonthembyrightmultiplication.Thisactionisgeneratedin“nitesimallybytheleft-invariantvector“eldsonSO(6)thataretangenttosubalgebra(3)attheidentity.Thus,suchvector“eldsgiveaframe“eldtangentto.Becauseis(3)-valued,(47)showsthatastheactionof(3)movespointsalong,thecorrespondingactiononthevectorwithcomponentsisisomorphictothestandardactionof(3)onThus,ineach“berthereisasubsetwhere=0,andtheunionofthesesubsetsisasmoothsubmanifoldofcodimension4within.(Notethattheconstructionofdoesnotdependonthechoiceoforthelocalcoordinateon=0,thenand.Thus,therestrictionof(44)toimpliesthattherearecomplex-valuedfunctionsf,ksuchthat=0,thesearetheonlylinearlydependenciesamongtheleft-invariant1-formsofSO(6)whenrestrictedto;thus,the1-forms(whichincludetherealandimaginarypartsof)formacoframeon)modandcomparingwith(45)showsthatontheimageunder ofasolutionofwemusthaveequalto1.Thus,inordertoconstructacandidateforsuchanimage,weneedtorestricttothesubsetisrealandpositive.Dierentiating(48)givesforsomecomplexfunction.Thisequationshowsthatthesubgroup(3)stabilizingcanbeusedtomakerealandpositive,providedthatarenotbothidenticallyzeroalonga“ber.Sincef,kgivethecom-ponentsofthesecondfundamentalformofasaholomorphicsubmanifold,then=0alongthe“berabovemeansthatisaplanarpoint.Thus,wewillrestricttononplanarpointsof.Then,ineach“berof