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Hyperkahler Manifolds

Permanent Link: http://ufdc.ufl.edu/UFE0041403/00001

Material Information

Title: Hyperkahler Manifolds
Physical Description: 1 online resource (80 p.)
Language: english
Creator: Fisher, Andrew
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: calabi, complex, cotangent, hyperkahler, projective, quaternion, reduction, symplectic, tangent
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The University of Florida is not alone in tracing its origins to 1853: important events in modern geometry (of curved spaces) and quaternions (a number system in which ab and ba may differ) may also be traced to this year. Regarding the former, Bernhard Riemann began preparing his inaugural lecture 'On the hypotheses which lie at the foundations of geometry' to the Gottingen faculty, delivered the following year. Regarding the latter, Sir William Rowan Hamilton presented to the public the results of his research in 'Lectures on Quaternions'. On the centenary of these foundations, Marcel Berger showed the special place occupied by hyperKahler manifolds: Riemannian spaces that incorporate quaternions in their structure. In this dissertation we examine the geometry of a particular manifold produced by the hyerpKahler reduction procedure of N. Hitchin, A. Karlhede, U. Lindstrom, and M. Rocek applied to quaternionic (n+1)-space, making quite explicit the identity with the cotangent bundle of complex projective n-space; the details of this identification are somewhat elusive in the literature. More generally, we present a scheme whereby the (co)tangent bundle to a suitable base manifold may be given a hyperKahler structure. We show that this scheme succeeds precisely when the base is a flat Kahler manifold and consider possible extensions. In developing this last result we make use of a special class of functions found in 'Tangent and Cotangent Bundles' by K. Yano and S. Ishihara. Our approach is more coordinate free in spirit which we feel makes some of the result of this text more transparent.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Andrew Fisher.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Robinson, Paul L.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041403:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041403/00001

Material Information

Title: Hyperkahler Manifolds
Physical Description: 1 online resource (80 p.)
Language: english
Creator: Fisher, Andrew
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: calabi, complex, cotangent, hyperkahler, projective, quaternion, reduction, symplectic, tangent
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The University of Florida is not alone in tracing its origins to 1853: important events in modern geometry (of curved spaces) and quaternions (a number system in which ab and ba may differ) may also be traced to this year. Regarding the former, Bernhard Riemann began preparing his inaugural lecture 'On the hypotheses which lie at the foundations of geometry' to the Gottingen faculty, delivered the following year. Regarding the latter, Sir William Rowan Hamilton presented to the public the results of his research in 'Lectures on Quaternions'. On the centenary of these foundations, Marcel Berger showed the special place occupied by hyperKahler manifolds: Riemannian spaces that incorporate quaternions in their structure. In this dissertation we examine the geometry of a particular manifold produced by the hyerpKahler reduction procedure of N. Hitchin, A. Karlhede, U. Lindstrom, and M. Rocek applied to quaternionic (n+1)-space, making quite explicit the identity with the cotangent bundle of complex projective n-space; the details of this identification are somewhat elusive in the literature. More generally, we present a scheme whereby the (co)tangent bundle to a suitable base manifold may be given a hyperKahler structure. We show that this scheme succeeds precisely when the base is a flat Kahler manifold and consider possible extensions. In developing this last result we make use of a special class of functions found in 'Tangent and Cotangent Bundles' by K. Yano and S. Ishihara. Our approach is more coordinate free in spirit which we feel makes some of the result of this text more transparent.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Andrew Fisher.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Robinson, Paul L.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041403:00001


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First,IwouldliketothanktheGodofAvraham,Yitzchak,andYa'akovaswellasHisanointedoneYeshua,withoutwhomIwouldnotbehere.Second,IwouldliketothankmyparentsforinstillinginmetheimportanceofschoolandwithoutwhosesupportIwouldneverhavemadeitthisfar.Third,IwouldliketothankmyadviserDr.PaulRobinsonforhisadvice,patience,andunderstanding.HeprovidedallthehelpIneededfromhowtoteachtohowresearchisconducted.Andtomycommitteemembersfortheirsuggestionsandsupport.Thankyou. 4

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page ACKNOWLEDGMENTS .................................. 4 ABSTRACT ......................................... 6 CHAPTER 1HYPERKAHLERMANIFOLDS ........................... 7 1.1HyperKahlerManifolds ............................. 7 1.1.1KahlerManifolds ............................ 7 1.1.2TheLevi-CivitaConnectionandKahlerManifolds .......... 8 1.1.3HyperKahlerManifolds ......................... 11 1.2TheReductionProcedure ........................... 13 1.2.1GroupActions .............................. 13 1.2.2SymplecticReduction ......................... 16 1.2.3KahlerReduction ............................ 17 1.2.4HyperKahlerReduction ........................ 20 2HYPERKAHLERREDUCTION:ASPECIALCASE ................ 21 2.1RepresentationsofM ............................. 24 2.2TangentSpacesandHyperKahlerForms .................. 26 2.3IntegralCurvesandtheBaseSpace ..................... 32 2.4TheFibersoftheProjection .......................... 36 2.5i,j,konVerticalVectors ............................. 39 2.6AdditionontheFiber .............................. 40 2.7TheCotangentBundletoCP(n)andtheStandardForm .......... 46 3GEOMETRYOFTANGENTBUNDLES ...................... 49 3.1Introduction ................................... 49 3.2TheGeometryoftheTangentBundle .................... 51 3.3SpecialfunctionsontheTangentBundle ................... 54 4HYPERKAHLERSTRUCTURESONTANGENTBUNDLES ........... 59 5CONCLUSIONSANDFUTUREDIRECTIONS .................. 72 REFERENCES ....................................... 78 BIOGRAPHICALSKETCH ................................ 80 5

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TheUniversityofFloridaisnotaloneintracingitsoriginsto1853:importanteventsinmoderngeometry(ofcurvedspaces)andquaternions(anumbersysteminwhichabandbamaydiffer)mayalsobetracedtothisyear.Regardingtheformer,BernhardRiemannbeganpreparinghisinaugurallecture'Onthehypotheseswhichlieatthefoundationsofgeometry'totheGottingenfaculty,deliveredthefollowingyear.Regardingthelatter,SirWilliamRowanHamiltonpresentedtothepublictheresultsofhisresearchin'LecturesonQuaternions'.Onthecentenaryofthesefoundations,MarcelBergershowedthespecialplaceoccupiedbyhyperKahlermanifolds:Riemannianspacesthatincorporatequaternionsintheirstructure.InthisdissertationweexaminethegeometryofaparticularmanifoldproducedbythehyerpKahlerreductionprocedureof[ 13 ]appliedtoquaternionic(n+1)-space,makingquiteexplicittheidentitywiththecotangentbundleofcomplexprojectiven-space;thedetailsofthisidenticationaresomewhatelusiveintheliterature.Moregenerally,wepresentaschemewherebythe(co)tangentbundletoasuitablebasemanifoldmaybegivenahyperKahlerstructure.WeshowthatthisschemesucceedspreciselywhenthebaseisaatKahlermanifoldandconsiderpossibleextensions.Indevelopingthislastresultwemakeuseofaspecialclassoffunctionsfoundin[ 14 ].Ourapproachismorecoordinatefreeinspiritwhichwefeelmakessomeoftheresultofthistextmoretransparent. 6

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ThepurposeofChapter1istointroducethenecessarydenitionsandsketchthebackgroundneededfortherestofthepaper.OurdiscussionwillcloselyparalleltheaccountofRobertBryant[ 4 ].Thereaderisinvitedtoconsult[ 4 ]foramorefulldiscussionofthedetailsifrequired.Foranalternativeaccountonecanconsult[ 12 ]. 1.1.1KahlerManifolds 2[(j)+(IjI)]. @xi=@ @yiandI@ @yi=@ @xi.Soa(realsmooth)manifoldwhich 7

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18 ]or[ 15 ]). Ametric(j)andcompatiblealmostcomplexstructureIgiverisetoatwo-form!Igivenby!I(j)=(Ij). 4 ].

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20 ]).ItwillbeimportantforafollowingtheoremtobringoutonemorecharacterizationoftheLevi-Civitaconnection: Thefactthatristorsionfreefollowssimilarly. ToseehowtheconnectionrelatestoKahlermanifolds,wenowletIbeanalmostcomplexstructurecompatiblewiththemetric.WethendenerI:Vec(M)!Vec(M)by(rI)=r(I)Ir.

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Wenowareabletorelatetheconnection,metric,Iandtheassociatedform!I. 1.1 .Oncewenotethatd!I(,,)=!I(,)+!I(,)+!I(,)+!I(,[,])+!I(,[,])+!I(,[,])=(Ij)+(Ij)+(Ij)(Ij[,])+(Ij[,])+(Ij[,]) WethenhavethefollowingCorollary. 1. 2. Proof.

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Theabovetheoremthenimpliesthatd!I(,,)=d!I(I,I,).Wewillthenhaved!I(,,)=d!I(,I,I)=d!I(I,,I)aswell(byusingthefactthatd!Iisalternatingandthenrelabeling). Thisnowallowsustowrited!I(,,)=d!I(I,I,)=d!I(I,I2,I)=d!I(I2,I2,I2)=(1)3d!I(,,)=d!I(,,). Thefollowingtheoremfollowsimmediately. Wewillrecallafewofthebasicfactsaboutquaternions;foramoredetailedaccountonecanconsult[ 9 ].FirstthequaternionsHcanbeidentiedwithasubsetofM22(C)byx01+x1i+x2j+x3k:=0B@x0+ix1x2+ix3x2+ix3x0ix11CA.

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[yjx] =1i+2j+3k. Onelastpropertyofquaternionsthatwewilluseinthefollowingisthefactthat(a+bi)j=j(abi)fora,b2R.Thusifjispassedacrossanelementc2span(1,i)(thatisciscomplexwithrespecttoi),thencisconjugated. Now,inordertocapturethequaternionicstructureintheKahlersetting,wemakethefollowingdenition: 1.4 ,wehave 12

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ThusourdenitionofahyperKahlermanifoldisequivalenttohavingthreecompatibleanti-commutingcomplexstructuresI,JandK(=IJ)suchthattheassociatedtwo-forms!I,!Jand!Kareclosed.Whenwewishtoemphasizetheforms,wewillwritethehyperKahlermanifoldas(M,!I,!J,!K)or(M,!1,!2,!3)where1,2,and3correspondtoI,J,andK,respectively. OnecanrecoverthecomplexstructuresfromtheformsintheKahlercase.However,itisnotenoughsimplytochoosethreesymplecticformsifonewantstoobtainahyperKahlermanifold.Asasimplecounterexample,wecouldchoose!2and!3tobeconstantmultiplesof!1.Thenthecomplexstructuresthatareobtainedwillcommute. AsarstexampleofahyperKahlermanifoldletM=Hn.Letv2HnTqHn(seeExample 1 )usingthestandardidenticationofHnwitheachofitstangentspaces.WethendeneI,JbyIv=ivandJv=jvrespectively.Forthemetric,wedeclaretheidenticationanisometry. Afteroneconsidersvectorspaces,itiscommontoconsidernexttheprojectivespaces.Then-thdimensionalquaternionicprojectivespaceisgivenbythecollectionofonedimensional(left)H-subspacesinHn+1.Theseturnouttobenon-examplessincetheexistenceofanalmostcomplexstructureentailscharacteristicclassrestrictionswhichquaternionicprojectivespacesdonotsatisfy(see[ 17 ]). 1.2.1GroupActions 20 ],[ 21 ]and[ 7 ].Adetailedknowledgewillnotbenecessaryforthepaper,sincewewilldealwithaconcreteexampleinwhichthepropertiescanbeseendirectly.However,therudimentsareusedtostatethefollowingtheorems. 13

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1 ]). 4 ]).

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4 ]). 15

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Proof. NextweshowthatisaLiealgebrahomomorphism.Wehavef,g=()()=()((()))=L()((()))=(L())(())+(L()())=([(),()])=(([,]))=[,] Associatedtothelinearmap:g!C1(M)wehaveamap:M!ggivenby:z()=(z) 4 ]). 11 ]or[ 16 ]).Tostartthetheorywegiveourselvesaconnectedsymplecticmanifold 16

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1.9 )thesetmap ThesubsetM=(

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4 ]. Wewillalsodeneaone-formbyz(vz)=1 2(z,v).Then!=d.Thus!isclosed.(SeethebeginningChapter2foradetaileddemonstrationofthesefactsinasimilarcase.)As!isalsonon-degenerate,MisaKahlermanifold. DeneanactionofthecircleS1onMby(eit,z)=eitzforz2M.Thisactionwillclearlypreservethecomplexinnerproductandhenceitsrealandimaginaryparts.Furthermore,since(eit):TzM!TeitzMisgivenby(eit)(v)=(eitv)eitz,isalsopreservedbytheaction.Indeed,wehave(eit)z(vz)=eitz((eitv)eitz)=1 2(eitz,eitv)=(z,v)=z(vz).

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1.8 ). Let=@ @tbethestandardvectoreldonthecircle.Then()z=(iz)z.ItnowfollowsfromtheproofofTheorem 1.8 thatwemaytakethelinearmaptobet(z)=((t))(z)=1 2(tiz,z)=1 2kzk2t. 2kzk2t.IfweidentifygwithR,thenwemaywritethismapas:Cn+1!R:z7!1 2kzk2. Sincethepreimagesofallnegativevaluesarespheres,it'sclearthateverynegativevalueisacleanvalue.Forexample,wecanlookatthepreimageof-1 2.Then1(1 2)=S2n+1andsothequotientM1 2isdiffeomorphictoCPn. Sincethecoadjointactionistrivial(becauseG=S1isabelian),wecanperformKahlerreductionat=1 2. Wewillcalculate!usinghomogeneouscoordinatesinstandardnotation.LetA0CPnbegivenbyA0=f[z0,...,zn]jz06=0g.Thenthereisabijectivemap:Cn!A0 Wemayexpress!onMas!=i zkwherethecoordinatesonMaregivenbyzk=xk+iykandthencalculate:(!1 2)=()(!)=i wk wk W3)=i wk

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wkweget:g1 2=(W2jk wk 13 ](seealso[ 10 ]). 4 ]). AsurprisinglyricharrayofhyperKahlermanifoldsisproducedwhenthisprocedureisappliedtoevensoelementaryaspaceasHn+1.InChapter2weworkoutthedetailsofaspecialexample:thatinwhichS1=feit:t2RgactsdiagonallyonHn+1ontheleft,inwhichcaseitturnsoutthatthereducedspacemaybeidentiedwithTCPn. 20

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InChapter2wewillinvestigatethehyperKahlerstructureobtainedbyperforminghyperKahlerreductiononHn+1withG=S1actingdiagonallyontheleft.Recallthatthequaternionicinnerproduct[wjz]= WethendenetheRiemannianmetricbygc(xc,yc)=(x,y)forallx,yandcinHn+1.AsinChapter1,wewilloftendenotethemetricby(j).Similarly,thetwo-formsmaybedenedby(!`)c(xc,yc)=`(x,y)for`=1,2,3. Toseethateachoftheseformsisclosed,wedenex2Vec(M)byxc=(x)c=c(x)forallc2Hn+1andxedx2Hn+1.Clearly,thesevectoreldsspanallthevectoreldsonHn+1.(Indeed,ife1,...,e4n+4isabasisforHn+1asarealvectorspacewithx1,...,x4n+4thedualbasis,thenthestandardtangentvectors@ @x`jcatthepointcaregivenby@ @x`jc=c(e`).)Thusitisenoughtoshowd!`(x,y,z)=0forallx,y,z2Hn+1and`=1,2,3(utilizingthefactthatd!`isC1(Hn+1)-linearineachvariable).Sinceourfollowingargumentdoesnotdependonwhichformisused(i.e.!1,!2or!3),wewillsuppressthesubscriptforthefollowingcalculations. Firstrecallthatd!(x,y,z)=x!(y,z)+z!(x,y)+y!(z,x)!(x,[y,z])+!(z,[x,y])+!(y,[z,x]).

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20 ]orTheorem4.3.3of[ 7 ])). Fortherstthreetermsontherighthandside,werstnotethatthedenition!c(x,y)=(x,y)forallc2Hn+1yields!(x,y)=(x,y).Inparticular,!(x,y)isaconstantfunctionandsoz!(x,y)=0.Thusthecyclicsumwillbe0whichgivesusd!(x,y,z)=0. Theforms!`thatariseareevenbetterthanclosed,theyareinfactexact.Forthis,wedeneavectoreld2Vec(Hn+1)viac=c(c)=cc(whichisjusttheEulereldincoordinatefreeterms).Thenwemaydenetheone-forms1,2and3by`=1 2y!`foreach`=1,2,3,sothat`c(v)=1 2`(c,v)for`=1,2,3(where`=`).Finally,weclaimthatd`=!`. Toseethis,westartwiththeowoftheEulereld.Sincec=c(c),anintegralcurvectforthroughcisasolutiontothedifferentialequationc0t=ct.Hencetheowisgivenbyt(c)=etc. Thisallowsustocomputet!`by(t!`)c(xc,yc)=!`etc(t(xc),t(yc))=!`etc((etx)etc,(ety)etc)=`(x,y)e2t=!`c(xc,yc)e2t. NextwedeneI,JandKsuchthatIv=vi,Jv=vj,andKv=vkforeachv2TcHn+1.Then!1,!2and!3aretheassociatedtwo-formsforI,J,andKrespectively. 22

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Nowwewillndthemomentummap:Hn+1!ggg. FollowingtheexampleinChapter1foreachofthethreeKahlerstructures(Hn+1,1),(Hn+1,2)and(Hn+1,3),weagainlet=@ @tbethestandardvectoreldonthecircle.Asbefore,wehave()q=(iq)q. Thelinearmap`associatedwith(Hn+1,`)canbetakentobe`t(q)=`((t))(q)=1 2`(tiq,q)=1 2`(q,iq)t. 2`(q,iq),onceweidentifygwithR. Wethenobtainthemomentummap:Hn+1!gggforthehyperKahlermanifoldbyidentifyinggggwithImHn+1andthenwriting(q)=1(q)i+2(q)j+3(q)k.Furthermore,since(qjiq)=0,wemaywritethismapas:(q)=[qjiq]. SinceG=S1isabelian,wecanperformhyperKahlerreductionati.Accordingly,wedeneM=1(i)andthequotientasN=GnM.Wewilldenotethequotientmapby:M!Nandinclusionby:M!Hn+1.ItisafactthatthismapisasubmersionsinceGiscompact(seepages262-267andinparticularTheorem4.1.23of[ 1 ]). 23

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4 ],[ 12 ])declaresNtobediffeomorphictoTCPnwiththeinducedhyperKahlerstructurecorrespondingtotheoneintroducedbyCalabi([ 5 ]).ItisthepurposeofChapter2toexplicitlyillustratethisrelationshipwiththequotientmanifoldNasitisgivenusingthemanifoldTCPnasaguide.Inaccordancewiththisaim,wewillidentifythebasespaceCPnasasubsetofN,ndanappropriateprojectionmapofNontothisspace,andshowhowthebersofthismapactonthetangentstothebase.Concurrentwiththispresentation,wewillndexplicitrepresentationsoftheformsandthecomplexstructuresobtainedbythereductionprocedureaswellasinvestigateafewoftheirproperties. SinceNisaquotientofthethemanifoldMaswellasMbeingasubsetofaEuclideanspace(makingiteasiertoworkwith),wewilloftenndthestructuresthatweneedbyrstlookingatM. OurrststepwillbetodevelopthevariousrepresentationofthesubsetMwhichwewilluse. Letq=a+bjfora,b2Cn+1,then bt)i(a+bj)=(

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2(qiqi)andq=1 2(q+iqi).Foruniqueness,supposeq+1+q1=q=q+2+q2wherethe+indicatescommutingwithiandtheindicatesanti-commutingwithi.Thenq+1q+2=q2q1.Thusitisenoughtoshowthatifqcommutesandanti-commuteswithithenq=0(fortheneachsideofthepreviousequationwouldbezero).Butthisresultistrivialsinceicommutingandanti-commutingwithqimpliesiq=qi=iqfromwhichq=0readilyfollows.ThusHn+1maybewrittenasadirectsumofthei-commutingandi-anti-commutingspacesH+andH,respectively. Usingthesecondrepresentationandthefactthat=[ajb]=0,[ajbj]=0,wegetM=fa+b2Hn+1:a2H+,b2H,kak2=kbk2+1and[ajb]=0g, Proof. 25

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Finally,supposec=c+(andsoic=ci)whichyieldsi=[cjic]=[cjci]=[cjc]i=kck2i.Hencekck2=1orkck=1. Afewnalobservations.Fromkc+k2=1+kck2and[c+jc]=0,onegets:2kc+k2=kck2+1,2kck2=kck21. [zjic]fromwhichweobtainTc=fz2Hn+1:[icjz]2Rg.

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Wewillhandlethetwo-formssimilarly,i.e.wedene!M1,!M2and!M3by(!M`)c(1cx,1cy)=`(x,y) TogetanexplicitrepresentationforthetangentspaceforN,wewillndthe(real)orthogonalcomplementtothekernelFcof:TcM!TcNwhere:M!Nisthequotientmap.SincethekernelcorrespondstothetangentstotheberwemaywriteFc=Ric.Thuswehave:Fc=fz2TcM:(icjz)=0g=fz2Hn+1:Re[icjz]=0and[icjz]2Rg=fz2Hn+1:[icjz]=0g=(ic)2 ItfollowsthatthecompositionTc1!TcM!TcNrestrictstoabijectionon(ic)2.Thuswecanidentify(ic)2withthetangentspacetoNatthepoint(c).WethenputametriconNbydeclaringthismaptobeanisometryforeachcanddenethetwo-forms!N1,!N2and!N3onNby(!N`)c(1c(x),1c(y))=`(x,y),sothat!N=!M.Theformsarewell-denedsincemultiplicationbyeitpreserves 27

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=1i+2j+3kandsimilarlyfor! FromTheorem 1.12 theformswegetonNandthemetrictintoahyperKahlerpicture.However,wecanshowthisexplicitly. Noticethatifq2Handx2TcN(=(ic)2),wehave[icjxq]=[icjx]q=0andsoxq2TcN.Thus,inparticular,ifx2TcN,thenxi,xj,xk2TcN.Sotherelations1(x,y)=(xijy),2(x,y)=(xjjy),and3(x,y)=(xkjy)onHn+1continuetoholdifweconsider`and(j)restrictedtoTcN.Itfollowsthat!N1(x,y)=(xijy),!N2(x,y)=(xjjy),and!N3(x,y)=(xkjy). Nextwewillshowtheseformsarenon-degenerate. (z,x)=0forallx2TcM.ToseethatFcK,letx2TcM.So[icjx]2R,whichinturnmeansthat (ic,x)=0.ItfollowsthatFcK. Fortheconverse,letz2K.Then[icjx]2R(equivalentlyx2TcM)implies (z,x)=0(sincez2K).Sowethengetforeach`=1,2,3:1(ic,x)=2(ic,x)=3(ic,x)=0)`(z,x)=0.

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Fromthenon-singularityof1onHn+1,z=icqforsomeq.(Infactq=13j+2k.)Now[icjicq]=[icjic]q=kck2q.Sincez2TcM,wemusthaveq2R.Thusz2FcandK=Fc. Thereverseinclusionisobviousandsotheresultfollows. SinceprojectionontoFckillsFc,itfollowsthattheformswillbenon-degenerateonTcN. Toseetheformsareclosedwerstnotethat0=d!M=d!N=d!N. 29

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1.5 ).ItturnsoutthatthisvectoreldisessentiallytheEulereldintheber. Soifwewanttocomparetheform!N2tothestandardsymplecticform,wemightstartbyndingavectoreld(onN)suchthattheone-formN2=y!N2canbeidentiedwiththestandardpotential.Tondavectoreld,recallthatinthecaseofHn+1wedenedthevectorsuchthatc=c(c)andthendened 2cy! 2 (c,z).Since(!N2)c(x,y)=2(x,y)whenwemaketheappropriateidentication(asintheabove),onemightattempttorestrict1 22(c,z)toTcN(identiedwith(ic)2)anddeneaone-formby2c(c(z))=1 22(c,z)forallz2TcN.However,cmaynotbeinTcN.Thisleadsustowanttoadjustthevectorcwhilekeepingthevalueof2(c,z)thesameforallz2TcN.Since,atthisstage,thereisnoreasontoexpect!N2tobetheappropriateformover!N1and!N3,wewilladjustthevectorcwhilekeepingthevalueof (c,z)thesameforallz2TcN(thushandlingallthreeformsatonce).Inordertoaddavectorvtocandhave (c+v,z)= (c,z),wewantvsuchthat (v,z)=0forallz2TcN.DeneKN=fv2Hn+1:[icjz]=0)[vjz]2Rg.

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2.1 aboveshowsthatwecanrewritethissetas:KN=fv2Hn+1:[icjz]=0)[vjz]=0g=(ic)22=icH. Sothevectorvthatwewantisv=ici 2(c+ici 2 (c+ici 2 (c,z). (c,z)thesameforallz2TcM.Inordertoaddavectorvtocandhave (c+v,z)= (c,z),weneedv2Fc.Soletv=ticfort2Randwecalculate:R3[icjc+v]=[icjc]+t[icjic]=i+tkck2. ForM2,wewantavsuchthat2(c+v,z)=2(c,z)forallv2TcMi.e.wewant2(v,z)=0forallz2TcM.ThusthesetKNbecomes:KM=fv2Hn+1:[icjz]2R)[vjz]2RRiRkg. 2.1 (withappropriatemodications),weagaingetv=icqforsomeq2H.Since[icjic]2R,theconditionforKMimplies[vjic]2

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Sowemayletv=icqforq2RRiRk.AndnowthecalculationR3[icjc+v]=[icjc]+[icjic]q=i+kck2q 2(c+ici Theone-formsN`andM`arealsorelatedfor`=2,3.InfactM`=N`for`=2,3.Itnowfollowsfrom!M`=dM`=dN`=dN EventhoughMy!Mdoesnotmatchtheconstructionsabove,itdoesstilldeneaone-formwhichwewillhaveoccasiontouse.SowewilldeneM1=My!M1. RecallthatonthecotangentbundleTPtothemanifoldPthevectoreldPsatisfyingP=Py!PisessentiallytheEulereldineachber.Itsowisgivenbyt(p)=etpforp2TpP.Notice,inparticular,thattremainsintheberTpPforalltime.Infact,astgoesto,tapproachesthezerotangentvectorintheberTpP.SincethezerosectionofthecotangentbundleTPcanbeidentiedwiththebasespaceP,weseethatlimt!t(p)givesusapointinthebase.Hence,ifweknow 32

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Inlikefashion,wewishtorecoverthebasespaceofNbyusingtheowofasuitablevectoreld.ComparingtherelationshipsP=Py!PwithN`=y!N`leadsonetoadoptthevectoreldasdenedintheprevioussectiontoplaytheroleofP.However,aswasmentionedabove,itiseasiertocalculateonMthanitisonN,sowewillperformtheaboveplanonMandthenfactorbythegroupaction.SowewillstartwiththevectoreldM. Itturnsoutthatonecanexplicitlycalculatetheintegralcurveswithouttoomuchdifculty.Wewillbreaktheproblemintothecommutingandanti-commutingpartsofM.Wehave:Mc=1 2(c+ici 2((c++c)+(ic+i+ici 2((c+c+ 2[ct+icti 2[ctjictcti 2kctk2[ictijict]1 2kctk2[ctjcti]=[ctjict]i=g(t)i. ThistellsusthatifctstartsinM,itremainsinM.Inparticular,itfollowsthatf(t)=[ctjct]1fromTheorem 2.1 .Wecanndamoreexplicitcharacterizationoff,whichwillbeusefulforlater,byndingadifferentialequationthatfsolves. 33

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2[ct+icti 2[ctjct+icti 2[ctjct]+1 2[icti 2[ctjct]+1 2[ctjicti [ctjct]=f(t)1 Sosupposef>1andsof0=f21 f21f0=1)1 2(1 Returningtotheproblemofndingct,weseeweneedtosolvethedifferentialequations:8><>:(c+t)0=1 2(c+tc+t 2(ct+ct 2.1 ,weseethatct=c+t,sothesecondequationreducesto(ct)0=0.Togetherwehavec0=0andconsequentlycc0.Wenowsupposekc0k>1.Sowehavef=p 34

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2(c+tc+t 2(ct+ct 2(1p 2(p Thissuggeststhatwedenethebasespacetobe:S=fc2M:kck=1g. 2.1 andwedenetheprojectionmaptobe(c)=c+

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SoweagainhavethesetS. Proof. Curve1:Denect=tu+etbwheret=p Curve2:Denect=u+eitb.Thenthederivativeatt=0yieldsthevectorib.Thisgivesthesecondterm. 36

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Subclaim:Foreachpairofunitcomplexvectorsu,vwith=0(whereisthecomplexinnerproduct),thereisaskew-adjointmatrixAsuchthatAu=v. [LetAx=vu.Then=<vjy><ujy>==uv>=. Itfollowsfromthissubclaimthatwecanndaskew-adjointmatrixthatwilltakethevectorb0toa(scalarmultipleof)anyotherorthogonalvectorinu.ThenwecanextendthismatrixbyrequiringAu=0.Thuswegetallofu\(b0),whichisthethirdterm. Sowehavea2n-(real)dimensionalspaceinTu+b1(u)andhenceisequaltothewholespacesinceTu+b1(u)isalso2n-(real)dimensional. Wecallthevectorstangenttotheberverticalvectors.Letuscalculatewhattheformsdotothesevectors. Firstwelookatthevectorskbk2 2.1 ).Thenwehave[kbk2

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Wecanwriteas1 2(c+ici 2.1 ).Asimplecalculationyields[kbk2 Nextwelookatib.Since[ibjiu+ib]=kbk2,ib=2TcNandweneedtoadjustthisvector.Thevectoribkbk2 Finally,weconsiderthevectorsvj2(u\(bj))jforv2C.Forthesevectors,wehave[vjjic]=j[vjic]=0and[vjjc]=0.Thus,wehave2[jvj]=[c+ici Conclusion:TheformsarezeroonalltheverticalvectorswiththeexceptionthatN1isnon-zeroon(kbk2 38

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Forthelatter,letvj2(u\(bj))jforv2C.Then[ujv]=0and[bjjv]=0.Thus[ujvi]=0and[bjjvi]=0.Sovji=vij2(u\(bj))j. Fortherstassertion,itiseasiesttostartwithibkbk2 Observation:IfwedidnotadjustthesecondsummandtoplacethevectorsintothetangentspaceofN,theninolongerleavesthespaceinvariant.Thisfollows 39

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Rightmultiplicationbyjalsodoesnotleavetheverticalvectorsinvariant.Forthis,noticethatinallthreesummandsoftheverticalvectors,thereisanon-zeropartthatanti-commuteswithi(providedweareoffthezerosection,i.e.b6=0).Letvjbeinthelastsummand.Thenvjjcommuteswithiandsothisisnotinthetangentspacetotheber. Multiplicationbykissimilar. Forsimplicity,considertheEulereldonavectorspaceV.Theowthroughthevectorv2Visgivenbyt(v)=etv.Thuswecanrecoverthe(positive)scalarmultiplicationm:R+V!Vbym(et,v)=t(v). RecallthattheowtforthevectoreldMisgivenbyt(c)=p

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r r Thisdenitionisnotatallobvious;however,thereisamorenaturalvectorspacethatcanbeidentiedwiththeberthatgivesrisetotheaddition.Let:1(u)!Cnbegivenby(u+b)=bj. t.Wewishtondu+bsuchthat=tandz t=bj.Inorderforu+btobeinM,weneedt2=1+kzk2 p

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p p p Wethenrecoverthescalarmultiplicationbyapplyingthemap.Forexample:(t(u+b))=q p p Sotheberisavectorspace.Undersuitablehypotheses,thisadditionisunique.Forthisweneedthefollowing: Foraproof,see[ 2 ].ThetheoremisstilltruefordimensiontwoifTisallowedtobeallorthogonaltransformations.AnaturalquestionforfurtherresearchistheextenttowhichtheunitarytransformationsonacomplexinnerproductspaceVdeterminetheaddition. Inordertosingleoutanaddition,wewillstartbyturningeachberintoa(real)innerproductspaceandthenshowhowtheoperatorsO(1(u))arisequitenaturallyfromthespaceHn+1(therebymakingtheadditiontheuniqueonepreservedbythese 42

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Toshowhowtheorthogonaltransformationsarise,webeginwiththemostnaturalclassofoperatorsonthevectorspaceHn+1:thelinearoperatorsonHn+1.Wethenrestricttothesubsetofoperatorsthatarelinearwithrespecttoournewscalarmultiplication,i.e.T(t(a+b))=t(Ta+Tb).Sincewewanttodescribeanadditionon1(u),weimposethelastrestrictionT(1(u))1(u). Tolearnmoreabouttheseoperators,letTbeareallinearoperatoronHn+1suchthatTpreservesthebercontainingc=a+bandTcommuteswithscalarmultiplication.(Hereasalways,a=c+andb=c). Itfollowsfromlettingt=0thatT(a k(Ta=kTak)k=a NowapplyingTtot(a+b)yields:q p p p p

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p p p p q q kbk=p kTbk. Noconditionwasplacedonbexceptthatb=cforsomec21(u).However,everyb2Hcanbeobtainedinthisway.ThusTisalinearisometrywhenweconsideritasanoperatoronHaftergivingHtheinherited(real)innerproductfromHn+1.ThusTpreservetheinnerproduct(j)restrictedtoH. SotheclassofoperatorsT2GL(Hn+1)thatpreservetheber1(u)andarecompatiblewiththescalarmultiplicationisT:=fT2GL(Hn+1):T(tu)=tuandT2O(H)g.

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Furthermore,everysuchorthogonaloperatorarisesinthisfashion.Indeed,ifweletT0beanorthogonaloperatorontheberthen:ka1kka2k(T0b1jT0b2)=(T0(a1+b1)jT0(a2+b2))F=(a1+b1ja1+b2)F=ka1kka2k(b1jb2).

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WehavethusshownthattheoperatorsO(1(u))arisefromtheclassofoperatorsT2GL(Hn+1)thatpreservetheber1(u)andarecompatiblewiththescalarmultiplication.ThisclassofoperatorsthensinglesoutauniqueadditionbyTheorem 2.2 Intheprevioussection,weidentiedthebersofwithCn.Wecanusetheinnerproductonthisspacetodenehow(u+b)actsasaform.Wedene(u+b)(z)=((u+b)jz)=(bjjz) Withthisidentication,wecanndthestandardpotentialonN.LetPbethestandardone-formonthecotangentbundle.BydenitionP(c)((z))=(c)((z)) 46

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2k+0k 2k+0k3 2kc+k3=z+ Wewillnextshowthat2isexactlythisform.Wedoapreliminarycalculationrst.Letz2TcNandso[icjz]=0.Splittingintorealandimaginarypartsandreorganizingwillgiveus:0=[ic++icjz++z]=[ic+jz+]+[ic+jz]+[icjz+]+[icjz]=()i+(+)ij. Nowtocalculate2,westartwith[jz]=[kck2

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1.1.2 ).Wewillusethisconnectiontosplitthetangentspacetothetotalspaceofthevectorbundle:TM!Mate21(p)asTe(TM)=VeHewhereVe=ker()=Te(TpM)aretheverticalvectorsandHe=fs(Xp):p=(e)andrXps=0g(forsasectionofTM)arethehorizontalvectors. NoticethatVeTpM.Thisisthecanonicalisomorphismbetweenavectorspaceandeachofitstangentspacesitwasdenotedbyc:V!TcVinChapter2whereVwastakentobeHn+1.Themapcwillnotbeexplicitlymentionedinthissection,althoughinsomelocations(mostnotableinthesecondproofofTheorem 3.3 )wemaystillemploythenotation()c:V!TcVforthismap.Usingthiscorrespondence,wedenetheverticallift_2Vec(TM)given2Vec(M)byVeTpM_e$p. ThemapisontoandVeisthekernelandso:He!TpMisanisomorphism.Wedenethehorizontallift2Vec(TM)given2Vec(M)byHeTpMe$p. 49

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1. Theintegralcurveof_througheisgivenbye(t)=e+tp. 2. Theowofisgivenbyparalleltransportt(e)(fore2TpM)alongtheintegralcurveforthroughthepointp. 20 ]orChapter10of[ 7 ](inparticularTheorem10.1.13). dte(t)=pand(_e)=pforallesuchthatp=(e). (2)Inordertoshowthis,foreaseofnotationletec(t)=t(e).Wewillshowthatec(t)istheintegralcurveforthroughe.First,ec(0)=0(e)=eand(ec(t))=c(t)(wherec(t)istheintegralcurvefor),hencet(e)2Tc(t)M.Soec(t)isaliftofthecurvec(t). Nextweshowthatec0(t)2Hc(t).Ifc0(t)=0forsomet,thencp.Soec(t)isconstant,thusec0(t)=02He.Thuswesupposethatc0(t)6=0forallt.TakeavectoreldYinaneighborhoodofc(t)suchthatt(e)=Yc(t).(Seepage233ofthesecondvolumeof[ 20 ]fordetails.)Sinceec(t)isparalleltransport,wehave:0=rc0(t)Y=Y(c0(t))=(Yc)0(t)=(0t(e))=(~c0(t)). 50

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Theeasiestis[_,_].Asimplecalculationshowsthat[_,_]isaverticalvectoreldsince[_j_]f=[_j_](f)=_(_(f))_(_(f))=0. 3.1 part1sincetheowsoftheverticalvectoreldsclearlycommute(seeforexampleCorollary5.12inVolumeIof[ 20 ]orTheorem4.3.3of[ 7 ]forwhythisissufcient). Wecanalsocalculate[j_].Again,itisasimplecalculationtoseethat[j_]isvertical:[,_]f=[,_](f)=(_(f))_((f))=(0)_(f)=0. Proof. 3.1 ,theowtofisgivenbyt(e)=t(e)along 51

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Wenowproceedtoshow[,_]=(r)_.Wehave[,_]e=L_(e)=d dtt(_t(e))j0=d dt(tt(p))_j0=(rp)_. Finally,weturntothestudyof[,].Wewillrstidentifythehorizontalandverticalpartsofthisvectoreld.Wecalculate[,](f)=[,](f)=(f)(f)=([,]f), AninterestingrelationshipbetweenV(,)andthecurvatureR(,)arisesfromconsideringtheJacobiidentityforthetriple,,_.Fromtwoapplicationof[,_]=(r)_,weseethat[,[,_]]=(rr)_.Similarlyfromanapplicationof[,_]=(r)_and[_,]=(r)_weobtain[,[_,]]=(rr)_.NowtheJacobiidentityyields0=[,[,_]]+[_,[,]]+[,[_,]]=(rr)_+[_,V(,)+[,]](rr)_

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ThisreadilygivesusthefactthatifV(,)isaverticalliftforall,,thenR(,)=0forall,,.SoweseethatifV(,)isaverticalliftforall,,thenMisat. ItisalsotruethatifR(,)=0forall,,,thenV(,)isaverticalliftforall,.Werstprovealemma. Proof. Lett(e)=e+tpbetheowof_.Thenwehave0=[_,X]e=(L_X)(e)=d dtt(Xt(e))j0=d dt0tu(t(e))pj0=d dt(u(t(e)))pj0. NowweseethatifR(,)=0forall,,,then[V(,),_]=0.SobytheLemma,V(,)isaverticallift. 53

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14 ],wedeneaspecialclassoffunctionsonTM.Givenaone-form21(M),wecanobtainafunction Wedepartfromtheusageofthesefunctionsin[ 14 ]inthatwewillnotspecializetotheone-formsdf(exceptintheproofofTheorem 3.3 below).Akeyfeatureofthesefunctionsisthattheyseparatethetangentvectors.Beforeprovingthis,wewillshowhowtheverticalandhorizontalliftsinteractwiththesefunctions. 1. 2. Proof. dt dtp(e+tp)j0=p(d dt(e+tp)j0)=p(p). dt dtc(t)(t(e))j0=(rp)(e)=r 20 ]. Wenowprovetheassertionthatthefunctions

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Lete=p2TpM(i.e.apointinTM).Supposethatp6=0andwriteXe=e+_e.AsapreliminarycalculationwehavefromTheorem 3.2 :Xedf =(f(g)+(f)(g)+(g)(f)+g(f)f(rg)g(rg)+gf+fg)(p). =((f)(g)+(g)(f))(p)=(pf)(pg)+(pg)(pf). =2r(pf)2. 55

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=(pf)(pg)+(pg)(pf)=(pf)(pf). ThusXe=0forallesuchthate6=0.ContinuitythenstepsintoshowthatXe=0foralle.SoX=0. Now(X,A)(x,a)df dtdf dtf0a+tA(x+tX)j0. Hencewehave(0,A)(x,a)(df dtf0a+tA(x)j0.Assumex6=0.Letfbequadratic,i.e.f(v)=1 2(vjLv)forL:V!Vsymmetricandlinear. Thenf0a+tA(x)=1 2(a+tAjLx)+1 2(xjL(a+tA))

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Toillustratetheusefulnessofthesefunctions,wecalculatethebracketsoftheverticalandhorizontallifts,whichwewillneedinChapter4. Westartwiththebracketoftwoverticallifts:[_,_] Forthesecondbracket,noticethat(f)=f.Wethenhave[,_] )=(()(r))=((r))=(r)_ 3.3 ,weobtain[,_]=(r)_. NextwecalculateV(,):V(,) 57

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1. 2. 3. 3.3 )thatV(,)0forall,ifandonlyifR=0.InotherwordsV(,)0forall,ifandonlyifMisatsinceonecanshowthat(R(,))()=(R(,)). 58

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ForChapter4,wepresentadifferentapproachtoturningthetangentbundletoaatKahlermanifoldintoahyperKahlermanifold.Let(M,g,i)beaKahlermanifoldwheregisthemetricandiisthecomplexstructure.Asintheprevioussection,themetricgivesustheLevi-Civitaconnection,whichinturnyieldsavertical-horizontalsplittingofthetangentspacestothetangentbundle.Wethendenetheverticalandhorizontalliftsofvectoreldsasbeforeandgivethetangentbundleametricasintheprevioussection.Finallywedene(almost)complexstructuresonTMasfollows. ForcalculatingI,weintroducesomenotationnewoperatorsonvectorelds. 1.

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3.4 For(1),wecalculateNI(_,_)=[I_,I_][_,_]I[I_,_]I[_,I_]=[(i)_,(i)_][_,_]+I[(i)_,_]+I[_,(i)_]. For(2),wewriteNI(,_)=[I_,I][_,]I[I_,]I[_,I]=[(i)_,(i)]+(r)_+I[(i)_,]I[_,(i)]=(rii)_+(r)_I(ri)_+I(ri)_=(rii+r+iriiri)_=((rii)+i(ri)). ThisleadstonecessaryandsufcientconditionsfortheintegrabilityofI.

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20 ]).Ifri=0,thenrandicommute,henceRandicommute.Wethenhave:(R(i,i)j)=(R(,)iji)=(iR(,)ji)=(R(,)j)=(R(,)j) 3.4 .SowehaveV(i,i)V(,)IV(i,)IV(,i)=0, Nowfortheproofofthetheorem. 4.2 4.1 andLemma 4.1 SosupposeNI=0.Dene(,,):=((ri)j).If(,,)werezeroforall,,,thenwewouldhaveri=0andNiwouldfollowfromthecalculationfollowingDenition 4.3 .Sowewillshow0.Butrstweshowthatissymmetricinthersttwovariablesandanti-symmetricinthesecond.Frompart(2)oftheorem 4.2 ,(rii)+i(ri)=D=0=D=(rii)+i(ri). 4.2 andlemma 4.1 (orthefactthatV(,)andIV(,)arevertical)weobtainDD=0.Hencewehave:D=D)DD=DD

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Next,sinceipreservesthemetricandi2=1,wehave(ij)+(ji)=0.Soweget0=0=(ij)+(ji)=(rij)+(ijr)+(rji)+(jri).=(rij)(jir)(irj)+(jri)=((ri)j)+(j(ri). Nowtoshowthat(,,)=0. Considerthepermutationoftheletters,,givenby(13)(i.eweinterchangeand).Thispermutationcanbeachievedinthefollowingtwoways:(13)=(23)(12)(23)(13)=(12)(23)(12). NowfortheintegrabilityofJ.

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3.4 .WebeginwithNJ(_,_)=[J_,J_][_,_]J[J_,_]J[_,J_]=[,]J[,_]J[_,]=V(,)+[,]J(r)_+J(r)_=V(,)+([,]r+r)=V(,). NextwehaveNJ(,_)=[J,J_][,_]J[J,_]J[,J_]=[_,](r)_+J[_,_]J[,]=(rr+[,])_JV(,)=JV(,). SoweseethatJisacomplexstructureifandonlyifV(,)0forall,(equivalentlyMatbybytheremarkfollowingTheorem 3.4 ). 63

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1. 2. 3. 3.4 ). LettingX=_,Y=_andZ=wehave:d!I(_,_,)=_((i)_j)+_((i)j_)((i)_j_)((i)_j[_,])((i)_j[,_])+((i)j[_,_]).

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3.4 .Finally,thesixthtermiszerobythesametheorem.Sotheequationreducesto:d!I(_,_,)=[(ij)+(ijr)(ijr)]=[(rij)(ijr)+(ijr)(ijr)]=[(rij)+(jir)]=(j(rii)). 3.4 andhenceisorthogonalto(i).Thesixthtermiszeroforsimilarreasons.Soallthatisleftisthefourthterm.Replacing[,]byV(,)+[,].Since[,]ishorizontal,itisorthogonalto(i)_.Soweareleftwithd!I(_,,)=((i)_jV(,)). LettingX=,Y=andZ=wehave:d!I(,,)=((i)j)+((i)j)+((i)j)+((i)j[,])+((i)j[,])+((i)j[,]).

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1.4 andDenition(refsec:symmetric))andthensimplifyingtheterms. Fromthisitisclearthat!IisclosedifandonlyifMisatandri=0. 1. 2. Proof. 3.4 66

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3.4 )andthusisorthogonaltoahorizontallift.Thethirdiszerosincetheverticalandhorizontalliftsareorthogonal.Thesixthiszerosincethebracketoftwoverticalsliftsiszero(Theorem 3.4 ). LettingX=_,Y=andZ=wehave:d!J(_,,)=_(_j)(_j_)+(j)+(,[,])(_,[,_])(_,[_,]).

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SoweseethatifMisatthen!Jisclosed.(WesawabovethatMmustbeatifwewantJtobecomplex,sothisdoesnotaddedanyextraconditions.) Puttingtheabovefactstogether,wehavethefollowingtheorem. 1. 2. Proof. 3.1 .)Wethencancalculate_e()=d dt(t)((t))jt=0=d dt(((t))j(t))jt=0=d dt(pje+tp)jt=0

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Forthesecondassertion,werecallfromTheorem 3.1 thattheowofeisgivenbyparalleltransportt(e)(fore2TpM)alongtheintegralcurvecp(t)forthroughthepointp.Wenowcalculatee()=d dtt(e)(t(e))jt=0=d dt((t(e)jt(e))jt=0=d dt(cp(t)jt(e))jt=0=d dt(1t(cp(t))je)jt=0=(rpje), Wenowdeneatwo-form!by!=d.Togainfurtherinsightinto!weinvestigate!onhorizontalandverticallifts.Theresultisthefollowing. 1. 2. 3. Proof. 4.1 above.Let,2Vec(M)ande2TpM.Fortherstassertion,wehave!(_,_)=d(_,_)=_(_)_(_)([_,_]). 3.4 Nowforthesecondassertion,wehave!(,)(e)=d(,)(e)=(()()([,])(e)

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Nowforthelastassertion.Wecalculate!(,_)=(_)_()([,_]) 3.4 )that[,_]isverticalandso([,_])=0.SowehavebyClaim 4.1 Wewillnowshowthat!=!J. Proof. Thedenitionofmayresemblethestandardpotentialonacotangentbundle.It'sdenitionisinfactinspiredbyusinganinnerproducttoidentifyTPMandTpMandlookingatthestandardpotentialunderthisidentication.Infact,ifthediscussionof 70

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71

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InChapter4,wehaveshownamethodforconstructingahyperKahlerstructureonthecotangentbundleTMofaatKahlermanifoldM.Thisislessthanideal,especiallysinceoneoftheclassicexamplesisCalabi'sexampleonTCPn(see[ 5 ]).(Note:themanifoldCPnhasnon-zerosectionalcurvature(seepages73-75[ 6 ])andsothecurvatureRisnonzero(seeCorollary4.9ofVolumeIIof[ 20 ]).) Thusonedirectionforfutureresearchistoconsiderhowtoadjustourconstructiontoavoidtheatnesscondition.TracingthroughtheresultsinChapter4,oneseesthatitwas!IbeingclosedorJbeingintegrablethatforcedtheatnessofM.Toimprovetheconditionon!IwewillneedtochangeeithertheoperatorIorthemetricweplacedonTMsincethesearethetwoingredientsneededtodenethetwo-form.OtherthanitbeingaconvenientwaytoproduceametriconTMfromoneonM,therewasnoparticularreasontodenethemetriconTMaswedid.However,ifweadjustIwemayloosetheneededconditionNI=0.Soasarstattemptinimprovingtheconstruction,itisreasonabletoconsiderchangingthemetric. TheintegrabilityofJisdifferentsincechangingthemetriconTMwillnotaffectNJ.HoweverJwassimplyaconvenientchoiceforanalmostcomplexstructure.SoitisreasonabletoconsiderhowtoaltertheconstructionofJ.ByusingChapter2asaguide,weseethatinthecaseofthecotangentbundletoCPn,Jwasthecomplexstructureassociatedwiththestandardtwo-form.ThustoobtainJwemightinsteadstartwiththetwo-formofTheorem 4.7 (whichisessentiallythestandardtwoformonacotangentbundlecastinthetangentbundlecase),thenusethemetricandthisformtogiverisetoanewoperatorJ0totaketheplaceofJ.Soagainweseethatadjustingthemetricmightgiveusawaytobypasstheatnesscondition.Asapreliminaryapproachonemightconsideradjustingthemetricbyaconformalfactor. 72

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Chapter2suggestsfurtherdirectionsofstudy.TherstistoseeiftheconstructionofCalabi(see[ 5 ])canbeseentocomeoutnaturallyonthemanifoldN.ToseewhythisisfeasiblewewilltakeacloserlookatCalabi'sexample.Hisconstructionstartswithann-dimensionalcomplexmanifoldMwithconstantholomorphicsectionalcurvature2K6=0.ThenaformisdenedoneachberofthecotangentbundleTM!Mby@@(+ut), 1+p 73

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8 ]). HyperKahlerstructuresarenottheonlywaytoincorporatequaternionsintothegeometryofamanifold;wecanconsidertherelatedeldofhypersymplecticmanifolds.Thesearemanifoldsaremodeledonthesplitquaternionsinsteadofthequaternions.Wewillgivetherelevantdenitions(followingthearticleHypersymplecticManifoldsbyA.DancerandA.Swannin[ 3 ])andstatethecorrespondingresults.ThesplitquaternionsBareafour-dimensionalrealvectorspacewithabasis1,i,s,tsatisfyingi2=1,s2=1=t2,is=t=is. Todeneahypersymplecticmanifold,westartbygivinga4ndimensionalmanifoldPasignature(2n,2n)metricg.Wewill,asalways,writethemetricas(j).WeagaingiveMacomplexstructureJthatiscompatiblewiththemetric.InsteadofgivingourselvesanothercomplexstructureI,thistimewegiveourselvesananti-commutingmapS:TM!TMsuchthatS2=+id.WestillrequireStobelinearoneachberTpMandtobeintegrable,whichisdenedbyNS=0whereNS(X,Y)=[SX,SY]+[X,Y]S[SX,Y]S[X,SY]

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WewillnowshowananaloguetotheconstructioninChapter4.LetMbeaneven-dimensionalmanifoldtogetherwithasignature(n,n)metric.Inplaceofthecomplexstructureiwegiveourselvesthemaps,i.e.welets:TM!TM,belinearoneachberTpM,s2=+id,and(sX,sY)=(X,Y)forX,Y2Vec(M).WeagainletrbetheLevi-Civitaconnectionandwritethetangentspaceatthepointe2TPMTMasTe(TM)=VeHewhereVeandHearetheverticalandhorizontaltangentsatthepointeasinChapter4.Thisinturn,allowsustodenetheverticalandhorizontallifts_and,respectively. Wenextdenethe(almost)complexstructureJandtheoperatorSbyJ=(s)_,J_=(s)S=(s),S_=(s)_ ThetensorsNJandNScannowbecalculatedinasimilarwayasinChapter4toobtain: 1. 2. 3. 4. 5. 6. AgainfromcalculationssimilartothoseinChapter4,weobtainthefollowingtheorem. 1. ThealmostcomplexstructureJisintegrableifandonlyifMisatandrs=0.

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TheoperatorSisintegrableifandonlyifNs=0. 1. 2. 3. 4. 5. 6. 7. 8. 4.8 ).Wedoinfacthavers=0)d!J=0. 5.3 arezero.Item3isclearlyzeroifrs=0andsoallthatisleftisitem4.TherstthingtonoteisthatV(s,s)=V(,).TheproofisessentiallythesameasinLemma 4.1 .Soitem3canberewrittenas((s)_jV(s,s))+((s)_jV(s,s))+((s)_jV(s,s)). 4.5 sinceweknowthat!JasdenedinChapter4isautomaticallyclosed(since!JwasexactbyTheorem 4.8 ). SoweareinthesamesituationaswithourhyperKahlerconstruction.ThediscussioninthehyperKahlercasealsoleadstohopeshereaswellthatanadjustmentinthemetricmightremovetheatnesscondition. 76

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14 ]ratherextensivelyinvestigatedthegeometryoftangentandcotangentbundlesviaalargelylocalcoordinateapproach.InChapter3ofthisdissertation,weadoptedoneoftheirviewpoints:specically,weconsideredone-formsonMasberwise-linearfunctionsonTM.However,weadaptedthisconstructioninthatweconsideredarbitraryone-formswhereasYanoandIshiharadealtprincipallywithexactone-forms.ThedistinctionwasespeciallyimportantfortheproofofTheorem 3.4 involvingthecovariantderivativeofaone-formthoughtofasafunctiononTM.Theapproachofthisdissertationalsodifferssignicantlyfromthatof[ 14 ]inthatouraccountiscoordinate-free;theutilityofwhichisborneoutinthesimplicityoftheproofs.Asfutureresearch,wewouldbeinterestedinrecastinganddevelopingmoreoftheresultsin[ 14 ]inacoordinate-freemanner. 77

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[1] R.Abraham,J.Marsden,FoundationsofMechanics,TheBenjamin/CummingsPublishingCompany,Inc.,Massachusetts,1978. [2] J.Aczel,J.Dhombres,FunctionalEquationsinSeveralVariables,CambridgeUniversityPress,Cambridge,1989. [3] D.Alekseevskii,H.Baum,RecentDevelopmentsinPseudo-RiemannianGeometry,EuropeanMathematicalSociety,Zurich,2008. [4] R.Bryant,AnIntroductiontoLieGroupsandSymplecticGeometry,RegionalGeometryInstitute(1991)116-152. [5] E.Calabi,Metriqueskahleriennesetbresholomorphes,Ann.Sci.Ec.Norm.Super.,12(1979)269-294. [6] J.Cheeger,D.G.Ebin,ComparisonTheoremsinRiemannianGeometry,ElsevierPublishingCompany,Inc.,NewYork,1975. [7] L.Conlon,DifferentiableManifolds,HamiltonPrintingCompany,NewYork,2001. [8] A.Dancer,A.Swann,HyperKahlerMetricsofCohomogeneityOne,JournalofGeometryandPhysics,21(1997)218-230. [9] H.EbbinghausH.Hermes,F.Hirzebruch,M.Koecher,K.Mainzer,J.Neukirch,A.Prestel,R.Remmert,Numbers,Springer-Verlag,NewYork,1991. [10] K.Galicki,AGeneralizationoftheMomentumMappingConstructionforQuaternionicKahlerManifolds,108(1987)117-138 [11] V.GuilleminandS.Sternberg,SymplecticTechniquesinPhysics,CambridgeUniversityPress,CambridgeandNewYork,1984. [12] N.Hitchin,SeminaireN.Bourbaki,748(1991)137-166. [13] N.Hitchin,A.Karlhede,U.Lindstrom,andM.Rocek,HyperKahlerMetricsandSupersymmetry,Comm.Math.Phys.,108(1987)535-589. [14] K.Yano,S.Ishihara,TangentandCotangentBundles,MarcelDekker,Inc.,NewYork,1973. [15] S.Kobayashi,andK.Nomizu,FoundationsofDifferentialGeometry,JohnWiley&Sons,Inc.,NewYork,2Vol.,1963-1969 [16] J.Marsden,A.Weinstein,ReductionofSymplecticManifoldswithSymmetry,Rep.Math.Phys.,5(1974)121-130. [17] W.S.Massey,Non-ExistenceofAlmostComplexStructuresonQuaternionicProjectiveSpaces,PacicJ.Math.,12(1962)1379-1384. 78

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A.Newlander,L.Nirenberg,ComplexAnalyticCoordinatesinAlmostComplexManifolds,Ann.Math.,65(1958)391. [19] S.Salamon,RiemannianGeometryandHolonomyGroups,PitmanResearchNotesinMath.No.201,LongmanScientic&Technical,Essex,1989. [20] M.Spivak,AComprehensiveIntroductiontoDifferentialGeometry,5Vols,PublishorPerish,Berkeley,1970-1975. [21] F.Warner,FoundationsofDifferentiableManifoldsandLieGroups,Springer-Verlag,NewYork,1983. [22] A.Weinstein,LecturesonSymplecticManifolds,C.B.M.S.RegionalConf.Ser.inMath.29,Providence,AMS,1977. 79

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AndrewN.FisherwasborninOgden,Utahin1982.HebeganhiscollegecareeratWeberStateUniversityinthefallof2000.HelatergraduatedasthevaledictorianofWeberState'sCollegeofLiberalArtsandSciencesintheSpringof2004withamajorinmathematicsandminorinphysics.HenextbeganhisgraduatestudiesinmathematicsattheUniversityofFloridainthefallof2004wherehereceivedhismaster'sdegreeinthespringof2006.HecompletedhisPh.D.inthespringof2010. 80