; TeX output 1999.04.17:14226Ytv#X;"VG cmbx10Spinors,SpOectralGeometryX,and\)CRiemannianSubmersions&̍'K晚ff  b> cmmi10.UX/./././././././././././././././././././././././././././././././ ffffuˍ "V cmbx10PETERLB.SGILKEY!", cmsy10ySK`y cmr10DepartmentnofMathematics,tUniversityofOregon,tEu- geneUUOr97403USA.email:qgilkey@math.uoregon.edu.JOHNN]V.NLEAHY^0ercmmi7?xDepartmentofMathematics,UniversityofOregon,EugeneOrUU97403USA.email:leahy@darkwing.uoregon.edu.JEONGHYEONG0P ARKzDepartmentofMathematics,φHonamUniversity*,506-090UUKwangGjuSouthKoreaemail:jhpark@honam.honam.ac.kr.fˍyqǫResearchUUpartiallysuppGortedbytheNSF(USA)?UUResearchpartiallysuppGortedbytheGARCzqǫResearch*partiallysuppGortedbyKOSEF971-0104-016-2andBSRI98-1425,theKoreanUUMinistryofEducation *aL\|{Ycmr8i6Y1ɧPPreface&̍'K晚ff .UX/./././././././././././././././././././././././././././././././ ffff TheseLHlecturenotesappGearedasLectureNotesSeriesNumber40intheRe- searchyInstituteofMathematics-GlobalAnalysisResearchCenter(SeoulNationalUniversity*,?SeoulD151-742Korea).ProfessorSangMoGonKim,whoisDirectorofGARC,pbhaskindlygivenhispGermissionforustopostthemontheEMISpwebserver. W*enotethatwiththepGermissionoftheGARC,thesenoteshavebGeencom-pletely`rewrittenandexpandedtoformChapterF*ourofthebGook`SpQectralqGe-ometry ,ZRiemannian<3Submersions,andtheGromo9v-LawsonConjecturebytGilkey*,;LeahyandPark.H|ThisbGookisexpectedtoappearinsummer1999fromCRCUUpress. W*eɳarearegratefultotheEMISɕformakingtheseLectureNotesavqailableontheUUworldwideweb.ii6Y1ʍ,Conrtents&6'K晚ff .UX/./././././././././././././././././././././././././././././././ ffff ,E IntroGduction IT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:1ۍ ChapterUUOne:qRiemannianSubmersions T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:3 ۍ/1.1UUIntroGduction \T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:83/1.2UUMeancurvqatureandintegrabilityUUtensors pT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:f-4/1.3UUNormalizationofloGcalcoordinates ±T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:4m5/1.4UUTheexterioralgebraanddeRhamcohomologygroups ЍT:qōT:T:T:b7/1.5UUIntertwiningthecoGderivqatives |T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:F910/1.6UUThederivqativeofthe bGervolumeelement T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T: F12/1.7UUIntegrablehorizontaldistributions ;T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:O13/1.8UUFibGerproducts 4T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:L14/1.9UUConnectionsandcurvqature cT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T: 16/1.10UUThegeometryofcirclebundles iЍT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:ی17/1.11UUThe rstChernclass T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:E19/1.12UULinebundlesoverUUthetorus T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:21/1.13UUTheHopf bration -lT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:(22/1.14UUTheHopfmanifold OT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:8 24/1.15UUThegeometryofspherebundles T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:R25/1.16UUPrincipalbundles ߍT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:[27/1.17UUIntegrationoverthe bGers T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:I˼29ۍ ChapterUUTwo:qOpGeratorsofLaplaceTypGe MaT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:31/2.1UUIntroGduction <T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:P31/2.2UUThesymbGolofanoperator T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:F<32/2.3UUSpGectralresolution nT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:*33/2.4UUSphericalharmonics ۥT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:Mb34/2.5UUSpGectralresolutionoftheHopfmanifold T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:pԼ36/2.6UUTheBoGchnerLaplacian XT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:м38/2.7UUPullback f^T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:42/2.8UUManifoldswithbGoundary T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T: ˼46/2.9UURiemanniansubmersionsofmanifoldswithbGoundary T:qōT:T:T:l52ۍ ChapterUUThree:qRigidityofeigenvqalues -eT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:"57/3.1UUIntroGduction <T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:P57/3.2UUThescalarLaplacian [T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:a58/3.3UUTheBoGchnerLaplacian XT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:м59/3.4UUTheformvqaluedLaplacian bT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:t61/3.5UUThecomplexLaplacian mhT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:%64/3.6UUOthersettingswhereeigenvqaluesarerigid )čT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:71/3.7UUThespinLaplacian T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:75 \tiii6Y/ɧ/3.8UUManifoldswithbGoundary T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T: ˼77 /3.9UUTheLaplacianwithcoGecientsina atbundle _T:qōT:T:T:T:T:T:T:T:T:ͼ80/3.10UUHeatContentUUAsymptotics bT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:84 ChapterUUF*our:qWhenEigenvqaluesChange ҍT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:91/4.1UUIntroGduction <T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:P91/4.2UUCirclebundles QT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:92/4.3UUPrincipalBundles moT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:+94/4.4UUComplexgeometry tT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:F96/4.5UUHermitiansubmersions PT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:°100/4.6UUSpingeometry TT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:x101 ChapterUUFive:qPositiveCurvqature _1T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:105/5.1UUIntroGduction T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:iټ105/5.2UUManifoldswithpGositivescalarcurvqature pݍT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:♼106/5.3UUManifoldswithpGositiveRiccicurvqature ۉT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:MF109/5.4UUUnsolvedproblems [T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:`118 AppGendix iT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:۶119/A.1UUIntroGduction T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:)ټ119/A.2UUSpingeometry 7T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:p119/A.3UUTheDiracopGerator PT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:µ122/A.4UUSpinorsonthetorusandsphere xT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:¼124/A.5UUComplexgeometry MjT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:'124/A.6UUHoGdgegeometry FQT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T: 127 References T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:M~129TT:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:q140 Index T:qōT:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:T:J1416iv6Y1ɧAbstract$̍'K虚ff .UX/./././././././././././././././././././././././././././././././ ffffK9Cfcmcsc8CAbstract.WJe@studythespproductsdiscussedareawayofconstructingnewRiemanniansubmer-sionsfromgivenones.GLetid:>Zi!YՠbGeRiemanniansubmersions.GInx1.8,we qǍde ne the bGerproductWc(Z1|s;Z2) andexpresstheintegrability tensorsW and{ChapterTOne:pRiemannianSubmersions46Y/ɧ!W forthe bGerproductintermsofthecorrespondingtensorsiand!iforthe submersionsUUiTL. ڍ In}αx1.9,we}discussconnectionsonrealandcomplexvectorbundlesandde nethe!curvqaturetensor.FInxE1.10,wediscussthegeometryofprincipalcirclebundles.TheselareRiemanniansubmersionswherethe bGersaretotallygeodesiccircles;theU1tensorNvqanishesandthetensor! isthecurvatureoftheassoGciatedcomplexlinebundle.NwInx1.11,IwediscusstheChernclassofacirclebundle.Inx1.12,IweconstructspGeci clinebundlesovertorithatweshallneedlater;ɠtheseareexamplesofFRiemanniansubmersionswithharmoniccurvqaturetensorsandtotallygeoGdesic bGers. ڍ Let8:@S^2n+1ba!CP6Tn?bGethe': cmti10Hopfdq br}'ation.W*egiveCPWTn6֫theF;ubini-Study metric|andsuitablynormalizetheroundmetriconthesphereS^2n+1!.QThenis6aRiemanniansubmersion.Inx1.13wediscussthegeometryofthe brationS^3 }!}S^2 _in _somedetail;fethisisaRiemanniansubmersionwithtotallygeoGdesic bGerhcircles.;Inx%11.14,wediscussacomplexanaloguede nedbytheHopfmanifoldS^1H8S^3. ڍ InRx1.15,BweRdiscussthegeometryofthespherebundleofavectorbundleofhigherOranktoconstructmoreexamplesofRiemanniansubmersions.aPrincipalbundles,=see7DZx1.16,are7alsoRiemanniansubmersions.gInthesealltheseexamples,the. bGersaretotallygeodesicsothetensorAKvqanisheswhilethetensor!Vgivesthecurvqature. InmJx1.17,sHwemJdiscussintegrationoverthe bGertode nethepushforward.of di erentialforms.\LetV8(y[٫)bGethevolumeofthe bGer[ٟ^1 M(y[٫)ofaRiemannian submersion.hW*e8Zshowthat5=dYlog(V8).Thusif5=0,>&thenthe bGershave qǍconstantUUvolume.:xZG!Îcю鍵1.2TMeancurv\ratureandin9tegrabilityTtensors Unlessotherwisenoted,allmanifoldsareassumedtobGecompact,connected,smoGoth,cjwithout`boundary*,and`Riemannian.LetTPMwdenotethetangentspacetoUUamanifoldMlpatapGointPofMandletT^cbPMdenotecotangentspaceofM.B1.2.1TDe nition.8(1)!IfY:ZF:!Y=isasmoGothmap,Zwehavepushforward(:TzZF:!T@Lz Y qǍ!andpullback[ٟ^u:$T^c፯@Lz Y]!T^c፯zsZ.MIf[issurjectiveandif&fissurjective, j!weUUsaythat.isasubmersion.LT8(2)!LetvR":Z~4!Y6bGeasubmersion.'qThe b}'erX?4of+istheinverseimageofthe ٍ!baseJpGoint;NgX:=[ٟ^1 M(y0|s).nPUptodi eomorphism,Mthe berisindependent!ofhtheparticularpGointy0ofYochosen.iIfydisanypGointofY8,mXthereexists !anneighbGorhoodOyofy:=gZp[([eiTL;fap];ei)f^a =^Z gWiia f^a2C^1 0(H).aTheFtensorcisthe qǍ!unnormalizedUme}'ancurvaturecovectorUofthe bGersof[٫;VMwehaveomitted !theUUusualnormalizingconstantofdim(X)^1ɫtosimplifylaterformulas.8(2)!Letp!:=DE!abisE= wx1wx&fes2 gZp[(eiTL;[fap;fbD])= wx1wx&fes2 (^Zp[abi j^ZMbai /).STheptensor!Iisthe d!inte}'grabilitycurvaturetensor.; TheUUfollowingobservqationisimmediatefromthede nitionswehavegiven:1.2.4TLemma.L}'et":Z~4!Y˺beaRiemanniansubmersion.8(1)!pThefollowingassertionsar}'eequivalent: Ǎ"㊫1-a)9pThe b}'ersofareminimal."UQ1-b)9pisaharmonicmap.#qī1-c)9p5=0.Ǎ8(2)!pThefollowingassertionsar}'eequivalent:"㊫2-a)9pThedistributionHisinte}'grable."UQ2-b)9p!"=0.ChapterTOne:pRiemannianSubmersions66YM}%LG!Îcю鍵1.3TNormalizationofloQcalcoordinates)1.3.1De nition.Letf^2%n eufm10HbGef^thehorizontalliftoperator.IfFisatangentvector eld?onY8,thenH 9F/Ϋisthevector eldonZ[whichischaracterizedbythefollowingtwoUUpropGerties:ɍ8(1)!Ifz2isanypGointinZ,ythenwehavetheintertwiningrelationH ޮFzb=F@Lz . 8(2)!W*eUUhavethatH OFisahorizontalvector eldi.e.qH F*2C^1 0H./I TheUUhorizontalliftisbracketpreserving.)1.3.2TLemma.L}'et":Z~4!Y˺beaRiemanniansubmersion.)8(1)!pIffid:=H FiTL,then[f1|s;f2]=[F1|s;F2].~ 8(2)!pWehave^Z Babc=[ٟ^(^YGabc 핫).Pr}'oof.pLet? \(t)and (t)bGethe owsofthevector eldsfandFonZD[and \ponXYrespGectivelyforw\=1;2.{F*orX xedvqaluesoftheparametert, \(t)isadi eomorphismofZX9and ޫisadi eomorphismofY8.UThecurvetEe7! \(t)(z0|s) 7isanintegralcurveforthevector eldfstartingatthepGointz0&2FZ;.similarlythe?curvetM7! \(t)(y0|s)?isanintegralcurveforthevector eldF astartingatthempGointy0S]2Y8.[Sincef 3=F\,!3TFintertwinesmthese ows;Ii.e.[wemhavetherelationship: qǍm[ \(t)= (t)[: 9W*eUUcanusethe owstocomputethebracket.qLetn:Jh(t):= 1|s(bpUWbfert ɫ) 2(bpUWbfert) 1(bpUWbfert) 2(bpUWbfert)(z0):JH(t):=[h(t)= 1|s(bpUWbfert ɫ) 2(bpUWbfert) 1(bpUWbfert) 2(bpUWbfert)([z0)forNz0y~2 Z.The rstassertionoftheLemmanowfollowsfromtheobservqation that: QG\q=bQ_<+hAx(0)=[f1|s;f2](z0)UUand\qϴ_h [۫(0)=x䍑_H (0)=[F1|s;F2]([z0):R LetkfFapgbGealocalorthonormalframeforthetangentbundleofYandlet ffapgCbGethehorizontallifttoanorthonormalset;Ifa:=H Fa.RW*ecancomputetheChristo elmsymbGolsintermsofthebracketanduseassertion(1)toproveassertion(2)UUbycomputing:,čKHZ!abc= K1K&fes2 )fgZp[([fap;fbD];fc)8gZ([fbD;fc];fap)8+gZ([fc;fap];fbD)g#V`= K1K&fes2 )fgYG([fap;fbD];fc)8gYG([fbD;fc];fap)8+gYG([fc;fap];fbD)gV`= K1K&fes2 )fgYG([Fap;FbD];Fc)8gY([FbD;Fc];Fap)8+gY([Fc;Fap];FbD)gjV`=Y _abc 핮: UV5 msam10yP .TGilk9ey,J.Leah9y,JH.P9arkt76Y/ɧ W*e\cannormalizetheloGcalcoordinatesasfollows.KLetO:ybeaneighborhood ofapGointy0ܱ2giY8.ZW*eusecoordinatesonOtoidentifyOwithEuclideanspace qǍR^m and toidentifyy0withtheorigin0ofR^m.W*eletf@^8y gandfdy[ٟ^ gbGethecoGordinateUUframesforthetangentandcotangentbundles.1.3.3!Lemma.L}'etR":Z~4!Y=6beRaRiemanniansubmersionwith berX.iChooseaŰp}'ointy0B#ofY8..ThereexistsaneighborhoodO ͺofy0B#andalocaldi eomorphismTvfr}'omX±8OtoZKsothatэ8(1)!pWehaveTc(x;0)=x..8(2)!pWehave[٫(Tc(x;y))=y[ٺ.8(3)!pWehaveT(@^8ya\)(x;0)=H (@^8ya)+܍Pr}'oof.pIfy[۱2R^m,@thenyldeterminesavector eldonOX2andthehorizontallift H3(y[٫)determinesavector eldon^1 MOG.XLet(t;y;x)bGethecorresponding ow qǍfromDRapGointx2X.lThereDRisaconstantCnsothatisasmoGothmapde nedforjty[ٱjC.qNoteUUthatwehave߯(ts;y[;x)=(t;sy;x):W*eW\de nethecoGordinatetransformationweneedbyde ningTc(x;y[٫)y:=(1;y[;x) ٍandUUrestrictthedomainofTtothosevqaluesofy.sothatjy[ٱjC^1 s. W NoteZthatthetwoZtensorsand!ofDe nition1.2.3giveobstructionstocon-structingncoGordinatesystemswiththenormalizationsofLemma1.3.3beingvqalidnotjustatthebasepGointy0 butinafullneighbGorhoodofy0|s.;uThefollowingobservqationUUisanimmediateconsequenceoftheLemma.1.3.4cYCorollary.ThetensorsLand!a]ar}'elinearinthe1jetsofthestructuresinvolve}'dandaretheonlytensorialexpressionslinearinthe1jetsofthestructures.9Q!ZG!Îcю鍵1.4TTheexterioralgebraanddeRhamcohomologygroups1.4.1yDe nition.IfMisaRiemannianmanifold,let^pR(M)bGethebundleof exteriorBdi er}'entialpformsponM*andletC^1 0^pR(M)bGethespaceofsmoothp ٍforms|%onM.6Ifup=(u^1|s;:::;u^m)|%isasystemofloGcalcoordinatesonM@andifA=fa1C<:::m IfFfapgisaloGcalorthonormalframeforTcM,Zletf[ٟ^a2Igbethecorrespondingdual]orthonormalframeforTc^sM.Thebundleofpformsinheritsanaturalinner [QproGductUUandf[ٟ^A :=[ٟ^a1 RI^8:::^[ٟ^ap eg:jAj=pǫisUUalocalorthonormalframe.Ѝ1.4.3&De nition.Let߮bGeacovector.>W*euseleftexteriormultiplic}'ationbytode nethelinearopGeratorext6([٫)onthebundleofexteriordi erentialforms^M:ݤexts([٫):=^8:LetintZ([٫)bGethedualoperationofinterior&fmultiplic}'ation;D0thesetwooperations qǍareUUrelatedbythedualityequation:Okk(ext t([٫)1|s;2)=(1|s;int q([٫)2):TheseopGeratorsare0^th 퉫orderoperators; incontrasttotheoperatorsdand`,they depGend\onlyonthevqalueofanexteriordi erentialformatthepointinquestion.ύ SuppGoseathat1:isaunitcovector.ThenawecanchoGoseanorthonormalbasis 4fiTLge۫forTcM|sothatisthe rstelementofthedualorthonormalbasisf[ٟ^i%gfor ٍTc^sM,UUi.e.qǮ"=[ٟ^1L.Relativetosuchanadaptedorthonormalbasiswehave:G>extTO([٫)(Ai):=^n i0mifvBKa1C=1;# i[ٟ^1,^8[ٟ^a1 RI^:::^[ٟ^apmifvBKa1C>1;andHintT([٫)(Ai):=^n i[ٟ^a2 RI^8:::^[ٟ^apkift쟮a1C=1;# i0kift쟮a1C>1:In:otherwords,t*inthisadaptedorthonormalbasis,exteriormultiplicationby addstheindex`1'toAwhileinteriormultiplicationby\removestheindex`1'fromUUA. Let;rbGetheLevi-Civitaconnectionandde netheChristo elsymbolsrelativetoUUthisloGcalorthonormalframebytheidentity:ݤra b\=abc 핮c:OjW*eUUextendtheLevi-CivitaconnectioninanaturalfashiontoC^1 0M. P .TGilk9ey,J.Leah9y,JH.P9arkt96Y/ɧ1.4.4TLemma.L}'et=A[ٟ^A 2C^1 0M.ThenU8(1)!pWehaver=[ٟ^ak) 8(@apA)[ٟ^A I+abc 핮[ٟ^a extT([ٟ^cn)int q([ٟ^b`))._8(2)!pWehaved=extc([ٟ^a2I)f(@apA)[ٟ^A I+8abc =ext4([ٟ^cn)int q([ٟ^b`)g.8(3)!pWehave`=int q([ٟ^a2I)f(@apA)[ٟ^A I+8abc =ext4([ٟ^cn)int([ٟ^b`)g.ǍPr}'oof.pIf&isafunctionora1form,Z0thenassertion(1)isimmediate.Sincer actsasagradedderivqationontheexterioralgebra,*zassertion(1)nowfollowsin qǍgeneral. ThehfollowingopGeratorsareinvqariantlyde ned rstorderpartialdi erentialopGeratorsUUontheexterioralgebrawiththesameleadingsymbolsasdand`:r\qSS^QldY:=(extG I)8rUUand\q>^`:=(int q I)r:Thus(yA1o:=&dŢ\q^d "andA2o:=&&\q=1^ ūareinvqariantlyde ned0^th 3&orderopGerators.ThisshowsthattheAic[areinvqariantlyde nedendomorphismswhicharelinearinthe3(1jetsofthemetricwithcoGecientswhicharesmoGothfunctionsofthemetrictensor.Given>apGointP2\M,ŸwecanalwayschoGosecoordinatessothe1jetsof qǍthemetricvqanishatPc.ZThisimpliesthatAiTL(P)=0.ZSincetheAieHareinvqariantlyde ned,UUweseetheyvqanishidentically*.qThisshowsthatTd=(extG I)8rUUand'=(int q I)r:TheUUremainingassertionsnowfollow. ڍ1.4.5TDe nition.TheUUdeRhamc}'ohomologygroupsarede nedbycэ>[HpoP(M;R):=<$?ker(dpfj:C^1 0^pRM3!C^1^p+1UM)Kwfem (֍image=(dp1ޫ:Cr1 0rp1ƮM3!Cr1rpRM):#o ThedeێRhamthe}'oremestablishesanisomorphismbGetweenthesegroupsandthe topGologicalJFcohomologygroupsofM;MtheHo}'dgedecompositionJFtheoremexpressesthe"NdeRhamcohomologygroupsofaRiemannianmanifoldintermsofthekernelofUUtheLaplaceopGerators.qLetMፑ`1ɍpM t:=pRdp2+8dp1Ʈp11on C1 0pM:lsIfUUisasmoGothpform,thenwehave\€2ker#(1ɍpM\)0(UX)d=0UUand`=0:LqThus'cE:=!abi ٨extvZ w(eiTL)int qZ ̫(fa)int qZ(fbӫ);#c:=intZ)<(G)8+E:O TheUUfollowingformulawillbGecrucialtoourstudy*.獵1.5.1TLemma.L}'et":Z~4!Y˺beaRiemanniansubmersion.荍8(1)!pWehaveZp[[ٟ^-8[ٟ^Y _=[ٟ^. f􍍍8(2)!pWehaveZp[[ٟ^-8[ٟ^Y _=fdZ;+dZg[ٟ^.ePr}'oof.pLetN2C^1 0^pR(Y8). W*eexpand=AFc^A 3anduseLemma1.4.4tocompute:y⍍IZp[[ٟ=p)Zp[[ٟ(A)fAji0 =s0 8int*Z(eiTL)ei([ٟA)fA(1.5.b)~s0 8int*Z(eiTL)Zp[iab ٨extvZ w(fbӫ)int qZ ̫(fa)[ٟ(1.5.c)s0 8int*Z(eiTL)Zp[iaj extZ!߫(ej6)int qZ ̫(fa)[ٟ(1.5.d)s0 8int*Z(fa)fap([ٟA)fA(1.5.e)s0 8int*Z(fa)Zp[abc =ext4Z! (fc&p)int qZ ̫(fbӫ)[ٟ(1.5.f)s0 8int*Z(fa)Zp[abi ٨extvZ w(eiTL)int qZ ̫(fbӫ)[ٟ:(1.5.g) 힟ChapterTOne:pRiemannianSubmersions[126Y/ɧSince#horizontalcovector eldsareannihilatedbyintuZЫ(e^iTL),WBthetermsin(1.5.b) andin(1.5.c)vqanish.]F*urthermorein(1.5.d)wemusthaveiJ.=j.]Byde nition, 銍.=r^Zp[iai f^a Rsohthetermsin(1.5.d)yieldintZZ(G)[ٟ^.Since^Y Uabc=^Z Wͫabc 핫,mthetermsUUin(1.5.e)and(1.5.f)yield[ٟ^YG.qNotethat+ۍn0߱int qZ ̫(fa)extGZw(eiTL)int qZ(fbӫ)=extÛZ (eiTL)int qZ ̫(fa)int qZ(fbӫ)~K=I8extTZE(eiTL)int qZ ̫(fbӫ)int qZ(fa):+ڍThusUUwemayanti-symmetrizetoseethetermsin(1.5.g)yieldI 717&fes2>bextLp֟ZR1(e^iTL)(^Zp[abi g8^Z;bai /)int qZ ̫(f^a)int qZ(f^bӫ)[ٟ^=E[ٟ^:iThe rstassertionnowfollows.ESincedZp[[ٟ^ի=[ٟ^dYG,/weprovethesecondassertion byUUcomputing:$ ݍ53mZp[[ٟ-8[ٟY _=QծdZp[Z[ٟ-+8ZdZ[ٟ8[ٟdYGY %'[ٟYGdY#w=QծdZp[(Z[ٟ-8[ٟYG)+(Zp[[ٟ[ٟYG)dYw=QծdZp[[ٟ-+8[ٟdY _=(dZ+dZ)[ٟ: UVOhLG!Îcю鍵1.6TThederiv\rativ9eofthe bQervolumeelement Let^;1ū:Z!YbGeaRiemanniansubmersion.xRecallournotationalconventionthatLfeiTLgandfe^igareloGcalorthonormalframesfortheverticaldistributionsand coGdistributionsUVҫandV}^ |andthatffapgandff^agarelocalorthonormalframes qǍforUUthehorizontaldistributionsandcoGdistributionsHn andH^. SuppGoseUUthe berX7of.isorientable.qLet7X a<:=e1S^8:::^edimX׍bGe9thevolumeformofthe ber.hZW*ecanexpressdZp[(X$)intermsofthetensorsandUU![٫.C1.6.1lLemma.L}'et~`":Z~4!YDbe~`aRiemanniansubmersionwithorientable ber o9X.ThendZp[(X$)=^8X !abi ٨extvZ w(f^a)extGZw(f^bӫ)int qZ ̫(e^iTL)X.ڍPr}'oof.pW*eTapplytheformulasofLemma1.4.4tothemanifoldZpandseparatetheindicesUUintohorizontalandverticalpartstocompute:;QdZp[(X$)=eijgk !0extZ!-(eiTL)extGZw(ek됫)int qZ ̫(ej6)X(1.6.a)Zh+8(ijga Hajgi ah)extGZw(eiTL)extGZ(fa)int qZ ̫(ej6)X(1.6.b)h+8ajgb extX|Z!׫(fa)extGZw(fbӫ)int qZ ̫(ej6)X$:(1.6.c) ʟP .TGilk9ey,J.Leah9y,JH.P9arkp136Y/ɧThehformX hasmaximalverticaldegree.Thusthetermsinequation(1.6.a) yieldL0fordimensionalreasons.UInequation(1.6.b),JthetermswithiE6=j3׫vqanish qǍandVowethereforeseti=j.uSinceVowearedealingwithanorthonormalframe eld,aii F =0._Thusiia Jːaii=iia andthetermsinequation(1.6.b)yield^ːX$.Finally*,UUthetermsfromequation(1.6.c)yieldMVnȢ 33133&fes2b٫(abjJ@8baj `)=!abj: UVCIZG!Îcю鍵1.7TIn9tegrablehorizontaldistributionsX1.7.1De nition.W*eZsaythataRiemanniansubmersion":Z~4!YN>isZ atifthe horizontalUUdistributionisintegrable. W*eUUcanimproveUULemma1.3.3ifthehorizontaldistributionis at.1.7.2Lemma.L}'etY:-Z!YٺbeYaRiemanniansubmersionwith berX"׺andinte}'grablehorizontaldistributionH.8(1)!pWec}'an ndlocalcoordinatesz7=(x;y[٫)onZKso(x;y)=y.Ξ8(2)!pInthesec}'oordinates,ds^2bZ 7s=gij (x;y[٫)dx^i,8dx^jo+habZ(y)dy^ak)dy^b`.8(3)!pIfwesetgX a<:=det(gij )^1=2 ʺ,then5=dYlnF(gX$):Pr}'oof.pChoGoselocalcoordinatesy:,=S(y[ٟ^a2I)de nedonaneighborhoodOCofsome GpGoint,y0inY8.d Let\qS@~fabethehorizontalliftofthecoordinatevector elds@^8yaޫfrom YtoZover[ٟ^1 M(OG);thesevector eldsdonotformanorthonormalframe.W*e qǍuseUULemma1.3.2toseethatwehaveэ}v[\q'!~fa Ǯ;\qɫ~fb C]=[@8ya\;@1ɍ8yvb]=0:PSincevwehaveassumedthatthehorizontaldistributionH,isintegrable,>wehave C[\q'!~fa Ǯ;\qɫ~fb C]R2H.Consequentlyw[\q'!~fa;\qɫ~fb C]=0.ChoGosewlocalcoordinatesxR=(x^iTL)forthe ٍ bGer9ĮX =CϮ[ٟ^1 M(y0|s)nearz0.W*eapplytheFrobGeniustheoremtoextendxtoa GsystemoofcoGordinatesz.=o(x;wD)onaneighborhoodofz06sothat\q~faͫ=o@^8wai.The qǍpro8jectionoftheintegralcurvesofthevector elds@^8wa aretheintegralcurvesof theVvector elds@^8ya\:Thereforey% =3[٫(x;wD)=w{andVqispro8jectiononthesecond factor.qAtUUapGointz7=(x;wD),wehaveZEdVzb=TxX=spanTf@8xig?Hz=Twy1Y=spanTf@8waig:QSinces isanisometryfromthehorizontaldistributionHhtothetangentbundleofUUTcY9ofY8,themetricloGcallyhastheformgiveninthesecondassertion.7ChapterTOne:pRiemannianSubmersions[146Y/ɧ Sinceʵ!摫=0, wemayapplyLemma1.6.1toprovethe nalassertionbycom- putingx卍㍍[~®X a<=gX$dx1S^8:::^dxdim @(XJ)$[~®dZp[(X$)=dYG(gX)8^dx1S^:::^dxdim @(XJ)KhE۫=dYG(gX$)g䍐[ٲ17sX Q-^8X a<=^X$: UV%덑 Let!®w:̮Z!YZbGea atRiemanniansubmersion.W*ecoverYZbyopGensetsO 9aanduseLemma1.7.2toconstructloGcaldi eomorphismsT fromX"YO to [ٟ^1 MO .soTLthat[T (x;y)=y%isTLpro8jectiononthesecondfactorandsothevector eldsl(T )@^8yacspanthehorizontaldistribution.ThenontheoverlapO L\HjO ъwe 7haveUUmusthavertTc1፯ X8T d(x;y[٫)=( ȫ(x);y[٫):D;In*thisequation,thedi eomorphism 7isindepGendentofy[٫.RFThusthe bGerbundleܮ":Z~4!Yisa at bGerbundle;Ztheglueingtransitionfunctionsarelocallyconstant.PwThusdthis bGerbundleisde nedbyarepresentationofthefundamental qǍgroup1|s(Y8)ofY8kintothegroupofdi eomorphismsofthe bGer.p]Inparticular,ifaY#Eissimplyconnected,thenthebundleadmitsaglobaltrivializationandthe RiemannianCsubmersionisgloballyproGductwiththemetrichavingtheformofLemmaUU1.7.2.1.7.3NDe nition.W*e saythata atRiemanniansubmersionhasSL:structur}'egr}'oup/ifthetransitionfunctions describGedabove/canbechosentopreservesomevolumeelementonthe bGersorequivqalentlyifthereexistsameasureContheV bGerssothattheLiederivqativeLH=s0vanishesforeveryhorizontalliftH;instead:ofthetangentmaptakingvqaluesinthefullgenerallineargroupofautomorphismsW3ofthetangentbundleofthe bGerX,WthestructuregroupreducestothespGeciallineargroupoftheautomorphismsofthetangentbundleofthe berrelative_Qtoasuitablychosen bGermetric.ThisisalwaysthecaseifthefundamentalgroupUUofY9is nite.6 ZG!Îcю鍵1.8TFibQerproducts The)1followingconstructionisaveryusefuloneforbuildingnewRiemanniansubmersions.1.8.1De nition.W*e^suppGosegiventwoRiemanniansubmersions @mappingU toQthesamebasemanifoldY8.pW*edenotethecorrespGondingverticalandhori-zontalUUdistributionsofthesesubmersionsbyH 7andV :LetPDѮW*:=fw =(u1|s;u2)2U1S8U2C:1|s(u1)=2|s(u2)g:bGethe b}'erEproduct.De nesmoothsubmersionsW ߫fromW*UtoYand yfrom WtoUUU 7by qǍQw"W (wD):=1|s(u1)=2|s(u2)UUand (w)=u :P .TGilk9ey,J.Leah9y,JH.P9arkp156Y/ɧTheUUverticalspaceofW n:isgivenbyy;VW (wD)=V1|s(u1)8V2|s(u2)withrespGecttotheusualembeddingofTcW'inTU1+TU2|s.Itisnowclearthat qǍW n:andUU(W )9aresurjectivesoWisasubmersion.qLet㍑0HW (wD):=f(1|s;2)2H1|s(u1)8H2|s(u2):(1|s)1C=(2)2gde neUUacomplementarysplitting.qW*ede neametriconWbyrequiringthat:8(1)!W*eUUhavetherelations:qDZHW ?V1|s,HW?V2|s,andV1C?V2ȫ. č8(2)!TheUUmetricsonV1ȫandV2areinducedfromthemetricsonU1andU2|s.8(3)!W*eUUhavethatW (wD)_:HW(w)!TcY8([٫(w))UUisanisometry*.tNotethatthemetriconHW իdi ersfromthesubspacemetricbyafactorof )1#&fe <]*pɟ*W s捬2;theWdiagonalinarightequilateraltrianglehaslengthPp PfeE2.xWiththesede nitions, weUUhave:L8(1)!W :W*!Y9isUUaRiemanniansubmersion.8(2)! y:W*!U 7isUUaRiemanniansubmersion.1.8.2TExamples.O8(1)!IfzctheUiίarevectorbundlesoverY8,thenthe bGerproductofU1֫withU2 qǍ!isUUtheWhitneysumvectorbundleU1S8U2|s. G8(2)!IfaUi.=ڲXi@Yaretrivialsubmersionswheretheidarepro8jectiononthe !second factors,=thenthe bGerproductisgivenbyW*=(X1$X2|s)YEand!theUUpro8jectionisagainprojectiononthesecondfactor.LLetj\ >andZ ڷbGethemeancurvqaturevectorsde nedbythesubmersions >and Zp[;UUlet! 7and!Z ŰbGethecorrespondingcurvqaturetensors.1.8.3oLemma.L}'etBͮ y:U !Y{b}'eBRiemanniansubmersions.~LetW\bethe berpr}'oduct'oftwoRiemanniansubmersions.ThenO8(1)!pWehaveW =^[ٲl11S+8^[ٲl22Zand!W=^[ٲl1!1S+8^[ٲl2!2|s.'8(2)!pIf_j2E(;1ɍpYG),andiffor tv=1;2wehave ޱ2E(zn+ ;1ɍpU ),then !pwehave^[ٲbW 2E(8+1S+2|s;1ɍpW):-Pr}'oof.pLet@fFapgbGealocalorthonormalframeforTcY8.4Letf^ a Yandf^Wam\bethe 銍horizontal1liftswithrespGecttothesubmersions LandW .=\Notethatf^Waťis also[5thehorizontalliftoff^ a !withrespGecttothesubmersion .hLetfeiTLgand fb:^ejޭgҖbGelocalorthonormalframesfortheverticaldistributionsof1O and2and Elet6fe^W;Zi gandfb:^e^W;Zj gbGehorizontalliftstoWHūwithrespecttothesubmersions1,ChapterTOne:pRiemannianSubmersions[166Y/ɧand2|s.Thenfe^W;Zi ; ^e^W;ZjkgisaloGcalorthonormalframeforV}(W),fe^W;ZigisaloGcal `TorthonormalframeforV}(2|s), yandfb:^e^W;Zj gisaloGcalorthonormalframeforV(1|s). W*eUUuseLemma1.3.2tocompute:#c% +ԮW=gW (b:^eWj ;[fWa ,t;fWb])=2=[ٲ፬2fg2|s(b:^ejޭ;[f2፯a;f2፯b])g=2=[ٲ፬2g^!abj?:lThisestablishesthe rstassertion.Thesecondassertionisanimmediateconse- quence5ofthe rstassertionandoftheintertwining5formulasofLemma1.5.1. 4ōLG!Îcю鍵1.9TConnectionsandcurv\rature1.9.1De nition.AGconnectionG\ronarealorcomplexvectorbundleV@overa manifoldM(isageneralizeddirectionalderivqative.Itisa rstorderpartialdi erentialUUopGerator qǍp r:C1 0(V8)!C1(TcsMO 8V8)qGwhichVisatis estheLeibnitzrule.uIfss2C^1 0VMisViasmoGothsectiontoVandiffڧ2C^1 0MlpisUUasmoGothfunctiononV8,thenwehave:𨍑3r(fs)=dUUfLo 8s+frs:h ThereUUisanaturalextensionoftheconnectiontoanopGeratorR!6r:C1 0(pR(M)8 V8)!C1(p+1U(M)8 V8)whichUUisde nedbysetting]yr(p2 8s)=dp 8s+(1)pRp^rs:MIncontrasttoordinaryexteriordi erentiation,5thesquareofthecovqariantderiv- ativeUUr^2ȫneednotvqanish.qHowever,86:r2|s(fs)=ddUUfLo 8sdfLo^rs+dfLo^rs+fr2|ss=fr2|ss:=P .TGilk9ey,J.Leah9y,JH.P9arkp176Y/ɧ1.9.25De nition.The#opGeratorr^2 isa0^th Ыorderpartialdi erentialoperator calledthecurvatur}'eF9.Ifsij]isaloGcalframeforV8,F@wede netheconnection1formUUbyexpanding ač.Frsid=A1ɍjˍio 8sj6:S4Inthenextsection,0wewilladdanextrafactorof$p [$fe 1H~tode nethenormalizedconnection[Ioneform.nF*orthemoment,Lhowever,we[Iomitthisnormalizingconstant.W*eUUcomputethatqr2|ssid=(dA1ɍjˍio8Aki$p^A1ɍjvk됫) sjandUUthusthecurvqatureisgivenbyk$mF1ɍ9jˍi=(dA1ɍjˍio8Aki$p^A1ɍjvk됫):㍫Ife~s =g1ɍ[ٯjˍisj6WisanotherloGcalframe,*@weshowthatthecurvqatureisaninvqariantly Dde nedUUendomorphismbycheckingthatx䍑~FJ=g[ٱF9g^1 M:z1.9.3TDe nition.AUUconnectionrisaRiemannianc}'onnectionUUifwehavej߇(rs1|s;s2)8+(s1|s;rs2)=d(s1|s;s2):nW*erestricttosuchaconnectionhenceforth. /RelativetoaloGcalorthonormalframe,thecurvqatureF@isskew-symmetric.NotethattheLevi-CivitaconnectionisUUaRiemannianconnectiononthetangentbundle.1.9.4&&Remark.Ifz8risaRiemannianconnectiononacomplexvectorbundle qǍoverY8,thereanaturalmetriconV.$sLetS(V)bGetheunitspherebundle.$sThen ":S(V8)!Y\isWxaRiemanniansubmersion.(Thecurvqatureoftheconnectionrandthetensor!&de nedinDe nition1.2.3encoGdeessentiallythesameinformation; werefertox4f1.15forfurtherdetails.Inthenextsection,weconsiderthespGecialcaseUUofcirclebundles.6t:ZG!Îcю鍵1.10TThegeometryofcirclebundles Let.LbGeacomplexlinebundleover.Y8.dW*esupposethatYgisequippedwitha [QsmoGothUU bermetricandaRiemannianconnection^L Qɱr.qLet{S(L):=f<߱2L:juDZj=1gbGeotheassociatedcirclebundleandlet<ū:S(L)!Y7Sbeothenaturalassociated ٍpro8jection.)Let|S^1=f2C:jj=1gbGetheunitcircle.)ThenS^1actstransitively on>the bGersofS(L)bycomplexmultiplicationsoS(L)isaprincip}'al/circlebundle; qǍconversely*,q\ofkcourse,everyprincipalcirclebundlearisesinthisfashion. W*enoteMChapterTOne:pRiemannianSubmersions[186Y/ɧthatߡtherearecirclebundleswhicharenotprincipalcirclebundles.W*ereferto xqɫ1.16UUforfurtherdetailsconcerningprincipalcirclebundles. LetUUsbGealocalorthonormalsectiontoLandlet?ݾrs=p ofe X1㎮AsF:sUde neEthenormalizedconnection1formAsF:.lW*ehaveEadoptedaslightlydi erent normalizationconventionfromthatgiveninDe nition1.9.2;Uthefactorof$p U)$fe 1willUUmakelatercomputationseasier. qǍ W*eUUintroGducelocalcoordinates(t;y[٫)onS(L)byԣ(t;y[٫)7!esIJpɟsĉW forv2|ݮS(L)andt2Risinvqariantlyede ned;@tGis the }assoGciatedunitverticalKillingvector eld.XAssertions(1)and(2)nowfollow. :NSinceEZ^L AαrEZisunitary*,HAsisareal1-form.lsLet~s r=eIJpɟĉW adim(Y8)andthat qǍLCP *canUUbGereplacedbytheHopfbundleinthisconstruction. w W*eusetensorproGducttogiveVect,n1R,nC(Y8)thestructureofanAbGeliangroup.>8The naturalmH-groupstructureonCPßT1givesthesetofhomotopyclasses[X:;CPT1x]agroupstructureandtheisomorphismofequation(1.11.2)isagroupisomorphism.1.11.33De nition.LetoԮH^2Lq(Y8;Z)denotetheintegercohomologygroupsofY; qǍthesecaneitherbGede nedusingsheafcohomologyorsingularcohomology*._^Letx generateUUH^2Lq(CP UVT1;;Z)=Z.qThetopGological rstChernclassisde nedtobe:h%ucZ፬1*(L):=fs(x)2H2Lq(Y8;Z): [ Thenc1isanaturaltransformationoffunctorswhichisanisomorphismfrom V*ectqȬ1RqC(Y8)UUtoH^2Lq(Y;Z).qW*eusetheuniversalc}'oecienttheoremUUtoseethattbH2Lq(Y8;R)=H2(Y8;R)8 ZcR;see,UUforexample,Spanier[178].qLet-([G]):=[]8 Zc1Rde neDanaturaltransformationoffunctorsmapfromH^2Lq(Y8;Z)toH^2(Y8;R). qǍTheFlinkbGetweenFalgebraictopologyanddi erentialgeometryisthengivenbythe identity: 썒o8cZ፬1=c1|s:wP .TGilk9ey,J.Leah9y,JH.P9arkp216Y/ɧ1.11.4Lemma.AssumeH^2Lq(Y8;R)" 6=0.07Ther}'eexistsaunitaryconnectionona qǍc}'omplex slinebundleLoverYEWsothatthecurvatureF isharmonicandnon-trivial.ԦPr}'oof.pSuppGose thatH^2Lq(Y8;R)6=0.XChoose L2V*ect81R8C(Y8)sothat06=(c1|s(L))in ٍH^2Lq(Y8;R).W*ecusetheHoGdgedecompositiontheorem(seeTheorem1.4.6)to nd06=2E(0;^2bYG)UUsothat7}kTi[ 133&fe2 PH]=(c1|s(L))UUinH^2Lq(Y8;R):4LetUU^Lx䍑 t~ QɱrubGeUUanyunitaryconnectiononL.qThen*0umL[F9(Lx䍑~trQ˫)]=[]UUinH2Lq(Y8;R):iThusUUwecan ndasmoGoth1-form onY9sothat=F9(^Lx䍑~trQ˫)8+d .qLetLr:=Lx䍑 n7~ÌrQë+8p 7fe X1UV :ThisPOconnectionisaunitaryconnectiononLandFQ=F9(^Ltr)=POisharmonic. 78ELG!Îcю鍵1.12TLinebundleso9verTthetorus 2 Give33T^2C:=S^1S^1C3the atproGductmetric.fgIfx^ifor0x^id2 aretheusual pGeriodicUUparametersonT^2|s,then5ds2C=(dx1|s)2S+8(dx2)2: Lemma[ 1.11.4showsthereexistsalinebundleoverT^2}withnon-trivialharmonic curvatur}'e.qW*eUUcanexhibitthislinebundlequiteexplicitly.1.12.1gLemma.Ther}'e<`existsaunitaryconnectionronacomplexlinebundleLoverT^2Zsothat Y9F9(L)= K1K&fes2 )(dx^1S^8dx^2|s):kPr}'oof.pW*eUUdecompGose5O^T2C=[0;2[٫]8S1=T͍+3=where-O(0;wD)T͍+3= UN(2[;wD):u\LetUUthelinebundleLbGede nedby:S΍#L:=[0;2[٫]8[0;2[٫]C=T͍+3=where-O(0;wD;zp)T͍+3= UN(2[;w;epɟʼnW ~>xGAQ=(x0|s;x1;x2;x3)=(zp0 ;zp1)=x0S+8x1|si+x2jk+x3kXRwhereLzp^0 O]=bSx^0Yѫ+^ix^1Ȇandzp^1=bSx^2Yѫ+^ix^3|s.VLet1(Ȯ~x)bS=i^&~xz,î2(Ȯ~x)bS=jo&~^xz,and 3|s(Ȯ~x)=kw~8x.qTheUUvectorsy-fȮ~x;1|s(Ȯ~x);2|s(Ȯ~x);3|s(Ȯ~x)gformҷanorthonormalbasisforR^4O*sof1|s;2;3gҷisanorthonormalframeforthe ٍtangentbundleoftheLiegroupS^3.}Letf^19V;^2;^3gbGethedualcoframeforthe ǍcotangentbundleofS^3.DThe^i andi"*areinvqariantunderrightmultiplicationand qǍareUUabasisfortheassoGciatedLiealgebraso (3).qW*ecompute:>"܍׵v91C=x1|s@0S+8x0@18x3@2+8x2@3;[2;3]=21|s;$92C=x2|s@0S+8x3@1+8x0@28x1@3;[3;1]=22|s;93C=x3|s@0S8x2@1+8x1@2+8x0@3;[1;2]=23|s;91n=x1|sdx0S+8x0dx18x3dx2+8x2dx3;d1n=22r6^39V;92n=x2|sdx0S+8x3dx1+8x0dx28x1dx3;d2n=23r6^19V;93n=x3|sdx0S8x2dx1+8x1dx2+8x0dx3;d3n=21r6^29V:P .TGilk9ey,J.Leah9y,JH.P9arkp236Y/ɧW*ekletthecircleS^1WWCactonS^3{bycomplexmultiplicationfromtheleft;ulet KbGethenaturalpro8jectionfromS^3tothequotientmanifoldS^1nS^3X=XCP讟T1y=XS^2; ٍthisUUistheHopf br}'ation.qIntermsofcoGordinates,":S^3!S^2eUisUUde nedby:܍n[٫(Ȯ~x)=:7(2Re(zp0ӫ zp1);2Im(zp0ӫ zp1);jzp0 j2S8jzp1j2|s)$2pm=:7(2(x0|sx2S+8x1x3);2(x1x28x0x3);x0x0+8x1x18x2x28x3x3):+ۍSinceY~|x :!e^itZ0~Rx de nes|the1-parameter owforthevector eld1|s,ʮ1'=0.If y"=(y[ٟ^0L;y[ٟ^1;y[ٟ^2)UUarethestandardcoGordinatesonR^3|s,thenh1[ٟ(dy[ٟ0,8dy[ٟ0+dy[ٟ1dy[ٟ1+dy[ٟ2dy[ٟ2L)=42r62+4339V: qW*eletgn pbGethestandardmetriconS^n .nThenisaRiemanniansubmersion dfrom^(S^3;g3|s)to(S^2; ۬1۟&fes4 g2|s)withverticaldistributionspannedby1andhorizontal ˍdistributionUUspannedbyf2|s;3g.qW*eUUcancheckthisnormalizationbycomputingh9<2[ٟ2d=vol7(S3;g3|s)=8(2[٫)=vol(S2; ۬1۟&fes4 g2|s)8vol(S1;g1|s):*AIfUU2ȫisthevolumeelementon(S^2; ۬1۟&fes4 g2|s),then[N [ٟ2C=2r6^839V:ZJNotepthatalthoughthevector eldsf2|s;3gpandcovector eldsf^29V;^3gparehori- ٍzontal,theyarenothorizontalliftsofvectorandcovector eldsonS^2.?Inx 1.14, weUUwillde neacomplexanalogueoftheHopf bration.  The@verticaldistributionisspannedby1|s;6thuse^1=$J^1Ɩandde^1=$J2[ٟ^2|s.W*euseUULemma1.10.1toseethatthecurvqaturetensortakestheform: D(1.13.a)xFQ=22|s: qW*eUUwillneedthisresultintheproGofofTheorem4.2.1.qWealsonotethatꍍ<$|1yDwfe  (֍2>cZx ySa2oGFQ=<$1Kwfe (֍ )Vds2C=0r680+11+22+339V:,W*eUUde neanalmostcomplexstructureonZ qbyde ning:8J9(0|s)=1;J9(1)=0;J9(2)=3;andqǮJ9(3)=2:^čSince+uJ!isunitarywithrespGecttothemetricde nedinequation(1.14.a),3ծds^2isa qǍHermitianUUmetric.qTheholomorphictangentbundleisspannedbyUIث0C:=<$K1Kwfe (֍2 -(0S8p 7fe X1UV1|s)UUand1:=<$K1Kwfe (֍2(2S8p 7fe X1UV3|s):NoteUUthatwehaveGnZ"4[0|s;1]=8p 7fe X1UVf[1|s;2]8p 7fe X1[1|s;3]g퍍}=i2pUWfe X1v3S822C=41|s: `FThusO*thecomplextangentbundleTC5Z:=gyspanuCf0|s;1gO*isintegrable;ifivare smoGoth6sectionstoTC5Z,nthen[1|s;2]6isasmoothsectionstoTC5Z.iThusthe qǍcomplexsMF*robGeniusintegrabilitysMconditionissatis edsotheNirenberg-Neulanderintegrabilityltheorem(seexN5A.5TheoremA.5.1)impliesthat(Z(;J9)isacomplexmanifold.џP .TGilk9ey,J.Leah9y,JH.P9arkp256Y/ɧ W*ecanalsoshowthat(Z(;J9)isacomplexmanifolddirectly.hThiscanalsobGe ٍseenUUdirectly*.qFixarealnumbGerUUr5>1.LetZactonC^2S8f0gby}.n:(z1|s;z2)!rGn(z1|s;z2):9ThisƄde nesa xedpGointfreeholomorphicrepresentationofZonC^2DZTf0gand wewletZbGetheresultingquotientmanifold.-W*eintroGducecoordinates(u;)onC^2S8f0gUUby oto(u;G)7!eu:brfor꭮u2R;52S3:TheIactiondescribGedaboveIthentakestheformn:(u;)7!(u!+ln v(r);)Isowitha"suitablechoiceofru5=.e^2 weseethatZJ>isisometrictoS^1rbS^3.+/Ifweidentify @rwithUU0|s,thenthecomplexstructureistheonegivenabGove:g=&J90C=i0=1ȫandCJ92=i2=3u W*eewgiveS^2 uwthestandardcomplexstructure.,Let|:S^3 !S^2bGeewtheHopf brationUUde nedinx1.13.qLet+}s~}(8(;p~xaī)=[٫(Ȯ~x):Z~4!S29de ne a brationfromtheHopfmanifoldtoS^2.Since;isaRiemanniansub-mersion,~"j*isfaRiemanniansubmersionaswell.]W*ehave~E0T=؈0.]Itiseasily checkedðthatK?~ٮ1@#isaholomorphictangentvectoronS^2.ThisshowsthatK?~Ӊisa(holomorphicRiemanniansubmersionandthemetricsinvolved(areHermitian.NoteUUthatthemetriconS^2eUisKaehlerbutthatthemetriconZ qisnotKaehler.7mZG!Îcю鍵1.15TThegeometryofspherebundles 1 InQ?x1.10Q?westudiedthegeometryofcirclebundles;Rthesewerede nedbycom-plex3linebundlesorequivqalentlybyorientablerealvectorbundlesofrank2.1`W*ecanUUalsostudythehigherrankcase.1.15.1De nition.LetY`VDbGearealvectorbundleofrankr53overaRiemannianmanifold\Y8.W*eassumethatV@isequippGedwitha bermetricandletS(V8)be qǍtheunitspherebundle.fW*euseaRiemannianconnectionronthebundleVtosplit b`Tc(V8)=V ]8H;seegnLemma1.15.2bGelowforfurtherdetails.W*eusethissplittingtode neaRie-mannianRmetricgV DKso":V!YfisRaRiemanniansubmersion.pTherestrictionofLtoS(V8)de nesapro8jectionS:S(V8)!Y)whichisaRiemanniansubmersion.FChapterTOne:pRiemannianSubmersions[266Y/ɧ W*e$GnowdevelopsomeoftheRiemanniangeometryofthissituation.amIntroGduce qǍloGcal(coordinatesy"=(y[ٟ^a2I)onY8.bLets=(siTL)bGealocalorthonormalframeforV8. ǍTheUUmap(x;y[٫)!siTL(y)x^iintroGducesUUlocalcoordinatesonV8.qLetP@rsid=Aaij ah(y[٫)dya2Isjwde ne;thecompGonentsoftheconnection1-formAofrrelativetothegivenloGcal frame;ёweRomitthe$p )$fe 1normalizingconstantofxě1.10.j?Let@1ɍ8yˍi and@^8xa bGethe [coGordinate)framesforthetangentbundleofthetotalspaceV8.jDThecurvqatureof theUUbundleisgivenbythetensor![٫.1.15.2TLemma.8(1)!pWehavegVɫ(@^8ya\;@1ɍ8yvb)=gYG(@ap;@bD)8+x^iTLx^j6Aaik LAbjgk &.38(2)!pWehavegVɫ(@^8ya\;@^8x;Zi)=Aajgi ahx^j6.H鍍8(3)!pWehavegVɫ(@^8x;Zi;@^8x;Zj)=ij .cC8(4)!pWehaveS:S(V8)!Y˺isaRiemanniansubmersion. y8(5)!pThe b}'ersofS ^aretotallygeodesic.8(6)!pIf\q~faƫ:=@^8ya8x^j6Aajgi ah@^8x;Zi,then\q~fa㕺isthehorizontalliftof@^8ya\. 8(7)!pL}'et RabijȵbethecurvatureoftheconnectiononV8.^Then2!abi =Rabjgi嬮x^j6:!ۍPr}'oof.pLetfв:#O@!OG(r)de nealocalgaugetransformation~s ﳫ=#f(y[٫)s.yFix qǍy0C2Y9andUUchoGosefhsothatkf(y0|s)=I7anddUUf(y0|s)=A(y0):P@Then~sh(y0|s)ln=s(y0)andrf:~s(y0)ln=0.cLet(~x;Z^~y)bGethenewsystemoflocalcoor- Ǎdinates;thevkidentityx^iTLs^iR=~=x^i o߫~ s^i\impliesy[ٟ^a0=~=y[ٟ^aandx^j4=~=x^i fij . ThusatapGoint Iz0C2[ٟ^1 M(y0|s)UUwehave:(1.15.a>)h@ ~8xi^4=@8xiqand^;@%{~8ya#׫=@8ya8Aaij ahxiTL@8xj:{荫SinceUUrf:~s(y0|s)=0,H(z0|s)=spanTf@^%{~8ya\gandV}(z0|s)=spanTf@^ ~8x;Zig.qTherefore Bō(1.15.bw)+aNgVɫ(@%{~8ya\;@1ɍ%{~8yvb)(z0|s)=gYG(@8ya\;@1ɍ8yvb)(y0|s);WzNgVɫ(@%{~8ya\;@8xi)(z0|s)=0;andqǮgV(@8xi;@8xj)(z0|s)=ij : (lAssertions^z(1-3)nowfollowfromequations(1.15.a)and(1.15.b).5Assertion(4)is immediate.cT*o^proveassertion(5),$normalizethechoiceofcoGordinatesonY4Bsothe1-jetsofthemetricvqanishaty0CmandchoGosetheframesors(y0|s)=0.ƶThentheF1-jetsofmetriconV~vqanishat(x;y0|s).lThusthecurve 8(t)=(xk+tx;y0|s)FisaϕP .TGilk9ey,J.Leah9y,JH.P9arkp276Y/ɧgeoGdesicinVΫandthe bersofVΫaretotallygeodesic.+NLet<߱2S0|s.Sinceorthogonal pro8jection7ofT`V0onTSʠiscontainedinS0|s,= S0isatotallygeoGdesicsubmanifold -ofScso(5)follows.E_SincegVɫ(\q'!~fa Ǯ;@^8xvk)=0andsincef@^8xvkgspantheverticalspace,\q~®fa Uisthehorizontalliftof@^8yadand(6)follows.WW*echoGosessoA(y0|s)=0andcompute ߍF,a2!abi /(x;y0|s)=hgVɫ([\q'!~fa Ǯ;\qɫ~fb C];eiTL)(x;y0|s)=(@1ɍ8yvb\Aajgi H8@8yaAbjgi <)(y0)xjCꍍ`=hRbajgi嬫(y0|s)xj6: ލSinceUU!.andRiaretensorial,assertion(7)follows. 9ZG!Îcю鍵1.16TPrincipalbundles m W*elrefertoEguchietal[47]forfurtherdetailsconcerningthematerialofthis -section.LF1.16.1De nition.LetFˮgG bGeabi-invqariantFRiemannianmetriconacompact LieUUgroupG.qW*esaythat":P*!Y9isUUaprincip}'alGbundleUUif8(1)!TheTbmanifoldPhasarightactionbyGthatpreserves[٫,i.e.!(z6g)=[٫(zp).m8(2)!TheUUgroupGactstransitivelyandwithout xedpGointsonthe bGersof[٫.z Letegn|i'˫bGeeabasisfortheLiealgebraofleftinvqariantevector eldsonGandlet 4g y^i (1bGelthecorrespondingdualbasisfortheleftinvqariantl1formsonG. LetgiTL(t) qǍbGeT$the1parametersubgroupofGcorrespondingtothevector eldsg ]i .n3ThenrightLimultiplicationbygiTL(t)de nesa owforthevector eldsg UionG.WRight multiplicationߑbythegiTL(t)alsode nesa owontheprincipalbundlePc.|Byanabuse_ofnotation,bwewillalsodenotethecorrespGondingvector eldsonPbyghi I; theseZvector eldsextendthegivenvector eldsg dMimfromGtoPc.CTheverticaldistributionUUV'ҫof.isspannedbythethevector eldsg^Οi .LF1.16.2TDe nition.AUUmetricgP DonPiscalledabundlemetricif8(1)!TheUUmetricgP DisinvqariantUUundertheactionofG.m8(2)!The7^restrictionofgP Mtothe bGersof7isthegivenbi-invqariantmetricgGګ.8(3)!TheUUmap.isaRiemanniansubmersion. 4NoteUUthatthe bGersof.arenecessarilytotallygeodesicifgP Disabundlemetric.1.16.3TExamples.Hz8(1)!TheUUcirclebundlesdiscussedinx1.10areprinciplebundleswith bGerS^1.8(2)!TheLAbundleS^4k+B+32!bHPF/Tk~discussedinx 1.13isaprincipalbundlewith ٍ!structureUUgroupS^3=SU(2).ᏟChapterTOne:pRiemannianSubmersions[286Y/ɧ8(3)!If@YxisaRiemannianmanifold,DRthenthebundleofallorthonormalframes qǍ!forwPthetangentbundleTcY4isaprincipalbundlewithstructuregroupthe!orthogonalUUgroupOG(dim(Y8)). 8(4)!If]_YCisanorientableRiemannianmanifold,athenthebundleoforiented!orthonormalgframesforthetangentbundleTcYisaprincipalbundlewith!structureUUgroupthespGecialorthogonalSO(dim(Y8)).8(5)!If֮YOisaHermitianmanifold,#Vthenthebundleofallunitaryframesforthe!complex"atangentbundleTC5Y[Eisaprincipalbundlewithstructuregroup!theUUunitarygroupU(dimC8(Y8)).b Let>8y"=(y[ٟ^1L;:::;y[ٟ^dim ݟ^(Y)&)bGeasystemoflocalcoordinatesonthebaseYwandlet sybGealocalsectiontotheprincipalbundlePc.2Themap(g[;y)!s(y[٫)LgjRgivesloGcalcoordinatestoPc.CThehorizontaldistributionHofޭisinvqariantunderthe GUUactionandisspannedbyvector eldsoftheform>fa:=@8ya+8iap(y[٫)g yi]Ů:t1.16.4TDe nition.TheUULie-algebravqalued1-formŦA:=iap(y[٫)dyak) 8gBYide nesaprincipalconnection. SuppGosethatGisamatrixgroup.LetVשׂbGethe assoGciated͌vectorbundle.DThentheprincipalconnectiononP1de nesalinearcon-nectiong(ronV andthecurvqatureofrisgivenbyA;theconnectionisRiemannian qǍifUUG=OG(kP),G=SOG(kP),orG=U(kP).Ѝ1.16.56eDe nition.The!curvqatureFZoftheprincipalbundlePѫistheLie-algebravqaluedUU2-formgivenby,78FNFQ:=[خdy[ٟak)^8dy[ٟb (@8ya\ibDg i@1ɍ8yvbiapg im+iap1ɍjvb6[g yi]Ů;g!j ͫ])~Sֺ=[خg[ٟijfѮ!abi /dy[ٟak)^8dy[ٟb gBYj y:Thus3inthissetting,kOthetensor!istheprinciplebundleconnection. Again,if Gisamatrixgroup,>FGgivesthecurvqatureoftheassoGciatedconnectionontheassoGciatedUUvectorbundle.V LetnH F%bGenthehorizontallifttoPofavector eldFonY8.If^Y 4~ Zs =/(F1|s;:::)is qǍaUUloGcalorthonormalframe eldforTcY8,letx(1.16.a>)axPg.~h#so:=(f1|s;:::;g!1 0;:::)UUwhere @fa:=H FabGethecorrespondinglocalorthonormalframe eldforTcP.QYThisshowsthatTcPandfTcYɫarede nedbythesametransitionfunctions;othiswillplayanimpGortantroleUUinx2.7whenwediscusspro8jectablespinors. [Q Let^P and^Y bGetheChristo elsymbolsofPtOandYIwithrespecttotheseloGcalUUframes.P .TGilk9ey,J.Leah9y,JH.P9arkp296Y/ɧ1.16.6TLemma.8(1)!pThe b}'ersofPvaretotallygeodesic,i.e.^P;Zijga(=0.J8(2)!pWehave[g yi]Ů;fap]=0.j 8(3)!pWehave^Pvabc=^Yvabc 핺,^Pvabi =^Pvaib=^Pviab=!abi /,and^P;Ziaj(=^P;Zaij=0.ԍPr}'oof.pIf;^G ez2g ,[let;ebGethecorrespondingverticalvector eldonPc. NxThe 1-parametergroupgeneratedbyeactsbyisometriessincethemetriconPjisinvqariantkunderrightmultiplication.e F*urthermore,thelengthofeisconstantonPc..Itthenfollowsthattheintegralcurvesofe,whichlieinthe bGersof[٫,aregeoGdesics;EthisQimpliesthe bersofaretotallygeodesicsothesecondfundamental [QformW^P;Zijga(=0;(thisprovesWthe rstassertion.[rLetFubGealocaltangentvector eld 葍onWY8.vThehorizontaldistributionisinvqariantundertheactionofG;thusthe qǍvector eldf:=.H ((Fc)isinvqariantunderthe owde nedbyeso[e;f].=0;the second1assertionnowfollows.eAs(H3Fc)=F,9 [fap;fbD]=[Fa;FbD].eThus,9 since1isUUaRiemanniansubmersion,EjgP([fap;fbD];fc)=gYG([Fa;FbD];Fc):nW*eUUexpressintermsoftheLiebracketUUtosee^Pvabc=^Yvabc 핫.qSince[g yi]Ů;fap]=0,C @Paij(=Piaj=Pijga=0UUandPabi =Paib=Piab /:YThisUUshows^PvabiUisskewsymmetricinaandb.qThush)Pabi = K1K&fes2 )(^Pvabi g8^Pvbai /)=!abi: UV?ZG!Îcю鍵1.17TIn9tegrationoverthe bQers Let0 :ٮZh!YbGeaRiemanniansubmersionwith berX(y[٫):=^1 My> over0a pGointy"2Y8.,QLetX=X(y0|s)bGethe beroverthebasepoint.,QLetX bethevolume qǍformZon bGer.UIfdvol3Z{handdvol3YTarethevolumeformsonZandYrespectively*,thenUUwehavedvolZ!:=X ^8[ٟ^edvolY. Let:AˮZ!YثbGeaRiemanniansubmersionwith berX.NPullbackde nesanaturalUUmorphism[ٟ^ի:C^1 0^pR(Y8)!C^1^pR(Z):rǍ1.17.1&]De nition.W*eaverageoverthe bGerstode nethepushԻforwar}'d.E+This qǍisamapEfromC^1 0^pR(Z)toC^1 0^pR(Y8).xLetY@2C^1 0^pR(Z).xLetF1|s,...,FpLYbGe ꬍtangent vectorsaty"2Y8.YLetV(y[٫):=ĴRwillusethe rstassertionofthisLemmainx3.5whenwediscussthecomplex Laplacian.*qW*eQwillusethesecondassertionoftheLemmainx3.10whenwediscusstheUUheatcontentUUasymptotics.B6Y1ɧulChapterTwro:8=1OpOeratorsofLaplaceTrype''K虚ff .UX/./././././././././././././././././././././././././././././././ ffffLG!Îcю鍵2.1TIn9troQduction In]thischapter,we]studypartialdi erentialopGeratorswhichareofLaplacetypGe. vInnx s72.2,lswende nethesymbolofanoperator. vInx s72.3,lswediscussthediscretefspGectralresolutionofanoperatorofLaplacetype."6Inxk2.4,_weusesphericalharmonicsbtoconstructthediscretespGectralresolutionofthescalarLaplacianonthep$sphere.4Inx2.5,vweusesphericalharmonicstoconstructthediscretespGectralresolutionmofthescalarLaplacianfortheHopfmanifold.MInx[62.6,wegiveanormalformR$foranyopGeratorofLaplacetypGeanddiscusstheBochnerLaplacian.h4W*estate'theBoGchnerandLichnerowiczvqanishingtheorems.v>W*ealsopresentabriefintroGductiontotheheatequationasymptotics;TifisaRiemanniansubmersion,thereMisnosimplerelationshipbGetweenMtheinvqariantsMofthebase,݋theinvariantsofUUthe bGer,andtheinvqariantsUUofthetotalspace. Innxຫ2.7,Xwengeneralizetheintertwiningformulasofxຫ1.5fromtherealtothecomplexA;category*.5yInthespinorsetting,|4thesituationisquitedi erentandweassume}5eg: Z!Yisaprincipalbundle.hW*echoGoseaspinorontheLiegroupinquestionOtode nethenotionofpullback;-Mthisistheconstructionofpr}'ojectablespinorseHofMoroianu.Oncethisisdone,iEwecandetermineanalogousintertwiningformulas?fortheDiracopGeratorsonY#andonZ.W*erefertothematerialoftheappGendixUUxA.2UUandxA.3forfurtherdetailsconcerningspinors. Inex).2.8,weediscussellipticopGeratorsonmanifoldswithboundary*.WeediscussGreen's(formulaandintroGduceabsolute,1relative,Dirichlet,and(NeumannbGound-aryRconditions.iFInxH2.9,wediscussRiemanniansubmersionsinthecategoryofmanifoldswithbGoundary*.LWeshowthatpullbackpreservesDirichletandabsolutebGoundaryconditions.)xW*egivenecessaryandsucientconditionsthatrelativeand,NeumannbGoundaryconditionsarepreserved.bW*eshowthattheNeumann ~ChapterTTw9o:pOpQeratorsofLaplaceTypQe#e326Y/ɧLaplacian4-onformsneednotbGepositivede nite;thisisasomewhatsurprising result.6pZG!Îcю鍵2.2TThesym9bQolofanoperator Letu=(u^1|s;:::;u^m)bGeasystemoflocalcoordinatesonaRiemannianmanifold ǍM.qLetUU@^8u;Zi=@8=@u^iTL.IfUU В=( 1|s;:::; m)isamulti-index,let"d@8x y:=(@8x፬1) 1 :::(@8xm) m V®;uǟ :=:uǯ 1l1 ɮ:::uǯ m፯m ̉;andqDZj zj:= 1S+8:::+ m:&#LetvVZandWbGesmoothcomplexvectorbundlesoverMёandletPmapping qǍC^1 0(V8)toC^1(Wc)bGeapartialdi erentialoperatorofordern.ChooselocalframesUUforV9andWtodecompGoselP*=P USj jn$\a (x)@^8x ɹwhereforanypGointx2M,thecoGecientsa (x)arelinearmapsfromthe bGerofVUΫoverxtothe bGerofWyoverx.^W*ede nethele}'ading`symbolofPbyreplacing di erentiationUUwithmultiplication:mqLt(Pc)(x;uǫ):=P USj j=n"Ca (x) (:If weidentifyaҫwiththecotangentvector8 :=FPPiOuiTLdx^i,then Lt(Pc)(x;uǫ)isan ʍinvqariantlyde nedmapwhichishomogeneousofdegreenfromthecotangentbundleUUTc^sMlptothebundleofendomorphismsfromV9toWc.qW*ehavenmLܮLt(Pcs)=(1)nq~L(Pc)9and#mLܮLt(P1S8P2|s)=L(P1|s)8L(P2|s): ThecleadingsymbGolissometimesde nedwithfactorsof$p 8$fe 10,ther}'eexistsanintegern()sothathzn/ 233#&fe#mFfn8n/ 233#&fe#mF+for&خnn():"7ChapterTTw9o:pOpQeratorsofLaplaceTypQe#e346Y/ɧ8(3)!pIf_Y2L^2|s(V8),Z=letcn =(;nq~):L2 (V)b}'etheF;ouriercoecients.Then :!p}ٺb}'elongstoC^1 0(V8)ifandonlyifthecn Warerapidlydecreasing,Ui.e. r!plim/n7!1F n^k됮cn8=0foranynatur}'alnumberkP. \䍍8(4)!pL}'etArbeaRiemannianconnectiononV %andletjjkѺbethesupnormofthe a!pkP^th c}'ovariant|derivativeof.TThenthereexistsj(kP)sothatjnq~jkб!n^jg(k+B) qǍ!pifnissucientlylar}'ge.V IfUUissmoGoth,itfollowsfromthisTheoremthattheseries̍Ȯ=X >n㉮cnq~n̍converges$absolutelytointheC^1 TtopGology*.aWe$willusethisobservqationinthe proGofUUofLemma3.5.7andintheproofofTheorem3.8.1.qLetȍczE(;DG)=f2C1 0(V8):D=gbGetheassociatedeigenspaces;|theseare nitedimensionalforallandnontrivialonlyforadiscretesetofeigenvqalues.ThenwehaveanorthogonaldirectsumdecompGosition jtL2|s(V8)=>:E(;DG):rNotethatthelinearspanoftheeigenfunctionsisdenseinC^1 0(V8)intheC^1 qǍtopGology*.8C~ZG!Îcю鍵2.4TSphericalharmonics Let{M|n=eSS^1 {bGetheunitcirclewiththeusualperiodicparameter. (8The Laplacianvonfunctionsisgivenby@^82v!andthediscretespGectralresolutiontakes RtheformfeIJpɟĉW W*edecompGosetheeigenspaceofthe LaplacianUUonS^3eUintoeigenspacesofthisactionƍL:-H(3;kP):=`H`(3;kP)UUwhere @(t)p=esIJpɟsĉW 9;Dqzp^1gggeneratethealgebraofallpGolynomials.Thusweseethatweneed k`kwhenUUstudyingH`(3;kP)soxrL2|s(Z)=:jT;j`jkjo8H`(3;kP):ThespacejY=0and`=0correspGondstofunctionswhichareinvqariantunderbGoth circleRactions;suchfunctionsarethepullbackofeigenfunctionsonthe2-sphere. cSinceUUS^2eUisKaehler,^0vSa2 y?=2:(0;0)vSa2.8rZG!Îcю鍵2.6TTheBoQc9hnerLaplacian ThereUUisanormalformforanyopGeratorofLaplacetypGe.6`2.6.1De nition.IfyrisaRiemannianconnectiononVH]andifEisanauxiliaryself-adjointUUendomorphismofV8,wecande nexpbDG(r;E):=(g[ٟwi 2crɱr+8E)BwhereVg[ٟ^widenotesthemetrictensoronthecotangentbundle.TheassoGciatedopGeratorUUg[ٟ^wi 2crɱriscalledtheBo}'chnerLaplacian.Ѝ W*eUUreferto[64]fortheproGofofthefollowingresult'P .TGilk9ey,J.Leah9y,JH.P9arkp396Y/ɧ2.6.2mLemma.L}'et ADT^beaself-adjointoperatorofLaplacetype.Thereexistsa uniqueRiemannianc}'onnectionronVandauniqueself-adjointendomorphismE qǍofMoVSsothatD5=DG(r;E).IfD=(g[ٟ^wi 2cIVɮ@ɮ@q+a^b@F+b),[thenMothec}'onnection D1formx䍑7x~AϺofrandendomorphismE'tar}'egivenby!ſnx䍑`˫~]:Al˫= K1K&fes2 )gwiA(a^2+8g[ٟ^ 0 ?W^9 IVɫ)]:EZ=b8g[ٟwi 2c(@x䍑Z~ɮAk+x䍑q~Ax䍑V;~ALx䍑q~Awwi ֊9 )"O W*eRwillusethisdecompGositioninxǫ3.10whenwediscusstheheatcontentasymp- totics.2.6.31Example.Let dbGetheexteriorderivqativeand'thecoderivqative.BZW*eformthePLaplaciani=d@+ `d.bThePW*eitzenbochformulasexpressintheformgivenbyLemma2.6.2._TheassoGciatedconnectionistheLevi-CivitaconnectionandUUtheendomorphismEisgivenby ˍKEZ= 33133&fes8bٮRijgk+Bnhc^lȫ(e^iTL)c^l(e^j6)c^rm(e^k됫)c^r(e^nq~)8 l1l&fes4Rijgji I"wherekc^l =BextRHlE&int q*l۫isde nedbyleftexteriorandc^rY=BextRHrint q*ruisde nedbyright[exteriorandinteriormultiplication.IfpЫ=0,]_the[endomorphismEQvqanishesandUUwehave }m0C=g[ٟwi 2crɱr\:lɍIf,p=1,theendomorphismEhisgivenbytheRiccitensor.GLetij :=Rik+BkjDbGethe 7RicciHtensor.^Ifisa1-form,thentheactionoftheRiccitensoronisde nedby: S w2S():=ijextR(eiTL)int q(ej6):TheUUW*eitzenbochformulasthenbGecome:61C=g[ٟwi 2crɱr+8:!2.6.4Example.Let]ER~:=Rijgji5bGethescalarcurvqature.LetD^C @bethespin \pLaplacian(seex׫A.2).^TheassoGciatedconnectionristhespinconnectionandthe LichnerowiczUUformula[123]showsW4ōX}DGCw=yT*rI(r2|s)8+ l1l&fes4R;i.e.E^C= 33133&fes4bٱR: 㰍 W*e[canusetheWeitzenboch[andLichnerowicz[formulastoprovethefollowingtheoremUUwhichwillbGeusedinx5.2.(ΟChapterTTw9o:pOpQeratorsofLaplaceTypQe#e406Y/ɧ2.6.5TTheorem.L}'etMbeacompactRiemannianmanifold8(1)!pSupp}'osetheRiccitensorofM#ispositivede nite.ThenH^1Lq(M;R)=0.38(2)!pSupp}'osekthescalarcurvatureRofMispositiveandMisspin.gThenthere qǍ!par}'enoharmonicspinorsonM.\̍Pr}'oof.pSuppGosetheRiccitensorispositive. W*ecan nd>0sothatwehavetheestimateg[٫(;)g(;).SuppGosethatisaharmonic1form.W*eusetheUUW*eitzenbochformulaandintegratebypartstocompute:" ލ[X0=hu(1|s;)L2 =(r;r)L2 t+8(;)L2#`hu(;)L2 0:"ThisrimpliesjjL2 =0andhence=0asissmoGoth.Q&Consequentlythereareno non-trivialharmonic1forms.ETheHoGdgedecompositiontheorem(Theorem1.4.6) ٍidenti esܮH^1Lq(M;R)withker0O(^1|s);the rstassertionnowfollows.FTheproGofofthesecondassertionissimilar.C{SuppGosethatthescalarcurvqatureRispositiveand thatUUMlpisspin.qChoGose>0UUsothatR.W*eUUintegratebypartstocompute:",43S0=cC(C0ޮ;)L2 =(r;r)L2 t+ l1l&fes4(R;)L2#[{ dvA1dvA&fes4i(;)L2 0:",TheUUsecondassertionnowfollows. Remark.Inx/2.8,wewillgeneralizeTheorem2.6.5(2)tothecaseofmanifoldswith9+bGoundary*.hdWewillimpGosespectralboundaryconditions.hdThiswillcreatean additionalUUbGoundaryintegrandwhichfortunatelyalsoispGositivesemi-de nite. AtJthisstagewedigressbrie ytoillustratetheuseofLemma2.6.2.n-LetMabGeasmoGothcompactRiemannianmanifoldwithoutboundaryofdimensionm.KLet qǍfnq~;ngBbGethediscretespectralresolutionofaself-adjointoperatorD>_ofLaplacetypGeactingonthespaceofsmoothsectionstoasmoothvectorbundleV8.UgThe [Qtrace3ofthefundamentalsolutionoftheheatequatione^tDchasanasymptotic traceUUast#0^+ ofUUtheform:c;YT*rE"L2O(etDN)=X >n㉮etn26X k+B0o(4[٫)m=2t(k+Bm)=2 ak됫(DG):%YThe;coGecientsak됫(D)arespectralinvqariants;oftheoperatorD.i Theyarelocallycomputable`gandvqanishforkoGddiftheboundaryofMwisempty*.We`grefertoGilkeyx[64]UUfortheproGofofthefollowingresult:)P .TGilk9ey,J.Leah9y,JH.P9arkp416Y/ɧ2.6.6Theorem.L}'etUMl7beacompactRiemannianmanifoldwithoutboundary. L}'etD߸beanoperatorofLaplacetypeonthespaceofsmoothsectionstoasmooth ve}'ctormbundleoverM.'DecomposeD3=RT*r (r^2|s)߱E.'Letm`;'denotemultiple qǍc}'ovariant;di erentiationwithrespecttotheconnectionrandtheLevi-Civitacon-ne}'ction.Letӫ b}'ethecurvatureoftheconnectionrandletRbethecurvatureoftheL}'evi-Civitaconnection.Wethenhave:rЍ8(1)!pa0|s(DG)=ĴRon"bήC1 0(p;q@L))(M):,jChapterTTw9o:pOpQeratorsofLaplaceTypQe#e446Y/ɧSinceUUpullbackcommuteswithbGothi^(p;q@L)>and[ٟ^(p;q@L)«,wecomputethat v(2.7.a=)n\q@?@8፯ZKQz[ٟ-8[ٟ\q8@8፯Y9=:[٬(p;q@L1)"Z6Zp[i:(p;q@L)"Z)[ٟ[ٟ:[٬(p;q@L1)"Y6YGi1ɍp;qY鍍C\=KQz:[٬(p;q@L1)"Z6(Zp[[ٟ-8[ٟYG)i:(p;q@L)"Y=:[٬(p;q@L1)"Z[ٟi:(p;q@L)"Y"eSon{EC^1 0^(p;q@L))(M).W*esuppresstheroleofi^(p;q@L).tocompletetheproGofofthe rst qǍidentity;UUweusetheidentitiesu\q[cZ_b@Zfڮ[ٟի=[ٟ\q8@Yvand'@:[٬(p;q@L)"Z\që®@Z R=\q @Z:[٬(p;q@L1)"ZRtogetherUUwithequation(2.7.a)tocompletetheproGofLemma2.7.2bycomputing:!Sn!:(p;q@L)"Z)[ٟH8[ٟ:(p;q@L)"Y=\q @Z(\qD@8፯Z x[ٟ-[ٟ\q8@8፯Y0!)+(\qD@8፯Z x[ٟ-[ٟ\q8@8፯Y0!)\qD@Y鍍>=\qGEҮ@ZQJ:[٬(p;q@L1)"Z6[ٟ-+8:[٬(p;q@L)"Z«[ٟ\q8@Y9=:[٬(p;q@L)"Z(\qD@Z x+\qD@Z)[ٟ: &A4 Thereisananalogueofthisintertwiningresultinspingeometry*.8Thema8jor technical@dicultyistoconstructasuitablenotionofpullback.ZLetS(R^m)moGdulestructure.HAlsothedecompositionformulassimplifyQgreatlywithrespGecttoproducts.pIfMiisaspinmanifold,RletCWMbGethe [QassoGciatedbundle.)LetA^CbM s betheassociatedDiracoperator.)LetD^CbM :=-U(A^CbM\)^2 *bGetheassociatedSpin6cL}'aplacian.LetcM NbetheleadingsymbolofA^C0ޮM.The endomorphismUUcM 5givesaCli ordmoGdulestructuretoCWMlpsinceu֓cM\(uǫ)2C=jj2S8IdUUon8CWM:TheTbundleCWMkǫinheritsanaturalconnectionr^C0ޫ.oRelativetoanorthonormalframekfF\gforTcMandrelativetotheinducedorthonormalframeforCWM,0the EconnectionUU1formA^C3ofr^C0ޫ,andtheopGeratorA^CbM 5arede nedby3(2.7.bv)n;AC:= V1V&fes4\Fc^0 8cM\(Fc^]X)cM(Fc^%)^Mwi82Tc^sMO End#(CWM);andj7!AC፯M t:=UѮcM <8rC:C1 0(CWM)!C1(CWM):ݍ LetYbGeaspinmanifoldandlet:>P ͱ!YbGeaprincipalGbundlewitha [Qprincipal^bundlemetricasde nedinx'1.16., W*eusethecorrespGondence^Y ~ ps罱7!^P ^~ rseof qǍloGcald]framesonYAtolocalframesonPdescribedinEquation(1.16.a)toidentifythetransitionfunctionsofthetangentbundleofP4withthepullbackofthe-՟P .TGilk9ey,J.Leah9y,JH.P9arkp456Y/ɧtransition~functionsforthetangentbundleofY8.S*ThespinstructureonY2bpGermits qǍusLRtoliftthesetransitionfunctionsfromSO(dimUVY8)toSpin(dimY8)Spin8(dimPc).Thisde nesanaturalspinstructureonPc.*Letg bGetheLiealgebraoftheLie groupN G.oYWithrespGecttothisnaturalspinstructureonPc,OwehavethefollowingrelationshipUUbGetweenthespinorbundles:R(2.7.c)CWP*=CYqı RCgЮ:SLetUU В2CWg%withj zj=1.qW*ede nethepullback(2.7.dv)Wn[ٲ፯ y:C1 0(CWY8)!C1(CWPc)UUby87!8 z:ThisQgivesrisetothenotionofpr}'ojectable^spinorsQintroGducedbyMoroianu[136, 137,B138,139,140];EQour=NpGointofviewisabitdi erentasweareworkingwiththe qǍbundleUUCꬫratherthanthebundleS,buttheconstructionisequivqalent. IfhjthestructuregroupGissimplyconnected,/then1|s(Pc)=1(Y8)hjandthiscorrespGondenceJofspinstructuresisabijectivecorrespondencebetweenJspinstruc-turesU?onPΫandspinstructuresonY8.qIf1|s(G)6=0,UChowever,theU?situationcanbGea_bitmorecomplicated;+ifwetwistthecanonicalspinstructureonGbytaking coGecientsina atlinebundleoverthe bGer,)wenolongerhavetheisomorphismgiven^CW^oԫ.]LetbetheparityopGeratorde nedinx)mA.2;ˮj=+1onCW^eࢮYandj:=1onCW^oԮY8.W*ehavetheUUfollowingintertwiningresults:KԮcYG(Fca9)8+cY(Fca9)=0UUandAC፯Y8+AC፯Y _=0:Letef!abi߫:= 1&fes2 D(^Pvabi rC^Pvbai /)bGethecurvqaturetensorde nedinx/1.2.W*ealsoadopt thenotationofx^׫1.16.5givingloGcalframesfFap;g!i mgandfFc^a9;g!^igforthetangentandUUcotangentbundlesofPc.qLetO(2.7.e)\>Espinn:= K1K&fes4 )!abi /cYG(Fc^a9)cY(Fc^bgӫ)8 cgy(g y^i]ū):D卫ThectensordoGesnotenter; wecareonlyconsideringprincipalbundleswithprinci-palbundlemetricssothe bGersaretotallygeodesic.4WiththenotationestablishedabGove,UULemma1.5.1hasthefollowinganalogue:NK2.7.3Lemma.L}'etX2;:bP9!YӮAC፯P(8 z)=AC፯YG8 BZ+ AC፯Gڮ +Espin+V( z):.SChapterTTw9o:pOpQeratorsofLaplaceTypQe#e466Y/ɧLet(H;Vɫ)bGehorizontalandverticaltangentvector elds./Undertheisomor- phismUUgiveninEquation(2.7.c),wehave_\خcP(H;Vɫ)=cYG(H)8 1+ cgy(Vɫ):ZW*e usethisidenti cationofthesymbGolandde nitiongiveninequation(2:7:b)to qǍseeUUthat:_3 4AC፯P(8 z)o4(AC፯YG)8 ~g=o(2Pabi g+8Piab /)cYG(Fca9)cY(Fcbgӫ) cgy(g yi]ū) N{+8(Paij H+2Pijga ah)cYG(Fca9) cgy(g yi]ū)cg(g yj @%) 捍{+8Pijgk v cgy(g yi]ū)cg(g yj @%)cg(g yk ) z:O:W*ejzuseLemma1.16.6toseethat2^Pvabi u+F1ɍpviabU=U!abizandthat0=^P;ZaijK=^P;Zijga ah. `TTheUU naltermsyield48 A^CbGڮ z. :ъZG!Îcю鍵2.8TManifoldswithbQoundary Zʍ Previously*,wephaveassumedthatMwaswithoutbGoundary*.YWenow,forthe moment,0dropthisassumption.W*esuppGose,instead,thatM isacompactRie-mannianmanifoldwithsmoGothboundary@8M.8NW*emustimposesuitableb}'oundaryc}'onditionsDtoensurethatouropGeratorsareself-adjoint.(LetNM randN^bMbGethe inwarde1unitnormalvectorandcovector eldsonthebGoundary@8M.\IfWN2ᇮTc^sM, qǍlet9extY(uǫ)9denoteexteriormultiplicationandletint+(uǫ)denotethedual,sinteriormultiplication.HThe followingassertionsarewellknown;fseeforexampleGilkey[64].^LemmaT2.8.1(Green'sF orm9ula).8(1)!p(dM\; ):L2 (M,)K =(;M ):L2 (M,)ұ8(ext t(N^bM); ):L2 (@n9M,)7i.ɍ8(2)!p(1ɍpM\; ):L2 (M,)K =(dM;dM ):L2 (M,)ҫ+8(M;M ):L2 (M,)i50+(int (N^bM\)dM; ):L2 (@n9M,)pI8(ext t(N^bM)M; ):L2 (@n9M,)7i.yb LetXwԱ2C^1 0^pR(M).|W*esaysatis esDirichletb}'oundaryconditionsXiftherestrictionoftothebGoundaryofMvqanishes,:i.e.>RtheDirichletboundaryop-eratoṟBD@ 2:=j@n9Mvqanisheson.,LetrbGetheLevi-Civitaconnection.W*esaysatis esNeumann%b}'oundaryconditionsūiftherestrictionofthenormalde- rivqative,WoftothebGoundaryofMCrvanishes,4i.etheNeumannbGoundaryoperatorBN:=rNmMLj@n9M(vqanishesUUon./P .TGilk9ey,J.Leah9y,JH.P9arkp476Y/ɧ ThereareothernaturalbGoundaryconditionsandoperators.Qtothissetting.jLetH^poP(M;R)bGetheabsolutecohomologygroupsandlet H^poP(M;@8M;R)UUbGetherelativecohomologygroups.M2.8.2 Theorem(HoQdge-deRham).L}'etM ͺbeacompactRiemannianmanifoldwithsmo}'othboundary.88(1)!pWehaveE(0;1ɍpMQ;BmAƊ)=H^poP(M;R).Ec8(2)!pWehaveE(0;1ɍpMQ;BmR«)=H^poP(M;@8M;R).׍ IfMisoriented,*theHoGdge?operatorintertwines1ɍpMQ;BmAand1ɍmpMQ;BmR«and >inducesUUthePoincaredualityisomorphismrڍ(;ܮHpoP(M;R)=E(0;1ɍpMQ;BmAƊ)E(0;1ɍmpMQ;BmR«)=Hmp(M;@8M;R):RelativenandabsolutebGoundaryconditionsareimportantinindextheory*.For example,UUtheEuler-Poinc}'arecharacteristicUUisgivenanalyticallyby:2ZF(M)=pR(1)pIdimPE(0;1ɍpMQ;BmAƊ);and덍ZF(M;@8M)=pR(1)pIdimPE(0;1ɍpMQ;BmR«): W*efshallneedthefollowingtechnicalLemma.6Althoughitiswellknown,*wegiveUUtheproGoftoillustratethetechniquesinvolved.0#ChapterTTw9o:pOpQeratorsofLaplaceTypQe#e486Y/ɧ2.8.3TLemma.L}'etB=BD@,BN,BA,orBRb.8(1)!p1ɍpMQ;B$isself-adjoint.I8(2)!pIf/JB36=XBN,"1ɍpMQ;Bsisnon-ne}'gative,i.e. k(1ɍpM\;)X0/Jforanyin ȍ!pC^1 0^pR(M)withBM۫=0.dPr}'oof.pW*eUUuseLemma2.8.1tosee'aፍ(2.8.a=)醍J(1ɍpM\; ):L2 (M,)ұ8(;1ɍpM ):L2 (M,)]Bث=J(int (N፯M\)dM; ):L2 (@n9M,)pI8(ext t(N፯M)M; ):L2 (@n9M,)Lֱ8(;int q(N፯M\)dM ):L2 (@n9M,)pI+(;extG(N፯M)M ):L2 (@n9M,)7i:' T*oa+establishassertion(1),d!wemustshowthatifand satisfythebGoundary qǍconditionsUUB0then٠(2.8.bv)i(pR; ):L2 (M,)K =(;p ):L2 (M,):ڍW*eQmustalsoshowthatifwearegiven sothatequation(2.8.b)holdsforallwithUUBM۫=0,thenBM۫ =0. Dirichlettb}'oundaryconditions: SIfand satisfyDirichletbGoundaryconditions,thenUUthebGoundarytermsinLemma2.8.1(2)vqanishandwehave+e(2.8.c)?G(1ɍpM\; ):L2 (M,)K =(dM;dM ):L2 (M,)ҫ+8(M;M ):L2 (M,):ڍEquation#F(2.8.c)issymmetricinand ;3weinterchangetherolesofand to seeѴthat(2.8.b)holds.Conversely*,suppGoseѴthat isgivensothat(2.8.b)holdsforUUallwithj@n9MZ=0.qW*euseequation(2.8.a)toseethat٠(2.8.dv)Ec(int (N፯M\)dM; ):L2 (@n9M,)pI8(ext t(N፯M)M; ):L2 (@n9M,)=0forUUallwithj@n9MZ=0.qNearthebGoundaryofM,wedecomposehP = 1S+8N፯M <^ 2ȫandC=1+8N፯M <^2|s:ҀW*eUUassumeiTLj@n9MZ=0.qThenequation(2.8.d)yieldsR(@m1|s; 1):L2 (@n9M,)pI+8(@m2; 2):L2 (@n9M,)=0:YSince̴wecanspGecifythenormalderivqativesofi!arbitrarily*,thisimpliesBD@ =0 ÌasLydesired.W3Bytakingb= Lyinequation(2.8.c),Bwesee(1ɍpM\;)b0Lywhich pWestablishesUU(2)forDirichletbGoundaryconditions.1'aP .TGilk9ey,J.Leah9y,JH.P9arkp496Y/ɧAbsoluteb}'oundaryconditions:qǫNoteUUthatnˍ[ð(ext t(N፯M\)M; )=(M\;int q(N፯M) ):Ifoand satisfyabsolutebGoundaryconditions,thenint`(N^bM\)dMoandint(N^bM\) AvqanishonthebGoundary*.TThustheboundarytermsinLemma2.8.1(2)vqanishand equation5(2.8.c)holds.IAsforDirichletbGoundaryconditions,;thisimpliesequation(2.8.b)2holdsandshowsthat(2)holds. Conversely*,j suppGosethat isgivensothatUU(2.8.b)holdsforallwithBA=0.qW*eUUuseequation(2.8.a)toseethat(2.8.e)DGg(M\;int q(N፯M) ):L2 (@n9M,)pI+8(;int(N፯M)dM ):L2 (@n9M,)=0: >T*akenadaptedcoGordinatesystemsxT=(y[;r)nsodr7q=TN^bM Noandsothat@r]=NM\;zrr ㍫isUUthegeoGdesicdistancetotheboundary*.qNeartheboundaryofM,letUe[;:=1፯I(y[٫)dyI+N+8rG2፯J(y)N፯M <^dyJ:ݍThenrG^2bJj@n9M=$0and@rm^1bIj@n9M=$0sosatis esabsolutebGoundaryconditions.NoteUUthatthereexistsanopGeratorQsothat:j#M\j@n9MZ=2፯J(y[٫)dyJ*i+8Q(1፯I):nˍDe neUUbytheequations:M1፯Idy[ٟI:=int(N፯M\)dM j@n9M(and&ұ82፯Jdy[ٟJ:=int(N፯M) j@n9M̳8Q(1፯I):W*eUUuseequation(2.8.e)toshowthatBA =0UUbycomputing:Ǯq"0="Y(2፯Jdy[ٟJ*i+8Q(1፯I);int q(N፯M\) ):L2 (@n9M,)pI+(1፯Idy[ٟIn;int q(N፯M\)dM ):L2 (@n9M,)368;="Yjjint q(N፯M\) jj2ٔL2 (@n9M,)pI+8jjint(N፯M)d jj2ٔL2 (@n9M,)7i:9tR}'elativeҾboundaryconditions:̫This_casefollowsfromabsolutebGoundaryconditions usingUUtheHoGdge?operator. 9Neumannab}'oundaryconditions:ULetl`;'denotemultiplecovqariantdi erentiationwithrespGecttoalocalorthonormalframe eld.LHW*eadopttheEinsteinconventionandsumoverrepGeatedindices.4W*eusetheWeitzenbochformulasofExample2.6.3toUUexpress Ì1ɍpM\=;ii +8Eowhere\E'isaself-adjointendomorphismoftheexterioralgebragivenbythecur-vqatureUUtensor.qW*ecomputenˍF\Q(;iil'; ):L2 (M,)K =(;iۮ; ;i):L2 (M,)ұ8(;m *; ):L2 (@n9M,)259ChapterTTw9o:pOpQeratorsofLaplaceTypQe#e506Y/ɧandUUwecomputect醍e(1ɍpM\; ):L2 (M,)ұ8(;1ɍpM ):L2 (M,)]^=e(;iil'; ):L2 (M,)ҫ+8(; ;ii):L2 (M,)^=h8(;m *; ):L2 (@n9M,)pI+(; ;m *):L2 (@n9M,)7i:#ctThebGoundarycorrectiontermsvqanishinthe nalequationifbothand satisfy NeumannbGoundaryconditions;converselyiftheseboundarycorrectiontermsvqan-ishtforallsatisfyingNeumannbGoundaryconditions,then satis esNeumannbGoundaryUUconditions. 3 TheoremUU2.3.1generalizestothissettingtobGecome:2.8.4KwTheorem.L}'etiM beacompactmanifoldwithsmoothboundary@8M.Let ByQdenote+vDirichlet,YNeumann,absolute,orr}'elativeboundaryconditions. `BThe Ìeigensp}'aces*E(;1ɍpMQ;BD=)ar}'e nitedimensionalspacesforallandwehavean >ortho}'gonaldirectsumpӍqZL2|s(pR(M))=>:E(;1ɍpMQ;BD=):Ther}'eߴareonlya nitenumberofnegativeeigenvalues.|Ordertheeigenvaluesso 1~Kخ2:::Uandr}'epeatUtheeigenvaluesac}'cordingUtomultiplicity.F;orany>0, qǍther}'eexistsanintegern()sothate hzn/ 233#&fe#mFfn8n/ 233#&fe#mF+for&خnn(): L}'etGbeasmoothpformwithBMۮ=0.rExpandG=nq~n MźinL^2 Xwher}'ethe Ìn82E(;1ɍpMQ;BD=).Thenthisseriesc}'onvergesintheC^1 top}'ology.2.8.5-Example.Let\(MsCbGetheintervqal[0;[٫].?ThespectralresolutionwithNeu- ٍmannbGoundaryconditionsisfcos c(nx);n^2|sgn0:uandwithDirichletboundarycondi- 4tions0Iisfsin G(nx);n^2|sgn>0.enOnfunctions,7absolutebGoundaryconditionscorrespond qǍtoNeumannbGoundaryconditionsandrelativeboundaryconditionscorrespond toDirichletbGoundaryconditions;.on1forms,߻therolesofthesetwobGoundaryconditionsUUarereversed.qThus"8(1)!H^0Lq(M;R)=E(0;^0bMQ;BmNIg)=18R.w8(2)!H^1Lq(M;R)=E(0;^1bMQ;BmD)=0:8(3)!H^0Lq(M;@8M;R)=E(0;^0bMQ;BmD)=0:8(4)!H^1Lq(M;@8M;R)=E(0;^1bMQ;BmNIg)=dx8R3 Theorem!2.6.6generalizestothissettingaswell.*W*eonlypresenttheresults forPheatequationasymptoticswithDirichletbGoundaryconditions;R@theresultsare3DvP .TGilk9ey,J.Leah9y,JH.P9arkp516Y/ɧsimilar,forNeumannbGoundaryconditions.d9Recallthese}'condnfundamentalform,is givenUUby: qǍL(X:;Y8)=(rX$Y9;@N) jwhereaXCandYKEaretangentialvector eldsand@N istheinwardunitnormal.[vOnthepbGoundary*,w^weletfeiTLgbealocalorthonormalframe eldsoem =y@N.vW*eletindicesa,-b,andcrangefrom1tom;1andindexthecorrespGondingorthonormalframeUUforthetangentbundleofthebGoundary*.Mэ2.8.6Theorem.L}'etMbeacompactRiemannianmanifoldwithsmoothbound-ary.We5imp}'oseDirichletboundaryconditions.LetDARbeanoperatorofLaplacetyp}'e4onthespaceofsmoothsectionstoasmoothvectorbundleoverM.[ZDecompose D=oӱT*r (r^2|s)|^E.L}'et `;'denotemultiplecovariantdi erentiationwithrespecttofthec}'onnectionfrandtheL}'evi-Civitaconnection.{Let bethecurvatureofthec}'onnection,randletRb}'ethecurvatureoftheLevi-Civitaconnection.fWethenhave:Mҍ8(1)!pa0|s(DG)=ĴR0ɫisUUde nedby>0whichispro8jectiononthepGositivespGectrumofBq.qW*eletS|=DGC፯MQ;B U:=AC፯MQ;B>0jAC፯MQ;B/|04RşChapterTTw9o:pOpQeratorsofLaplaceTypQe#e526Y/ɧbGetheassociatedoperatorofLaplacetype.nW*enotethatifthescalarcurvqature [QR[ȫispGositive,]ethenker;(Bq)=f0g[ȫbyTheorem2.6.5. ThusA^CbM <$isself-adjointwith this,'bGoundaryconditioninthissetting.:E(;DGC፯MQ;BD=):U!pOr}'derhtheeigenvalues01C2:::hwher}'eeacheigenvaluesisrepeated qǍ!pac}'cordingRtomultiplicity.F;orany|>0,ther}'eRexistsanintegern()so!pthat ؍hzn/ 233#&fe#mFfn8n/ 233#&fe#mF+for&خnn():!pL}'etTbeasmoothpformwithBMۮ=0.ExpandT=nq~n-inL^2"wher}'ethe [Q!pn82E(;D^GCbB).Thenthisseriesc}'onvergesintheC^1 top}'ology.W8(2)!pIfR>0,thenE(0;D^GCbB)=0,i.e.ther}'earenoharmonicspinors.K0Pr}'oof.pTheU rstassertionfollowsfromworkofGrubbandSeeleyy[86,V$87,88]Uandwerefertothemforfurtherdetails.2W*eprovethesecondassertiontoillustrate theuseofspGectralboundaryconditions;}thiswas rstestablishedbyBotvinnikand!GilkeyE3[28].ֻLetDb=D^GCbM bGethespinorLaplacian.SuppGosethatDf/-=0 Aand1that0 t(fj@n9M ӫ)W=0:1W*emustshowthatfkk=W0.v\W*egeneralizetheproGofof "Theoremx=2.6.5tothissetting.~W*eintegratebypartsandapplytheLichnerowiczformulaUUdiscussedinx2.6toseethat$I(2.8.f)bE30=VlcZ[ yMdQ(DGfV;f)=cZUR yM5f(T*r (r2|s)fV;f)8+<$l1lwfe (֍4 GR(f;f)gL=VlcZ[ yMdQf(rfV;rf)8+<$l1lwfe (֍4 GR(f;f)g8+cZ y@n9MZ(@NfV;f):%JSinceUU(A^CbMQ;BD=)^2|sfڧ=0,wehaveA^CbMQ;BD=fڧ=0andconsequentlyʍ-@Nfj@n9MZ=Bq(fj@n9M ӫ):=SinceUU0 t(fj@n9M ӫ)=0,wehaveYLcZ_ y@n9MlY(BqfV;f)0 sorcZ$* y@n9M1(@Nf;f)0:Since~weassumedRwaspGositive,gallthetermsappGearinginequation(2.8.f)are qǍnon-negative.qThusUUwemayconcludethatfڧ=0asdesired. 5dQP .TGilk9ey,J.Leah9y,JH.P9arkp536YOn썟ZG!Îcю鍵2.9TRiemanniansubmersionsofmanifoldswithbQoundaryX2.9.1 De nition.Let'߮":Z~4!Y8.bW*esuppGoseY`ëandZhavenon-emptybGound- ary*.qWeUUsaythat.isaRiemanniansubmersioninthissettingif:Ս8(1)!W*eUUhave.isaRiemanniansubmersionontheinteriorofY8.O8(2)!W*eUUhave[ٟ^1 M@8Y=@Z. 8(3)!W*eUUhave":@8Z0!@8Y9isaRiemanniansubmersion. LethѱfFiTLgbGealocalorthonormalframe eldforthetangentbundleofYon theibGoundarysothatFm 6ԫ=9NY °istheinwardiunitnormalandsothatfFapgfor qǍ1Vam1{isaloGcalorthonormalframe eldforthetangentbundleofthe [QbGoundaryUU@8Y8.qLetL^Y Abethese}'condfundamentalformUUonY8;甍`LYab !̫=LYG(Fap;FbD):=gY(rFa FbD;Fm);~ thesecondfundamentalformonZHisde nedsimilarly*.$LetiY };andiZ ObGethe inclusionsof@8Y"and@ZinY"andZ./.W*ehaveͮڮiZ .=iY ![٫.LetͮF"2Tc^sY8.SinceUU.isaRiemanniansubmersion,o[ٟ-8int*Y(Fc)=intZ)<([ٟF)8[ٟandltĮ[ٟ-8extTY(Fc)=extcZ([ٟF)8[ٟ:!卫ThefollowingLemmasummarizessometechnicalresultsthatweshallneed.lLetUUbGetheChristo elsymbolsoftheLevi-Civitaconnection.X2.9.2TLemma.L}'et":Z~4!Y˺beaRiemanniansubmersion.X8(1)!pL^Yvab !̫=L^Zvab and^Z;Zmaio=2!ami78Lai. 8(2)!pi^bYintYt(N^bYG)Y _=(m;p)8@n9Y ؞i^bY .onC^1 0^pR(Y8)for(m;p)=1. #8(3)!pBRb=0Z(UX)BAp?Y %'=0(UX)i^bYG=0andi^bYY=0.8(4)!pWehavei^bZp[(Z[ٟ^-8[ٟ^YG)=0ifandonlyif:+8a)9pIfp=0,thenther}'eisnoconditiononG.If1p0,lNeumann 銍!pb}'oundaryconditionsarepreservedifandonlyif^Z;Zmaio=08i;a.愍Pr}'oof.pAssertion`(1)isimmediate.(SuppGosethatsatis esabsoluteboundary qǍconditions.qThenUUweargue: 獑 i^bYintYt(N^bYG)=0UUandi^bYint(N^bY)dY=0,<)UU[ٟ^i^bYintYt(N^bYG)=0,and[ٟ^i^bYint(N^bYG)dY=0,)UUi^bZp[[ٟ^eint.Y}u(N^bYG)=0,andi^bZp[[ٟ^eint.(N^bYG)dY=0,)UUi^bZint ̟Z}'(N^bZp[)[ٟ^=0andi^bZint ̟Z}'(N^bZp[)dZ[ٟ^=0,L)UU[ٟ^satis esabsolutebGoundaryconditionsonZ.7P .TGilk9ey,J.Leah9y,JH.P9arkp556Y/ɧThis40provesassertion(2).fIfweassumeZp[[ٟ^ի=[ٟ^YG,:theproGofofassertion(3)is qǍtheUUsame.qW*enotethatTD)rNmZ c®[ٟ8[ٟrNmY 4=extcZ(eiTL)int qZ ̫(fa)ZmaiW[ٟ:JIfsp=0,thisvqanishesautomatically*."Ifp>0,thisvqanishesifandonlyif^Z;Zmai vqanishesUUonthebGoundaryofZ.  W*e6concludethissubsectionbyshowingthatLemma2.8.3(2)issharp;Aifp1, qǍthenUUtheNeumannLaplacianneednotbGepositivede nite.T2.9.4Lemma.L}'et0~0.rmIncontrasttothescalarvqaluedcase,Eeigen- vqaluesBcanchange.However,>weBshallshowinTheorem3.4.1thattheycanonly Ìincrease, i.e.,if2E(;1ɍpYG)andif[ٟ^2E(;1ɍpZp[), then.,Inthenext chapterqinTheorem4.2.2,wewillshowthatthistheoremissharpbyshowingeigenvqaluescanincreaseifp 2;wedonotknowifthisresultissharpifp =1andOifthebGoundaryofMjisempty*.However,inOTheorem3.6.1,weshoweigen-vqalues7arerigidifp=17andif5=0.QhInTheorem3.4.2,weshowthat[ٟ^preservesalltheeigenfunctionsifandonlyifbGoththemeancurvqaturecovector andtheintegrabilityUUtensor!.vqanish. Inx3.5,ӈweturntothecomplexsettingandproveanalogousresultsforthe ȍcomplexLaplacian^(p;q@L)).FW*eshowinTheorem3.5.1thateigenvqaluescanonlyincrease.W*eyshallalsoshowthatifq =0,'theneigenvqaluesarerigid.InthenextchapterinTheorem4.4.1wewillshowthatthisresultissharpbyshowingeigenvqaluesmcanincreaseifq"1andphf+q2;wemdonotknowifthisresultissharp qǍif-(p;q[٫)=(0;1).7In-Theorem3.5.2,hwegivenecessaryandsucientconditionsthat allUUtheeigenformsarepreservedoncethebidegree(p;q[٫)is xed. Inx x3.6,wex turnourattentiontoexampleswhereeigenvqaluesarerigid.(W*e showinTheorem3.6.1thateigenvqaluesarerigidforSLsubmersionsandforspherebundlesHwith bGerdimensionatleast2.W*ealsoshowthateigenvqaluesarerigid for principalGbundleswhereH^1Lq(G;R)=0.In thenextchapterwewillshow ٍeigenvqaluesUUcanchangeforaprincipalGbundleifH^1Lq(G;R)6=0. qǍ In&x3.7,we&studytheanalogoustheoremsinthespinsetting./Thegeneralizationof1Theorems3.4.2and3.5.2isabitdi erent.WfW*eshowthatif^[ٲ፯ preservesallthe jeigenspinors,thentheeigenvqalueisshiftedbyaconstant.=ThishappGensifandonlyiftheintegrabilitytensor!"=0.OSinceweareworkingwithprincipalbundlesand)principalbundlemetrics,themeancurvqaturecovector)"Falwaysvqanishesbyassumption. Inbx&+3.8,%webstudymanifoldswithbGoundaryandextendTheorems3.2.1,3.2.2,3.4.1,iZandeY3.4.2tothissettingwithDirichlet,absolute,orrelativebGoundarycon- ȍditions.LTheobservqationthattheopGerators^pand^(p;q@L)arenon-negativeop-eratorsplaysacrucialroleintheproGofofTheorem3.4.1;|asnotedinx2.9,NtheNeumannYLaplacianforamanifoldwithbGoundaryisnotanon-negativeoperatorandUUeigenvqaluescandecreaseinthissetting. In^"x3.10,we^"discusstheheatcontentasymptotics.aLetMu=bGeacompactmanifoldwithPsmoGothboundary*.pUWePimposeDirichletboundaryconditions.pULettheinitial qǍtempGerature9ofMTbe1andlethM\(t)bethetotalheatenergycontent9ofMTfor pGositiveGtimet.9Ast#0,JhM\(t)GhasanasymptoticexpansionwherethecoecientsareݩloGcallycomputable. Let':NZaj!YbeݩaRiemanniansubmersion. Ifk=N0, thevolumeofthe bGersisconstantbyLemma1.17.2.gW*euseTheorem3.8.1toshowgthatif+=0,kqthentheheatcontentasymptoticsonZ!andonYarerelatedbyUUtheidentity qǍꌮhZp[(t)=hYG(t)8v[ol2`(Fc):N*Notethattheheatequationasymptoticsde nedinxch2.6andx2.8donotsatisfythisUUpropGerty*.;P .TGilk9ey,J.Leah9y,JH.P9arkp596YOn썟ZG!Îcю鍵3.2TThescalarLaplacian TheUUeigenvqaluesofthescalarLaplacianarerigid.$3.2.1&Theorem.L}'et(Y:Z!Y be(aRiemanniansubmersion.ZIf2E(;^0bYG) do}'esnotvanishidenticallyandif[ٟ^2E(;^0bZp[),then=.Pr}'oof.pLetxu0K6=2E(;^0bYG)andlet=[ٟ^.'SuppGose2E(;^0bZp[).'By pLemmaUU1.5.1, qǍxK(8)=intZ)<(G)[ٟdYG:ChoGose y0so(y0|s)ismaximal.)ThendYG(y0)i=0 so(g)(z0)i=0 where[z0C=y0|s.bvBy'breplacingbyifneedbGe,0wemayassumethemaximalvqalueof UUispGositiveandconclude=.  This}showsthatasingleeigenvqaluecannotchange.X*W*ecanalsogivenecessaryand4sucientconditionsthatalltheeigenfunctionsarepreserved.eW*erecallthede nitionUUoftheunnormalizedmeancurvqaturergiveninDe nition1.2.3:Pu5:=gZp[([eiTL;fap];ei)fa=Z 7siia fa2C1 0(H):5~3.2.2Theorem.L}'etf7:E^Zz!YкbefaRiemanniansubmersion.Thefollowingc}'onditionsareequivalent:$8(1)!pWehave^0bZp[[ٟ^ի=[ٟ^^0bYG.l{8(2)!pF;orall,wehave[ٟ^E(;^0bYG)E(;^0bZp[). Ȣ8(3)!pWehave5=0.Pr}'oof.pThe equivqalenceofassertions(1)and(2)followsfromTheorem2.3.1which ٍgivesadiscretespGectralresolutionfor^0bYG.^If4k=N0,1thenLemma1.5.1implies ^0bZp[[ٟ^n= [ٟ^^0bYG.Conversely*,ifthisidentityholds,thenintpZ;(G)[ٟ^dY = 0.Since ㍫isUUahorizontalco-vector,thisimplies5=0. ;0ZG!Îcю鍵3.3TTheBoQc9hnerLaplacian The=BoGchnerLaplacianwasintroGducedinxJ2.6.~LetrbeaRiemanniancon-nection/onacomplexvectorbundleVhmoveraRiemannianmanifoldMFwithoutbGoundary*.e(WeQ usetheconnectiononVandtheLevi-CivitaconnectiononthetangentbundleTcMŪtocovqariantlydi erentiatetensorsofalltypGes;+let`f;uv ?'de- 7notebthecompGonentsofthesecondcovqariantderivqativeofatensor eldf.TTheBoGchnerUULaplacianisde nedbytheformula:YWzdDr.:fڧ=f;ii 3?=T*r r2|sfV:<ChapterTThree:pRigidit9yofeigenv\ralues@606Y/ɧThisXqisaself-adjointellipticopGeratorofLaplacetypGeonC^1 0(V8).{F*orexample,Y8if qǍwetakeV7tobGethetrivialvectorbundleMCandifwetakertobGethetrivial connection,7the0BassoGciatedBochnerLaplacianisthescalarLaplacian^0bM\.ekInthis section,UUwegeneralizetheresultsofx3.2tothissetting. Y> Let:V:D®Zޱ!Ys:bGeaRiemanniansubmersionwith berX8andletDY &beaBoGchnerqLaplacianonavectorbundleV%UoverY8.NW*ecangivethepull3hb}'ackbundle[ٟ^VoverڮZ?thepullbackconnectionandpullbackmetric.-W*eusethesestructurestoUUde netheBoGchnerLaplacianDZp[.qPullbackinducesanaturalmap}z[ٟի:C1 0(VYG)!C1(VZp[):TheUUfollowingresultgeneralizesTheorem3.2.1tothissetting. 3.3.1"dTheorem.L}'etZ:ȮZ`!Y>beZaRiemanniansubmersion.LetDY bea Bo}'chnerLaplacianoverY˺andletDZ BbetheinducedBochnerLaplacianoverZ.$C8(1)!pWehaveDZp[[ٟ^-8[ٟ^DY _=intZ)<(G)[ٟ^rYG.Y>8(2)!pIf2E(;DYG)isnon-trivialandif[ٟ^2E(;DZp[),then=.8(3)!pThefollowingassertionsar}'eequivalent: Y>&8߫3a)9pWehaveDZp[[ٟ^ի=[ٟ^DYG.%3b)9pF;orall,wehave[ٟ^E(;DYG)E(;DZp[).&3c)9pWehave5=0.dPr}'oof.pSinceqthecalculationsareloGcal,8wemayassumeVUistrivial.SLetfFapgbGe ٍaUUloGcalorthonormalframeforTcY8.qW*eexpandrYGandr^2bYintheform:HIxαrYG:=;a R߱ 8Fca Tandr2፯Y:=;ab # 8Fcar߱ FcbgӮ:h*DecompGose^L2C^1 0(V8)intoitscompGonents;kthuswehave^L=(1|s;:::;rm).W*e qǍde neDvthevectorvqaluedderivativeofbydi erentiatingthecompGonents;this meansUUthat pFap():=(Fa(1|s);:::;Fa(rm)):}ThiscorrespGondstotakingthe atconnectionde nedlocallybytheframein [Qquestion.PLet4^Y {A4bGetheconnection1-formofrYG;?wemayexpand^Y {A:=^Y _AapFc^awhereA^Y mAaWisAanrEV(r^matrix.W*emaythenexpressthecompGonentsofrand r^2|sUUintheform: ,;a =Fap8+Y %'Aa;andq;ab e[=FbD;a R߫+Y %'Ab;a R߫+Y %'bca 핫;cVp:ThisUUthenleadstothefollowingexpressionfortheBoGchnerLaplacian: XDYG=(Fap;a R߫+8Y %'Aa;a+8Y %'aca ;cVp):=P .TGilk9ey,J.Leah9y,JH.P9arkp616Y/ɧLetӌs=[ٟ^.lNotethat^Z CA=[ٟ^^Y A.lThusrZp[=[ٟ^rYG.lSincethevertic}'al [Qc}'ovariant derivativesk;i .=0andsincebyLemma1.3.2,[^Z !abc=[ٟ^^Y abc 핫,[we \pproveUUassertion(1)bycomputing:Dn"ԮDZp[=(fap;a R߫+8Z;Aa;a+8Z;aca ;cP+8Zici y;cVp);j"ԫ(DZp[[ٟ-8[ٟDYG)=Zici y;c =intZ)<(G)(rZ)=intZ)<(G)[ٟrYG:㍑ W*eNapplyessentiallythesameargumentasthatwhichwasusedtoproveThe- qǍoremQ3.2.1inourproGofofassertion(2).Let06=2E(;DYG).SuppGoseQthat [ٟ^2E(;DZp[).qLetUU:=[ٟ^.qW*eUUusethe rstassertiontoseethatvG(8)=intZ)<(G)[ٟrYG:KȍSincerisaRiemannianconnection,%wemaytaketheinnerproGductwithtosee^F[?V(8)jj2C={1int"ٟZ4(G)[ٟ(rYG;)= K1K&fes2 ԙintbZ6()[ٟ^djj^2#qJ= z1z&fes2A(G;[ٟ^djj^2|s):XChoGoseUUy0ȫsojj^2|s(y0)UUismaximal.qChoosez0ȫso[z0C=y0|s.qThendjj^2(y0)=0UUsoڍ9(8)jj2|s(z0)=0:}ڍSinceUUjj^2|s(z0)=jj^2|s(y0)6=0,weconclude=. TheLequivqalenceofassertions(3-a)and(3-b)followsfromthediscretespGectralresolutionforDYG;ʝthefactthatassertion(3-c)impliesassertion(3-a)followsfromassertion(1)./SuppGosethatassertion(3-a)holdssoDZp[[ٟ^=ٮ[ٟ^DYG.Thenassertion(1)showsthatint}Z(G)[ٟ^rY _=0..Sinceܫisahorizontaldi erentialform, qǍthisUUequalityontheopGeratorlevelimpliesthat5=0. 8bZG!Îcю鍵3.4TTheformv\raluedLaplacian W*e?shallshowinTheorem4.2.2thateigenvqaluescanchangefortheformvqaluedLaplacian.qHowever,UUTheorem3.2.1doGeshaveUUatleastapartialgeneralization:73.4.1&Theorem.L}'et(Y:Z!Y be(aRiemanniansubmersion.ZIf2E(;1ɍpYG) isnon-trivialandif[ٟ^2E(;1ɍpZp[),then.ɍPr}'oof.pW*efusethe* bGerproductde nedinx/1.8.\wLet":Z~4!YNJbefaRiemannian Ìsubmersion.ЂLet>06=2E(;1ɍpYG)andlet[ٟ^2E(Ы+;1ɍpZp[).ЂLetZ(0)=Z andUUinductivelylet qǍm Z(n):=Wc(Z(n81);Z(n1))>@ChapterTThree:pRigidit9yofeigenv\ralues@626Y/ɧbGeIthe berproductofZ(n"1)Iwithitself.mLetn8:Z(n)!YЫbGeItheassociated UTpro8jection.qByUULemma1.8.3,e_y|[ٲ፯nq~2E(8+2n;1ɍp6Z}(n)۫):ISincetheLaplacianonZ(n)isanon-negativeopGerator,׮~{+2^nq~u0.Sincethis qǍholdsUUforalln,0UUasdesired. @h W*e|canalsogivenecessaryandsucientconditionsthatalltheeigenforms arepreserved.QSinceeigenvqaluescanchange, thestatementisjustabitmorecomplicatedUUthaninthecasep=0.3.4.2ATheorem.L}'et :2ZON!Y͈beaRiemanniansubmersion.Fixanindex1pdimq(Y8).Thefollowingc}'onditionsareequivalent:-8(1)!pWehave1ɍpZp[[ٟ^ի=[ٟ^1ɍpYG.bP8(2)!pF;orall,wehave[ٟ^E(;1ɍpYG)E(;1ɍpZp[).8(3)!pF;orall,ther}'eexists=()so[ٟ^E(;1ɍpYG)E(;1ɍpZp[).8(4)!pWehave5=0and!"=0.3.4.3Remark: DNote:thatifcondition(2)holdsforanypwith00=Mܮ()(18HY)[ٟ#Eʾ=M(18HY)(dZp[(int ɟZb$(G)+E)+(int ɟZ(G)+E)dZp[)[ٟ:)SinceUUthespanoftheeigenspacesE(;1ɍpYG)isdenseinC^1 0^pR(Y8),wehave*(3.4.a)/C(18HY)(dZp[(int ɟZb$(G)+E)+(int ɟZ(G)+E)dZp[)[ٟի=0UUon8C1 0pR(Y8):ӍFixapGointz0ۍ2_Zgandlety0=_[z0|s.fChoGoseF©2C^1 0YnsothatFc(y0|s)=0.fLet r۫:= dFc(y0|s).Since~intpZݫ(G)Tx+Ecis~a0^th forderoperator,weapplyequation(3.4.a) qǍtoUUFcandevqaluateatz0ȫtoseethatdnW~0=g (18HY)fext tZ ϫ([ٟuǫ)(int ɟZb$(G)+E)#iE+8(int ɟZb$(G)+E)extGZw([ٟuǫ)g[ٟ((y0|s)):?ٟP .TGilk9ey,J.Leah9y,JH.P9arkp636Y/ɧNoteUUthatd70=(18HY)fext tZ ϫ([ٟuǫ)int qZ ̫(G)+int*Z()extGZw([ٟuǫ)g[ٟ:+SinceUUE:MalwaysintroGducesaverticalcovector,weconcluded0=fext tZ ϫ([ٟuǫ)Eث+8Eext,Zo([ٟ)g[ٟ:T*oUUsimplifythenotation,wetempGorarilyde ne>*Ea1:=extcZ(fa);Ei:=extcZ(eiTL);andqǮIafj:=intZ)<(fa):FixUUcandchoGosesothat[ٟ^<߫=f^c&p.qW*ecompute:(p0=7Ѯ!abi /fEcnEiٮIaRIb+8EiIaIb&Ecng=!abi /EiٱfEcIaRIb+8IaIb&Ecg~07=7Ѯ!abi /EiٱfIaREcnIb+8IaIb&EcNaciQIbg07=7Ѯ!abi /EiٱfIaRIb&EcN+8IaIbEcN8aciQIb+bc%IaRg=2!cbi kqEiٮIb:+Since:p@1, thisimplies!=0andhencethehorizontaldistributionHisinte- grable.3 W*e|nowapplytheresultsofx1.7. LetdX denoteexteriordi erentiationalong qǍtheUU bGer.qW*esetE=0anduseequation(3.4.a)toseedl0=dX D̫int6Z(G)[ٟJon-C1 0pR(Y8):This9impliesthatthemeancurvqaturecovector9>Visconstantonthe bGerssowe may_expressE=([ٟ^asthepullbackofagloballyde ned1-formonthebase.SincethehorizontaldistributionH5ϫisintegrable,O weuseLemma1.7.2togivealoGcalUUdecompositionofZ qsothatwehave+~`y5=[ٟ=dYlnF(gX$):LetUU [٫(y)bGethevolumeofthe bers.qLetd^ex^9betheEuclideanmeasure.qThen0ybd0dYG [٫(y)=[dYcZ%) yXigX$(x;y[٫)dex=cZUR yXv(gXg䍘17sX MdYGgX)(x;y)dexS=]8cZ yX gX$(x;y[٫)G(x;y)dex=(y)cZ8 yX}gX$(x;y)dexS=]8(y[٫) (y):0B@Thus qthegloballyde nedfunction [٫,kwhichgivesthevolumeofthe bGers,kde nesG,UUi.e.qwehave Tͮ5=[ٟdYln [:@|ChapterTThree:pRigidit9yofeigenv\ralues@646Y/ɧ W*e*de neaconformalvqariationofthemetricontheverticaldistributionand leaveUUthemetriconthehorizontaldistributionunchanged:Š]fg[٫(t)Z 7s= 2tݢds2፲VX+8ds2፲HY:,ꍫThen]0:ʮZ(t)!YqisaRiemanniansubmersionwithintegrablehorizontaldistri- qǍbution.W*eUUuseLemma1.7.2toseethat,ꍑ63G(t)=(18+tdimUV(X)):ConsequentlyUUwehaves*+-L1ɍp6Z}(t)[ٟ-8[ٟ1ɍpY _=(18+tdimUV(X))(dZint ̟Z}'(G)+int*Z()dZp[)[ٟ񍍍x鐫=(18+tdimUV(X))(1ɍpZp[[ٟ-[ٟ1ɍpYG):s)ThisUUshowsthatGĮ[ٟE(;1ɍpYG)E(8+(1+tdimUV(X))();1ɍp6Z}(t)):FSinceUUtheLaplacianisanon-negativeopGerator,wehaveu8+(1+tdimUV(X))()0:,ꍫSinceUUthisidentityUUholdsforarbitraryt,()=0.qThisUUshowsthatlU(dZint ̟Z}'(G)8+int*Z()dZp[)[ٟի=0:ThisUUpGermitsustoconcludethat5=0. 3.4.4Remark.Thisisexactlywheretheargumentfailswhenweconsiderthe NeumannbyLaplacianforformsofdegreeatmost1.2ThisopGeratorneednotbenon-negativeandeigenvqaluescandecrease.سW*erefertoLemma2.9.4forfurtherdetails.6GtZG!Îcю鍵3.5TThecomplexLaplacian Inthissection, weextendTheorems3.2.1and3.3.1totheholomorphicsetting.W*erefertoxsA.5foradditionalinformation.vInx2.7.1,,wede nedthenotionofaDHermitianosubmersion.>ThismeantthatisaRiemanniansubmersionfromZto6Y8,oJthatZandYoʫarecomplexmanifolds,oJthatiscomplexanalytic,andthatthemetricsonZ8andonYareHermitian.W*erefertoworkofJohnson[105]andW*atson[194]foradiscussionofsomeofthegeometrywhichisinvolved;AP .TGilk9ey,J.Leah9y,JH.P9arkp656Y/ɧthese0authorsalsoconsiderthealmostcomplexandtheKaehlercategories.ZXW*e complexifyUU[ٟ^Jtode neImi[ٟի:C1 0p;q (Y8)!C1p;q (Z):W*eUUthenhaveUUtherelationsPeq[ٟ1ɍ[ٯp;qY ث=1ɍ[ٯp;qZ E[ٟJandܮ[ٟ\q8@Y9=\q @Z[ٟ:2W*e)extendinteriormultiplication,^exteriormultiplication,^and!tobGecomplexlinear.LetJbGethealmostcomplexstructure.W*ethenhaveJ9H7cH.If!厫isa2UUform,wede ne: qǍwiJ9![٫(1|s;2):=!(J91|s;J2):\3.5.1TTheorem.L}'et":Z~4!Y˺beaHermitiansubmersion.r8(1)!pIf2E(;1ɍp;qY )isnon-trivialandif[ٟ^2E(;1ɍp;qZ ),then.8(2)!pIf2E(;1ɍp;0Y ū)isnon-trivialandif[ٟ^2E(;1ɍp;0Z ū),then=.RIPr}'oof.pTheproGofgivenofTheorem3.4.1extendswithoutchangetothecomplex qǍsetting3'toestablishthe rstassertion.fbTheproGofofassertion(2)isquitedi erentinUUthecomplexcasehowever.qLetXMV06=2E(;1ɍp;0Y ū)UUand[ٟ2E(8+";1ɍp;0Z ū):ꍫSinceUU^p;0 ݫ=\q @8^\qR =Q@,weuseLemma2.7.2toseethat"[ٟ=1ɍ[ٯp;0Z W[ٟ\q8@Y0!:2SinceUUE:Mhasanon-trivialverticalcompGonent,J^0=1ɍ[ٯp;0Z WE[ٟ\q8@Y0!UUsor"[ٟ=1ɍ[ٯp;0Z FintZdj(G)[ٟ\q8@Y0!:W*eUUapply9toseethat B֍Ϯ"=1ɍ[ٯp;0Y FintYV()\qD@Y ;d:W*eͅde neavqariationofthemetricwhichleavesthemetriconthehorizontal distributione|unchangedandisaconformaldeformationontheverticaldistribution:"g[٫(t)Z 7s:=V82t ds2፲VX+8ds2፲HPThen_4:ϮZ(t)!YګisaHermitiansubmersionandEDtransformsconformally*.We useUULemma1.7.2toseeG(t)=(18+tdimUV(X))rand!`[ٟ2E(8+(1+tdimUV(X))";1ɍp;06Z}(t)):PThusbyassertion(1),}gF+(1+tdimUV(X))"0forallt2R.NThisshows"=0: g9 W*etNcanalsogeneralizeTheorem3.4.2tothissetting.&TheproGofisquitedi erent andUUtherearemorecasestobGeconsidered.BChapterTThree:pRigidit9yofeigenv\ralues@666Y/ɧ3.5.2Theorem.Fix(p;q[٫)with0p;q"dimqƟCY8.>Thefollowingc}'onditionsare qǍe}'quivalent:⍍8(1)!p1ɍp;qZ [ٟ^ի=[ٟ^1ɍp;qY.CǍ8(2)!p[ٟ^E(;1ɍp;qY )E(;1ɍp;qZ ):8(3)!pF;orall,ther}'eexists=()so[ٟ^E(;1ɍp;qY )E(;1ɍp;qZ ):8(4)!pThe b}'ersofareminimaland: "㊫4-a)9pifp=0andifq"=0,ther}'eisnoconditionon![ٺ."UQ4-b)9pifp>0andifq"=0,thenJ9^!"=![ٺ.#qī4-c)9pifp=0andifq">0,thenJ9^!"=!i.e.H1;0lͺisinte}'grable. "UQ4-d)9pifp>0andifq">0,then!"=0i.e.Hisinte}'grable.xr3.5.33DRemark.Inxi4.5,ѲwewillprovideexamplesofRiemanniansubmersions qǍwhereڮ!"6=0andJ9^!=![٫.^Thustheconditionsinassertion(4)aredistinctand non-vqacuous. TherremainderofthissectionisdevotedtotheproGofofTheorem3.5.2."Thereare`anumbGer`oftechnicalLemmastobeestablished.gLet9beaHermitiansub-mersion.xThencthehorizontalandverticaldistributionsH|FandV6 areinvqariantunderUUthealmostcomplexstructureJ9.qThecanonicaldecompGositionof{ QTcZ 8C=TZ1;0 Ʊ8TZ0;1S7thereforeUUinducesadecompGositionL HQ 8C=H1;0 ƱH0;1.;and!V ] C=V1;0 ƱV0;1 :ChoGose:3alocalorthonormalframe eldforthehorizontaldistributionHRofthe formVff1|s;:::;fɮ;J9f1;:::;J9fɱgVwherel2=0;andifq">0,weassume!=0andE=0. If?p>0andifq"=0,wweneedonlyconsidertheactionof1ɍ[ٯp;0Z WE[ٟ^on^p;1 ū(Y8).2Thus pWonlyLthetermsin(3.5.c)and(3.5.d)abGoveLarerelevqantandthesevanishsinceweassumedZxJ9^!+=ϧ![٫./Ifp=0andq+>0,[thenweneedonlyconsidertheactionof B֍1ɍ[٬0;qZ "E[ٟ^onZ^0;q@L+1 (Y8).Thusonlythetermsin(3.5.a)arerelevqant.ThesevqanishsinceweassumedJ9^!k=![٫.=Thisshowsassertion(4)impliesassertion(1).=Itis immediate٬thatassertion(1)impliesassertion(2)andthatassertion(2)impliesassertion;9(3).iTheremainderofthissectionisdevotedtotheproGofthatassertion(3)UUimpliesassertion(4);Theorem3.5.1willplayacrucialroleintheproGof. qǍ W*eUUbGeginwithatechnicalLemmainthetheoryofPDE's.D6(ChapterTThree:pRigidit9yofeigenv\ralues@686Y/ɧ3.5.5TLemma. OU8(1)!pL}'etJRbeanylinearoperatoronC^1 0^p;q (Y8)sothat(Rǫ;)L2 ,=0forall !pinC^1 0^p;q (Y8).ThenR߫=0. r8(2)!pL}'et P!bea1^th ȸorderpartialdi erentialoperatoronC^1 0^p;q (Y8).Suppose!pthatٮP'hisnon-ne}'gative,i.e.)m(Pc;)L2 0forall2C^1 0^p;q (Y8).)mThen!pPvisa0^th or}'deroperator,i.e.if(y0|s)=0,thenPc(y0)=0.=UPr}'oof.pLet5"bGearealparameter.g4Since(Rǫ(1u+q"2|s);1+"2|s)=05forall",;we qǍhave uڤ(Rǫ1|s;2)8+(R2|s;1)=0:ύW*eUUreplace"by$p $fe 1qˮ"toseethat_Ǎuڤ(Rǫ1|s;2)8(R2|s;1)=0:ThisUUshowsthat(Rǫ1|s;2)=0UUforalliTL;wetake2C=Rǫ1ȫtoseeR߫=0.  W*evusethemethoGdofstationaryphasetoprovevthesecondassertion.Decom- pGoseUUtheoperatorPintheformVP*=apPca9@8ya+8Q:W*eFmustshowthatPc^a R=GS0foralla.XLet 2C^1 0(Y8)andlet0Ʊ2C^1 0^p;q (Y8). qǍW*eUUde ne (NQ$(t):=esIJpɟsĉW ::1NoteUUthat OaT[ٟPcR="()>::&Thus2fE(;1ɍp;qY )gareeigenspacesofPc.`^Sincetheseeigenspacesareorthogonal andtheeigenvqaluesarereal,8weseethatPfisself-adjoint.NByTheorem3.5.1, qǍ"()Ԯ0]|soP isanon-negative rstorderself-adjointdi erentialopGerator.;Thus PhasUUorder0.qIf2C^1 0^p;q (Y8),UUwemayexpand&mgx=>:forR2E(;1ɍp;qY ):1W*e5useTheorem2.3.1toseethatthisseriesconverges5intheC^1 etopGology.g;ThenPc=>:"()so rlQ(1ɍp;qZ [ٟ-8[ٟ1ɍp;qY)=1ɍ[ٯp;qZ E(\qD@Z x+\qD@Z)[ٟ=>:[ٟ"()R=[ٟPc():*SincekP0isa0^th vNorderopGerator,3Pc(F)=FcP()kforanyFx2C^1 0(Y8)sothederivqativesUUofFdonotenterintothisequation.qThisimplies9(3.5.e)I$1ɍ[ٯp;qZ E(ext tZ ϫ([ٟ\q8@Y0!Fc)8+extGZw([ٟ\q8@YFc))[ٟ=0:RecallJthat=intZ)<(G)"t+E/whereJint;ZC()JdoesnotinvolveJanyverticalcovectors andNwhereE3ǫdoGesinvolveNverticalcovectors.oThusequation(3.5.e)decouplesintotwooseparateequationsinvolvingointa[ZѶ(G)oandETseparately*.~IfYisahorizontal covectorgatz0|s,wecanchoGoseFso[ٟ^dYGFc(z0|s)@=uǫ.LTheng[ٟ^\q8@Y0!Fc(z0|s)=uǟ^0;1and qǍtheUULemmafollows. OaPr}'oof^ofThe}'orem^3.5.2.pW*eImustshowassertion(3)impliesassertion(4).mRecallthatJH3intT%vZZѫ(uǟ1:)int qZ ̫(uǟ2)8+int*Z(uǟ2:)int qZ(uǟ1)=0;$H3extU!Z\@|(uǟ1:)extGZw(uǟ2)8+extTZE(uǟ2:)extGZ(uǟ1)=0;andH3intT%vZZѫ(uǟ1:)extGZw(uǟ2)8+extTZE(uǟ2:)int qZ ̫(uǟ1)=g[٫(uǟ1:;uǟ2):"Recallwlthat~gZ$=istheextensionofgZ ǫtobGecomplexbilinear. Supposetheas- sumptionsofTheorem3.5.2(3)hold. 3 T*osimplifynotation,Plet/=kVuǟ^0;1 N,letGfWP .TGilk9ey,J.Leah9y,JH.P9arkp716Y/ɧE^:=q\ext ПZ~+([٫), let~I^u:=q\intc%ZӀ(G),letE^iY5:=q\ext ПZ~+(e^iTL),andletI^a :=q\intc%ZӀ(f^a).CBy UTLemmaUU3.5.7, 9^<0=1ɍ[ٯp;qZ E(ErI+8IOE)=1ɍ[ٯp;qZ N~gZ{%(G;[٫);]ThisUUimplies5=0sincerishorizontal.qW*ealsocompute:-w(3.5.f)э80=G !abi /1ɍ[ٯp;qZ EbErEiٮIaRIb+8EiIaIb&Erb G[ٟ~?=G !abi /1ɍ[ٯp;qZ EbEiٮErIaRIb+8EiIaIb&Erb G[ٟ?=G !abi /1ɍ[ٯp;qZ EbEiٮIaRErIb+8EiIaIb&E}n~8gZ nE([;fa)EiIbb }[ٟ?=G !abi /1ɍ[ٯp;qZ EbW~gZ 5e([;fa)EiٮIb+n~8gZ nE(;fbӫ)EiٮIaRb t/on^p;q (Y8).W*etakeʱ=\qRnخ^ >.Thedualof\q^ CwithrespGectto~gZ]is .Thusby equationUU(3.5.f),ā(լ0=extcZ([ٟ0;1 4![٫( ; d))int qZ ̫( !ɫ)8+extTZE(1;0 4!( ;\qq o))int qZ ̫(\q ʫ)onUU^p;q (H^)forall ^ϫand .qTheseequationsdecoupleandwehave:j0=|1wextZ>F([ٟ0;1 4![٫( ; d))int qZ ̫( !ɫ)(3.5.g)j0=|1wextZ>F([ٟ1;0 4![٫( ;\qq o))int qZ ̫(\q ʫ):(3.5.h)If-pԫ=0,Lthenwecandrawnoconclusionfromequation(3.5.h);~ifql=0,Lthen weWcandrawnoconclusionfromequation(3.5.g). Ifp>0,WthenWequation(3.5.h) showsd[ٟ^1;0 4![٫( ;\qq o)=0forall nand ;lFbyLemma3.5.4,hsthisimpliesJ9^!)0, thenequation(3.5.g)shows[ٟ^0;1 4![٫( ; d)=0andhence![٫( ; d)=0 7forall ہand ;`byLemma3.5.4,4thisimpliesJ9^!ī=![٫.Ifp>0andq>0,4we qǍcombineGthesetwoidentitiestosee! -=T0.9Thisshowstheconditionsof(4)aresatis ed. e Inñx!4.5,^wewillgiveexampleswhere!6=]0andJ9^!=]α![٫.W*epGostponethe discussionUUuntilthatpGointtoavoidinterruptingthe owofourdiscussion.7aKZG!Îcю鍵3.6TOthersettingswhereeigen9v\raluesarerigid Generically*,thepullbackofaneigenformwillnotbGeaneigenform;xitisquite ÌasspGecialsituationwhen0E6=2E(;1ɍpYG)sand[ٟ^E2E(;1ɍpZp[).V W*essayeigen- valuesochangeMif6=.ӰTheoremM3.2.1showsthateigenvqaluescannotchangeifpJ=0._F*urthermore,9ifrp>0andifalltheeigenformsarepreserved,9theneigen-vqaluescannotchange.fThereareothercircumstancesunderwhichevenasingle o9eigenvqalue(wcannotchange;7ltheeigenvqaluesarerigid.bLetH^k(M;R)denotethede qǍRhamUUcohomologygroupsofamanifoldM.HvChapterTThree:pRigidit9yofeigenv\ralues@726Y/ɧ3.6.1S@Theorem.L}'et+خ":Z~4!Ydbe+aRiemanniansubmersion.vWesupposegiven Ì06=2E(;1ɍpYG)sothat[ٟ^2E(;1ɍpZp[).'<8(1)!pIf":Z~4!Y˺is atwithstructur}'egroupSL,then= ٍ8(2)!pIf":Z~4!Y˺isaprincip}'alGbundlewithH^1Lq(G;R)=0,then=. 8(3)!pIf":Z~4!Y˺isaspher}'ebundleof berdimensionr52,then=.8(4)!pIfthe b}'ersofareminimalandifp=1,then=.u3.6.2z}FlatRiemanniansubmersionswithstructuregroupSL.Let@̫bGe aoRiemanniansubmersionfromZ&toYwithintegrablehorizontaldistribution.Assume4_thereexistsameasureUonthe bGerssotheLiederivqativeLx:H3x:F ڮȫ=:0 UwherefH `Fisfthehorizontalliftofavector eldonY8.2Expanddv[olZ 7s=e^ ^dvolY qǍtode neasmoGothfunction В2C^1 0(Z)._=W*eapplyLemma1.7.2tochoosealocaldecompGositionUUofZ qsothat]CUbrds2፯Z 7s=gij (x;y[٫)dxi,8dxjo+habZ(y)dyak)dyb`:+{Expand L=hVe^ \(x;y@L)S^exwhere^exisEuclideanmeasure.SinceLw@ߍn9ya  L=hV0andsince `Lw@ߍn9ya ^ex=0,UU qisindepGendentofy[٫.qW*eexpandeXsdv[olZ 7s=gX$exdvolY _=e \+ XexdvolYG:d\ThisshowsgX a<=e^ \+ Bso5=dYlnF(gX$)=dYG( '+$ z)=dY( z).CSuppGosethatύQ06=2E(;1ɍpYG)UUand[ٟ2E(8+;1ɍpZp[):P(AsUUintheproGofofTheorem3.4.2,weconsiderthec}'anonicalvariation~ds2ٔZ}(t)|˫=e2t ds2፲VX+8ds2፲H 7toUUsee ca[ٟ2E(8+(1+tdimX);1ɍp6Z}(t)):aThisUUshows=0UUandcompletestheproGofofTheorem3.6.1(1). ߐ3.6.3=PrincipalGbundles.Let{GbGeacompactLiegroupwithabi-invqariant metric. AssumeîH^1Lq(G;R)[=0.Letî:PZ!YbGeaprincipalRiemannian GJbundle.OLetԽ2^TeKK(G)andletg[٫(t)bGethe1-parametersubgroupofGwitht_g (0)S=uǫ.oMultiplicationbyg[٫(t)de nesa owonP Mwhichisanisometry*.oLet qǍbGe theassociatedKillingve}'ctor eld;,Ыhas constantlengthsincethe bershave thebi-invqariantmetric.Consequentlytheintegralcurvesof,aaregeoGdesics;E[ٟ=:jAj=p2!mext/ P5( A)fA:IsP .TGilk9ey,J.Leah9y,JH.P9arkp736Y/ɧTheUU A areverticalco-vectorswhichareGinvqariant.qSincedYG=0UUand5=0,#Y*(8)[ٟ=1ɍpP[ٟ-8[ٟ1ɍpY _=dPE[ٟhasnoverticaldepGendence.Thusthevertic}'alWderivativeoftherestrictionof A tonthe bGersvqanishes.SinceH^1Lq(G;R)=0nandsincethe A XareGinvariant,u$this qǍimpliesYtherestrictionofthe A itothe bGersvqanishesandhence A onthemetric;wedonotassumethesubmersionisRiemannianinthe followingUUtheorem.ua3.6.7Theorem.L}'et":Z~4!Ylbeasubmersion.QwSupposethat06=2E(0;1ɍpYG) isnon-trivial.Supp}'osethat[ٟ^2E(;1ɍpZp[).Then"-8(1)!pIfp=1,then=0. 덍8(2)!pIfthe b}'erofisanevendimensionalsphere,then=0.Pr}'oof.pW*ePusemethoGdsofalgebraictopologytoprovePTheorem3.6.7;R`werefertoSpanierMn[178]fordetailsconcerningtheresultswhichwewilluse.ZLetMdbGea connected9manifold.tTheAbGelianizationofthefundamentalgroup1|s(M)is rstintegerUUhomologygroupH1|s(M;Z).qBytheuniversalc}'oecienttheorem,i(3.7.b)@0=<$I+;1I+;wfe (֍2Q cZVQ yY_.@!abi /(cYG(Fca9)cY(Fcbgӫ);AC፯Y)dycZ yGN(cgy(g yi]ū) z; )dgJ0+<$l1lwfe (֍2 IcZ) yYo!abi /(cYG(Fca9)cY(Fcbgӫ);)dycZ yGN(cgy(g yi]ū) z;AC፯Gڮ )dg[:)+SinceUUcgy(g y^i]ū)^_=cg(g y^i),E(cgy(g yi]ū) z; )=( z;cgy(g yi]ū) )=(cg(g yi]ū) z; )=0:=NSinceUU(cYG(Fc^a9)cY(Fc^bgӫ))^_=cY(Fc^a)cY(Fc^bgӫ),UUthesecondtermalsovqanishes. k W*eHconcludethissectionbygeneralizingTheorem3.4.2.Weusecomplexspinors toUUde nethepullback;Lemma2.7.3extendstothissettingwithoutchange.3.7.22Theorem.L}'etV":P*!YJ:beVaprincipalbundle.nLet В2CW^c8g andlet2R.Thefollowingc}'onditionsareequivalent:a8(1)!pD^GCYrcbP 7?^[ٲ፯ y=^[ٲ፯ (D^GCYrcbY p+8)._8(2)!p80,9()0so[ٟ^E(;D^GCYrcbY 7?)E(();D^GCYrcbP 7?):8(3)!pThehorizontaldistributionde ne}'dbyisintegrableand В2E(;D^GCYrcbG 7?).MljP .TGilk9ey,J.Leah9y,JH.P9arkp776Y/ɧPr}'oof.pTheNimplication(3))(1)followsfromLemma2.7.3andthecommutation 銍relationXD^GCbY 3ҫ+GD^GCbYGkī=0. 3Assertion(1))(2)isimmediate.SuppGosethat Aassertion(2)holds.KAsweareworkingwiththecomplexi cationofrealopGerators, qǍweUUhaveM=ni0E(;DGCYrc፯Y 7?)=E(;DGC፯YG)8 RCUUandqʮ;΍i0E(;DGCYrc፯G 7?)=E(;DGC፯Gګ)8 RC:Thus wecanrestricttoreal.DecompGose ֫=\ 1/ի+b$p $fe 1خ 2wherethe iadarereal.~WBytakingrealandimaginaryparts,:E(;MQ;BD=):{F*orcexampleasnotedinxګ2.8.5,fifM=[0;[٫]thenthediscretespGectralresolution qǍofUUtheDirichletandNeumannLaplacianstaketheform\Zfcos c(nx);n2|sgn0Gand&ufsin G(nx);n2gn>0:L-3.8.1eTheorem.L}'et:6Z8R!YٺbeaRiemanniansubmersion.uLetBкdenote DirichletorNeumannb}'oundaryconditions.8(1)!pIf2E(;^0bYn9;B s)andif[ٟ^2E(;^0bZI;B <),then=.X8(2)!pThefollowingc}'onditionsareequivalent: ٍ&8߫2a)9pF;orevery,wehave[ٟ^E(;^0bYn9;B s)E(;^0bZI;B <). %2b)9pWehave5=0. Pr}'oof.pTheMwproGofisessentiallythesameasthatgiveninx@3.2withappropriatemoGdi cations8totakeintoaccountthepresenceofthebGoundary*.Suppose8that 0!>6=2E(;^0bYn9;B s)landthat[ٟ^!>2E(\+;^0bZI;B <). W*eluseLemma1.5.1tosee Tthat (3.8.a)Vk[ٟ=0፯Zp[[ٟ8[ٟ0፯Y _=intZ)<(G)dZp[[ٟ:ҍByreplacingbyifnecessary*,wecanassumethemaximalvqalueofispGositive;.letthismaximalvqaluebeattainedaty02lzY8.yIfy05isintheinteriorof qǍY8,UUthendYG(y0|s)=0.qChoGoseUUz0ȫso[٫(z0)=y0.qThen s1 dZp[[ٟ(z0|s)=[ٟdYG(y0|s)=0soa equation(3.8.a)implies[ٟ^(z0|s)ڟ=0a andhenceڟ=0.Ifa B(z=BD@,cthencannotZattainitsmaximumonthebGoundary@8Y>andthe rstassertionfollows.@LetOꔟP .TGilk9ey,J.Leah9y,JH.P9arkp796Y/ɧB(=:MBN.A'Sinceuy02@8Y8,d@n9Y (y0|s)=0.SinceuNYG()(y0|s)=0,dY(y0|s)=0uand qǍ=0;UUthiscompletestheproGofofthe rstassertion. CD ThebGoundaryconditionsarealwayspreservedonfunctionsbypullback.W*euseLemma1.5.1toseeassertion(2b)impliesassertion(2a).qQConverselyassume ٍthat?assertion(2a)holds.0Let0M6=2E(;^0bYn9;B s)?andletM:=[ٟ^.W*e?use TLemmaUU1.5.btoseethatˍx(8)=intZ)<(G)dZp[[ٟ:ˍBytassertion(1), {=tandthusint=Z(G)[ٟ^dYG=0.#Lett 2C^1l0 0(Y8)bGeasmoGoth9functiononYwithcompactsupport.6ByTheorem2.3.1,ǥwecanuniformly approximate intheC^1 topGologyby nitesumsofeigenfunctions.;SThuswehave;int‹Z2(G)[ٟdYG =0onuC^1l0 0(Y8).`Sinceisahorizontalco-vector,}thisimpliesC߫=0ontheinteriorofM.qContinuityUUthenyields5=0onthebGoundaryaswell. ̍ Theorems3.4.1and3.4.2extendtothecategoryofmanifoldswithbGoundary ifDweimpGoseDirichlet,GAbsolute,orDRelativebGoundaryconditionsonthespaceofsmoGothpforms;Twerefertox s2.9forthede nitionoftheseboundaryconditions.W*ehavealreadynotedinx+2.9.4thatTheorem3.4.1failswithNeumannbGound-ary2conditions;l!wedonotknowifTheorem3.4.2failswithNeumannbGoundaryconditions.3.8.2eTheorem.L}'et:6Z8R!YٺbeaRiemanniansubmersion.uLetBкdenoteDirichlet,Absolute,orR}'elativeboundaryconditions.Letp>0.J8(1)!pIf2E(;1ɍpYn9;B s)isnon-trivialandif[ٟ^2E(;1ɍpZI;B <),then. 8(2)!pThefollowingc}'onditionsareequivalent: &8߫2a)9pF;orall,wehave[ٟ^E(;1ɍpYn9;B s)E(;1ɍpZI;B <). aэ%2b)9pF;orall,ther}'eexists=()so[ٟ^E(;1ɍpYn9;B s)E(;1ɍpZI;B <)  &2c)9pWehave5=0and!"=0.~׍ T*oprovethe rstassertion,weusethesameargumentemploying bGerproductsTasusedintheproGofofTheorem3.4.1.o[IfBC=phBD }orB=phBA,RthebGoundaryconditions-arepreservedautomaticallyandthereisnothingnew.dIfB=BRb,5thenbyUUassumptionBRb=0UUandBR[ٟ^=0.qThus;Ti፯Zp[(Z[ٟ-8[ٟYG)=0ˍsovwehavei^bZint ̟Z}'(G)[ٟ^SN=0vandi^bZp[E[ٟ^SN=0.n)Lemmav1.8.3thenshowsinduc- tivelyUUthat ;R1TiٔZ}(n)Lint>L:Z}(n)0'(G)[ٲ፯nq~=0UUandiٔZ}(n)۱E[ٲ፯n=0:PhChapterTThree:pRigidit9yofeigenv\ralues@806Y/ɧItfollowsthatBRb^[ٲ፯nq~=0so^[ٲ፯nsatis esthegivenbGoundaryconditionsandthe argumentUUgoGesthrough.qThisprovesassertion(1). ItW4isimmediatethatassertion(2a)impliesassertion(2b).SuppGoseassertion(2c) qǍholds.SMDirichletandAbsolutebGoundaryconditionsarepreservedautomatically; RelativeibGoundaryconditionsarepreservedsince5=0and!"=0.M#Thusassertion(2a)Lholds.PSuppGosethatassertion(2b)holds.TheproGofthatE=0isexactlythesameasbGeforeandusesthefactthattheeigenspacesoftheLaplacianaredenseC^1l0 0^pR(M).qInUUtheproGofthat5=0,weusedavqariationinwhich63G(t)=(18+tdimUV(X)):a[DirichletandabsolutebGoundaryconditionsarepreservedautomatically;relativebGoundary*conditionsarealsopreservedowingtotheparticularformofthisvqaria-tion.\W*esuppGoseBR =0.Sincei^bZp[[ٟ^ի=[ٟ^i^bYG,"Ywehavei^bZp[[ٟ^=0.\SinceE=0, seeUUthat:/QBR=A[ٟ=0ㅱ(UX)i፯Zp[Z[ٟ=0ㅱ(UX)i፯Zint ̟Z}'(G)[ٟ=0:SinceG(t)=(1^a+tdimUV(X)),thisconditionispreserved.M]Theremainderofthear-gumentisunchanged.:KNotethatthislineofargumentfailsforNeumannbGoundaryconditionsUUsincetheLaplaciancanbGenegativeaswasshowninLemma2.9.4.8ZG!Îcю鍵3.9TTheLaplacianwithcoQecien9tsina atbundle1.3.9.1RDe nition.Letx䍑|l~MN*bGetheuniversalcoverofM. Thefundamentalgroup &1|s(M)actsonx䍑T~Mpbydecktransformations<~x !ië~x @2gfor<~x2x䍑ī~M+andg"21|s(M).3LLet ̫bGeaunitaryrepresentationofthefundamentalgroup1|s(M)ofaRiemannianmanifoldYM;-[forsomek5wehave:1|s(M)!U(kP)Yor:1|s(M)!OG(kP):YThe qǍassoGciatedUU atvectorbundleVyisgivenby:؍gVm<:=x䍑~MʢRk@orV:=x䍑~MCkˍwhereUUweidentifya[\(~x;G@~ve)(~x8g[;(g)~v4)UUforall g"21|s(M):hLet11^kbGethetrivialrealorcomplexvectorbundleof berdimensionkȫoverx䍑^~1Mn۫. CLet\q|v~UUf S2C^1 0(1^k됫);\q|v~UUf isUUasmoGothmapfromx䍑~MuTtoC^k.qW*esay\q|v~f ise}'quivariantifۍ\qXG~V f\(~x;G@~ve)=\q9~f(~x8g[;(g)~v4)UUforall g"21|s(M):QRP .TGilk9ey,J.Leah9y,JH.P9arkp816Y/ɧClearlysuchafunction\q~fpreservestheglueingsgivenabGoveanddescendsto ayOsectionfF2C^1 0V$.ݶConversely*,NeveryyOsmoGothsectionfޫtoC^1V sde nesan YequivqariantUUfunction.qDe nethe atconnectionon1^k@by:1x䍒~ir\q~]f=d\q'!~f:Itisthenimmediatethatx䍑4~ri?descendstoaconnection^0ronVwithzerocurvqa-ture[tensor.Sinceisaunitaryrepresentation,\thenaturalinnerproGductonC^k descendsMtoa bGermetriconVWqandwehave^WqrisaRiemannianconnectionas 7discussedUUinx1.9.3.A ThereisanalternateviewpGointwhichissometimesuseful.Insteadofstartingwitharepresentationofthefundamentalgroup,Ewecouldstartwithavectorbundle!pthenhave.F;urthermor}'e,ifp=0,then=. V8(2)!pFixpwith0pdimq(Y8).Thefollowingc}'onditionsareequivalent:㍍&8߫2a)9pWehave1ɍpZI;mZ[ٟ^ի=[ٟ^1ɍpYn9;mY*o.Y%2b)9pF;orall,wehave[ٟ^E(;1ɍpYn9;mY*o)E(;1ɍpZI;mZ).&3c)9pF;orall,ther}'eexists=()so[ٟ^E(;1ɍpYn9;mY*o)E(;1ɍpZI;mZ).捍%3d)9pWehave5=0.Ifp>0,thenwealsohave!"=0.3.9.10Remark.ThereYaresimilarresultsinthecomplexsettingandformani- foldsUUwithbGoundary*.9 IZG!Îcю鍵3.10THeatCon9tentTAsymptotics  W*erefertovqandenBergandGilkeyV[20,21]forfurtherinformationconcerningthe^heatcontent^asymptoticswhicharedescribGedinthissection;cseealso[18,`19,127,128]Dforrelatedwork.@LetM\bGeasmoothcompactRiemannianmanifold qǍofʫdimensionmwithsmoGothboundary@8M.W*etakeinitialtemperatureandpumpheatintothemanifoldacrossthebGoundaryataconstantratedeterminedbybthe uxfunction .*Let@N ~bGetheinwardunitnormalderivqativeandlet [QbGeZthescalarLaplacian.TheresultingtemperaturefunctionH^Nb; (y[٫;t)forthe TNeumannhe}'atpumpUUisthesolutiontotheequations:UDP .TGilk9ey,J.Leah9y,JH.P9arkp856Y/ɧ8(1)!(@t6+8)H^Nb; =0y(3.10.a)n8(2)!lim.㑟t#0<(H^Nb; (x;t)=x(3.10.b)8(3)!@NH^Nb; (x;t)= (x)UUforx Ӊ2@8Mw(3.10.c)UEquationd(3.10.a)istheevolutionequation,hequation(3.10.b)istheinitialcondi- tion,UUandequation(3.10.c)isthebGoundarycondition. qǍ LetNFݫbGeanauxiliaryfunctionwhichisusedtomeasurethetemperaturepro le. F*oriexample,.Fcouldrepresentthesp}'eci c`heatiatapGointxe2M.W*eiconsidertheUUweighte}'dheatcontentUUenergyfunction֍_ Nͫ(; ;Fc)(t):=cZUR yMVHN፬; (t;x)F(x):4֍OneUUcanshowthatthereisanasymptoticexpansionast#0UUoftheform]t Nͫ(; ;Fc)(t)Sf1 X n=0A Nn(; ;Fc)tn=2 ծ: T Thisproblemhasconsiderablephysicalrelevqance.KLCarslawandJaeger[35,p75] qǍnotedUUthat:!\The,b}'oundaryconditionofconstant uxisofconsiderablepracticalim-!p}'ortance.Itwappearsifheatisgeneratedbya atheatingelementcarrying!ele}'ctricYcurrent,uifheatisgeneratedbyfriction,uandasanapproximation!inthee}'arlystagesofheatingafurnaceorroom.Italsohasimportant!applic}'ations$toproblemsondi usion.KThecoolingoftheEarth'ssurface!after5dsunsetonacle}'arwindlessnightisverynearlythatduetoremoval qǍ!ofhe}'atataconstantrateperunitareaperunittime,thus[this]givesthe!wayinwhichthesurfac}'etemperaturefallsaftersunset.")OW*eUUalsorefertoBrunt[31]forotherphysicalapplications. 銍 ThereisananalogousproblemfortheDirichletʆhe}'atpump.#LetH^Db; (x;t)bGe TtheUUsolutiontotheequations:ٍ8(1)!(@t6+8)H^Db; =0=@(3.10.d)n8(2)!lim.㑟t#0<(H^Db; (x;t)=ŕ?(3.10.e)8(3)!H^Db; (x;t)= (x)UUforx Ӊ2@8M)(3.10.f)܍W*eUUde netheweightedUUheatcontentUUenergyfunction:֍_;N D( ;;Fc)(t):=cZUR yMVHD፬; (x;t)F(x);VS^ChapterTThree:pRigidit9yofeigenv\ralues@866Y/ɧthereUUisanasymptoticexpansionast#0UUoftheform:^) D( ;;Fc)(t)%-X n0 Dn( ;;Fc)tn=2 ծ:  Thezheatcontentzasymptotics ^Nn ϫand ^Dn Barelo}'cally*computable.LetzLbGe jtheެsecondfundamentalformonthebGoundary*,letRsbetheRiemanncurvqature tensorofM andletij |:=~Rik+BkjذbGetheRiccitensor. ULet`;'and`:'denote 7covqariant!di erentiationwithrespGecttotheLevi-CivitaconnectionsofM0UUinprovingTheorem3.10.3. qǍ Equations(3.10.a)-(3.10.c)andequations(3.10.d)-(3.10.f)canbGedecoupled;5wecanHconsiderthetempGeraturefunctionde nedbytheinitialconditionwithho- mogeneous bGoundaryconditionsandthetemperaturefunctionde nedby0initialconditionDaninhomogeneousbGoundaryconditionsseparately*.%ThusforanypGointy.ofUUthemanifoldY9wehave:(3.10.g)ƍXHHB፬; (y[٫;t)=HB፬;0 ʫ(y;t)8+HB፬0; (y;t)UUsoXH B፯ny6(; ;Fc)= B፯n(;0;Fc)8+ B፯n(0; ;Fc):lWLetdfU^B፯n 5;^B፯ngbGethediscretespectralresolutionofY Q:withgivenboundarycon- 銍ditions.qLetUUu^B፯n2:=[ٟ^U^B፯n 5.Expand퍑X=X >n㉮cB፯n()UB፯n ^andTF*=X >ncB፯n(Fc)UB፯n 5;theseUUseriesconvergeUUinL^2|s.qItisthenimmediatefromthede nitionthatp fwHB፬;0 ʫ(y[٫;t)=X >n㉮etrBncB፯n()UB፯n 5(y):_ԍW*eUUcomputetherefore:*0Ik By6(;0;Fc)(t)=dX tDͯi;jծetrBicB፯i()cB፯j(Fc)cZ8 yMUB፯i 5UB፯jB=dX AnծetrBncB፯n()cB፯n(Fc):*/RecalloNthatwede nedf:=a[ٟ^Fݫandwede neda:=[ٟ^.W*eoNpGerformasimilarcalculationUUonZ qtoexpand_ԍ\fm=X >n㉮cB፯n()uB፯n Goand9fڧ=X >ncB፯n(Fc)uB፯n:XqChapterTThree:pRigidit9yofeigenv\ralues@886Y/ɧW*e!haveĴR<n㉮etrBncB፯n()uB፯n(z)qandUUasimilarargumentusingthefactthatĴR9,bGeaharmonicfunctionwithboundaryvqalue .b[ThenUsXH^DbU;0Yϫsatis es theFequationsde ningH^Db0; IandconsequentlyH^Nb0; =U}߱fĮH^DbU;0 2.NThisimpliesthat-fa(3.10.i)USfO- (0; ;Fc)=cZUR yMVUFo8 (U;0;Fc);&fO- D፬0(0; ;Fc)=0;URandjfO- Dn(0; ;Fc)= nq~(U;0;F)UUif n1:+:Let-u=[ٟ^U;;thisisaharmonicfunctionwithbGoundaryvqalues [٫.dThusasimilar argumentUUshows:ݑ(3.10.j)ni&s D፬0(0; [;f)=0;URandji&s Dn(0; [;f)= nq~(u;0;f)UUif n1:ݐTheXEassertionofTheorem3.10.3concerningDirichletbGoundaryconditionsnow qǍfollowsUUfromequations(3.10.g),(3.10.h),(3.10.i),and(3.10.j). The]situationisslightlymorecomplicatedwithNeumannbGoundaryconditions. SuppGosethatĴR_<@n9Mo OO=0.fW*ecanthen ndaharmonicfunctionU+sothat@N onUU@8M.qW*ecanthenarguethatthefunctionUO8HNU;0@satis esUUtheequationsde ningH^Nb0; UXandconsequentlyōHN፬0; =UO8HNU;0 2:ƍSimilarly*,UUsincetheaverageUUvqalueof .iszeroon@8Z qwehave}HN፬0; .=u8HNu;0 ի;Y~P .TGilk9ey,J.Leah9y,JH.P9arkp896Y/ɧtheԯremainderoftheargumentisthenthesametoseethattheheatcontent asymptoticsUUtransformpropGerly*. T*oJcompletetheproGof,Mwemustremovetheassumptionthattheaveragevqalueof3 vqanishes.`ChoGoseasmallpointy0>intheinteriorofYandchoGose|>03sothatd+thediskofradiusabGouty0isdisjointfrom@8Y8.HChooseasmoothfunction &x䍑>;~F (so`thatx䍑 ~FpagreeswithF2near@8YDandsothatx䍑 ~FvqanishesidenticallynearthebGoundary^ofthisdisk. W*eletx䍑~Y:=հYw>B?(y0|s)andx䍑x~Z r=[ٟ^1x䍑 ۫~ MYj. Extend sothat ԍtheaveragevqalueof onthebGoundaryofx䍑j~Yիvanishes.UITheargumentgiveninthe previousUUparagraphshowsthat(3.10.k)mh Nnͫ(0;\qЫ~ ;\qɫ~f)CJ~Z 7s=V8 Nn(0;x䍑7~ qƮ;x䍑~F |p)C7~YG:cTheRrheatcontentRrasymptoticsre ectthe owacrossthebGoundary*.iTheinitial [QvqalueIofthefunctionsH^Ne(0; [٫)Y 6andH^N(0; [٫)C7~Y 6iszero.O?Theprincipalofnot RfeelingUUthebGoundaryshowsthatꬍmx (0; ;Fc)YG(t)8 (0;x䍑7~ qƮ;x䍑~F |p)C7~YG(t)vqanishesitoin niteorderast#0isincetheheatwhich owsacrosstheadditional bGoundaryZcomponentisweightedwithzeroneartheadditionalbGoundarycompo-nent.qThus 4ԍr|߮ nq~(0; ;Fc)Y _= n(0;x䍑7~ qƮ;x䍑~F |p)C7~YG:m荫W*eUUmaythereforeuseequation(3.10.k)toseethat-mh Nnͫ(0; [;f)Z 7s=V8 Nn(0; ;Fc)YG:5؍TheUUdesiredresultnowfollows. 򯍑 In[21],vqandenBergandGilkeycomputed ^Dl5(1;0;1).EThiscorrespGondstoa qǍheatconductionprobleminasolidwithconstantspGeci cheatandconstantinitialtempGeraturexissuddenlyimmersedinicewater.W}V*andenBergandGilkeyusedH.W*eyl's[199]theoryofinvqariantstoshowthatthereexistuniversalconstantssothat! ^Dl5(1;0;1)= Z1K&fet*240鿲pɟ鿉W oALĴR#<@n9M0nfb1|smm;mm!^۫+8b2Laa ,mm;mF@+b3LabZRammb;m(++b4|s^2፯mm+8b5RammbRammbʫ+b6Laa ,Lbbmm+b7LabZLabmm]V++b8|sLabZLaciQRmbcm+-b9Laa ,Lbc%Rmbcm+-b10xRammbRaccby+b11Laa ,Lbc%Rbddc򯍑++b12xLabZLaciQRbddc#i+8b13LabLcdqwWhenEigenrv`aluesChange)􂍍'K虚ff .UX/./././././././././././././././././././././././././././././././ ffffLG!Îcю鍵4.1TIn9troQduction r㍑ In#ChapterThree,-weestablishedrigidityresultsandgaveconditionstoensure thateigenvqaluesdidnotchange.h`Inthischapter,westudywheneigenvqaluescanchange.RIn>x 4.2,we>studythegeometryofcirclebundles.W*e rststudythe spGectralEgeometryoftheHopf brationS^3!S^2.W*eusethisexampletoshowthat.Theorem3.4.1issharpifp2;;given.0,5we.can nd":Z~4!Yfand Ì06=2E(;1ɍpYG)UUsothat[ٟ^2E(;1ɍpZp[). : W*evZthengeneralizetheHopf brationandgiveexamplesduetoMuto[143, 144]kofprincipalcirclebundleswhereeigenvqalueschange.W*eusetheseexamples qǍto7ibuildexamplesofholomorphicHoGdgemanifoldswhereharmonicformsoftype(p;p)UUpullbacktogiveeigenformswhicharenolongerharmonic. 䪍 InZNx3.6,[weZNshowedthatifP2!bY2wasaprincipalbundlewithstructuregroup G~csothatH^1Lq(G;R) =0,then~ctheeigenvqaluestructurewasrigid;ifthepullbackofpaneigenformwasagainaneigenform,thentheeigenvqaluedidnotchange.In xqɫ4.3,weZshowthishypGothesiswasessential.TSuppGoseGisacompactconnectedLiegroupwithH^1Lq(G;R)6=0.KLet>0bGegiven.KInxdޫ4.3,weshowthatthere ٍexistsţaprincipalbundlePӱ!DS^3գwithstructuregroupGand2E(4;^2vSa3 ')so UNthatUU2E(48+^2|s;^2bP);thisgivesmoreexampleswhereeigenvqaluescanchange. %ƍ In2xZ3.5,)we2turnourattentiontocomplexgeometryandestablishthateigen- vqalueskcanchange.nW*ehaveshownpreviouslyinTheorem3.5.1thateigenvqaluescannotchangeifq"=0.P2F*urthermore,wehaveshownthateigenvqaluescannotde- qǍcrease.i3W*e;willshowthattheseresultsaresharpexceptpGossiblyif(p;q[٫)=(0;1);this caseisleftunsettled.`+Let0 bGegiven,+letq"1begiven,+andletpbe\hChapterTF our:pWhenEigen9v\raluesChange*926Y/ɧgivenwithp<+q"2.;W*ewillshowthereexistsaHermitiansubmersion":V!U ÌandDaneigenform06=2E(;1ɍp;qU )Dsothat[ٟ^2E(;1ɍp;qV ).ʔTheDcase(0;2) requiresUUspGecialargumentasthisisabitexceptional. The!statementofTheorem3.5.2givingnecessaryandsucientconditionsthatallLtheeigenvqalueswerepreservedinvolvedstudyingtheconditionsJ9^!*=cQ![٫.Inx \4.5,weconstructexamplesofHermitiansubmersionswhereJ9^!3=Ү!landJ9^!"=!.forUUnon-trivial![٫. In xzɫ4.6 wediscusstheanalogoustheoremsinspingeometry*.IfthestructuregrouppkisAbGelian,0givenany0pkand0,0wepkcan ndaprincipalbundle v: P~!T^2with!NstructuregroupGover!Nthetwo!NdimensionaltorusT^2sothat rthere'~exists%Z2E(;D^GCYrcvT2 7?)'~with^[ٲ፯ <2%ZE(;D^GCYrcbP 7?)whereD^GCYrcbM ^denotesthespinor #thesituationisabitmorecomplicatedandwecanshowthatUUeigenvqaluescanincreaseif>>.6TLG!Îcю鍵4.2TCirclebundles MutoL[143,144]gaveLexamplesofRiemannianprincipalS^1Lbundleswhereeigen- ٍvqaluesN.change.W*ebGeginbyconsideringtheHopf brationꊮ":S^3!S^2^.considered din6x1.13;note6thatS^26isidenti edwiththesphereofradius i1i&fes2 wEandhasvolume whileUUS^3eUisidenti edwiththesphereofradius1andhasvolume2[ٟ^2L.'4.2.1Theorem.L}'etp 2~bethevolumeelementonS^2.Then2C2E(0;^2vSa2 ').L}'et UN":S^3!S^2b}'etheHopf bration.Then[ٟ^2C2E(4;^2vSa3 ').ÍPr}'oof.pSince`thevolumeelementofanyRiemannianmanifoldisharmonic,Kitis immediateUUthat2C2E(0;^2vSa2 ').qW*euseLemma1.10.1toseethatk/yFE=extGS#(e1|s)[ٟeint.Y}u(F9)9wherecFb0isthecurvqatureoftheHopf brationande^1jisthedual1form.W*euseequation(1.13.a)toseeF2=22|s.5^ThusE[ٟ^2>l=2e^1|s.5^Sincede^1>l=2[ٟ^2|s,~the qǍdesiredresultfollowsfromtheequationsofstructuregiveninx`ɫ2.7;5=0sincethe bGersUUaretotallygeodesic.  W*e¨cannowshowthatTheorem3.4.1issharpifp}M2.Note¨thatwedonotknowUUifeigenvqaluescanchangeifp=1.4.2.2 Theorem.L}'et/0andletp2b}'egiven.!mThereexistsaprincipal Ìcir}'cle2bundle":P*!YkӺover2somemanifoldYandther}'eexists06=2E(;1ɍpYG) sothat[ٟ^2E(;1ɍpZp[).Pr}'oof.pIf.1ʫ=,e\thentheTheoremisimmediate.W*eassumetherefore<.LetT,poҫ=2+rIforroҫ0.nLIf=0,byrescalingthemetricsinvolved,wemayassumeXwithoutlossofgeneralitythatǫ=4.LetXT^rMbGethetorusofdimension]P .TGilk9ey,J.Leah9y,JH.P9arkp936Y/ɧrG.$MW*e;letY:=FS^2RRT^rm,u Zҫ:=S^3RT^rm,u and;let[٫(u;t)F=(H();t);whereH qǍisztheHopf bration. 7Letr HbGethevolumeelementonthetorus; thisisa harmonic5~form.CSincetherolesofthespheresandthetorusdecouple,mwethen Ìhave#Z2Q^^r42E(0;1ɍpYG)and[ٟ^(2^rm)=^[ٲbH2^r42E(4;1ɍpZp[);4this#Zprovesthe theorempif=0.%Ifp6=0,jagain,byrescalingthemetricsinvolved,jwemayassume qǍ=++4.o|ByNsrescalingthemetriconthetorus,Owemay nd06=T2E(;^rlT*). ׍TheUUtheoremnowfollowsfromtheobservqations:ꍍnKQn2S^8T2E(;1ɍpYG)UUand#KQn[ٟ(2S^8T*)=[ٲ፯H2^8T2E(+4;1ɍpZp[): UV"x# W*e23remarkthatwedonotknowifTheorem3.4.1issharpifp=1;=we23knowof no4exampleswhereeigenvqalueschangeifp=14norcanweshowthattheydonot. s W*e#[canuseTheorem4.2.2toconstructRiemanniansubmersionswhereseveraleigenvqaluesUUchange:DX4.2.39Theorem.Fix0p2.oF;or1rG,Wlet0ȱ *gb}'egiven.oThere ÌexiststOaRiemanniansubmersionw:]Z!Y3andtO06= Wg2E(ɮ;1ɍpYG)sothat [ٟ^2E(ɮ;1ɍpZp[)for1jrG.&vPr}'oof.pW*euseTheorem4.2.2toconstructRiemanniansubmersions:Z!Y Ìanddi erentialforms0滱6=2E(ɮ;1ɍpY -)sothat^[ٲ፯2滮E(;1ɍpZ \)forthe indices1C3)rG.LetY|:=Y1ı7Q:::Yrm,2kletZO:=C3Z1ı:::Zrm,2kandlet":=1Kx:::rm.J6W*eޠusepullbacktoregardtheiasdi erentialformsonY8;2theconclusionsfoftheTheoremthenfollowsincewehave1ɍpY --R=31ɍpYG/andsince weUUhave1ɍpZ \^[ٲ፯ɫ=1ɍpZp[[ٟ^. &v TheEHopf brationgeneralizeseasily*.%ThefollowingessentiallyfollowsfromcalculationsUUofMuto:DX4.2.4Theorem.L}'et^LސrbeaunitaryconnectiononacomplexlinebundleoverY [Qwithasso}'ciatedcurvature2-formFQ=F9(^Ltr)andassociatedprincipalcirclebundle QōSy=S(L).L}'et2E(;1ɍpYG).AssumethatdY=0,IthatdYintYt(F9)=0,and thatextGY3c(F9)int qY(F)=forc}'onstant.Then[ٟ^2E(8+;1ɍpS).&vPr}'oof.pW*eUUuseLemma1.10.1.qSince5=0andsincedYG=0,!ȝ[71ɍpS[ٟ_^[ٟ1ɍpYG=dSE[ٟ=dSfe1S^8[ٟ(int ɟY(F9))g#W@=a>8[ٟfext tY(F9)int qY(F)g=[ٟ: * W*ecanusethisTheoremtoconstructmoreexampleswhereeigenvqalueschange.^ChapterTF our:pWhenEigen9v\raluesChange*946Y/ɧ4.2.5Theorem.L}'et.YgbeahomogeneousmanifoldwithH^2Lq(Y8;R)c6=0.iTher}'e 銍existshWac}'omplexlinebundleLoverY;andaunitaryconnection^L d˱ronLwith normalize}'dcurvatureF suchthat:8(1)!pWehave:=jF9j^2Zisc}'onstant.[8(2)!pWehave06=FQ2E(0;^2bYG).w8(3)!pWehave^[ٲbSFQ2E(;^2bS).-Pr}'oof.pW*eZuseLemma1.11.4to ndaunitaryconnection^L αronacomplexline bundleUULoverUUY9suchthatPr#06=:=F9(Ltr)2E(0;2፯YG):鍫Letu06==jj2C=int qY(F9);#ЍweUUhavewNϱextGY3c(F9)int qY(F)=:Since%misharmonic,YsdYG!=0.By%mTheorem4.2.4,itsucestoshowdYG!=0 or[Eequivqalentlythatisconstanttoshowthat[ٟ^{2E(;^2bS).Let[EGbGethe connectedpcompGonentoftheidentityintheisometrygroupofY8.1&PullbackinducesanaturalactionofG|onthedeRhamcohomologygroupsH^2Lq(Y8;R). SinceGis qǍconnected,ttheԺhomotopyɸaxiomforcohomologyimpliesthisactionistrivial.!DW*eusethedeRham-HoGdgeTheorem1.4.6toidentifyH^2Lq(Y8;R)withE(0;^2bYG).HW*e mayH=nowconcludethatE(0;^2bYG)is xedbythisactionofGѫ.mjThus "=foranyZ 좱2Gf+andthusjj^2Jͫis xedbyGѫ.SinceY>isahomogeneousspace,웱Gf+acts ٍtransitivelyUUonY8.qThisshowsjj^2ȫisconstant. |ލ4.2.6iGRemark.thegener}'alized CHopf br}'ationS^1!S^2n+1!CP؟Tn:Qprovidesan example^ofthisphenomena.W*ehave^H^2Lq(CP UVTnԫ;R)^=R6=0.The^assoGciatedlinebundleEistheHopflinebundleandthenormalizedcurvqaturegeneratesH^2Lq(CP UVTnԫ;R).J̍ This%phenomenaofchangingeigenvqaluesalsoappGearsincomplexgeometry*. LetLbGeaholomorphiclinebundleoveracomplexmanifoldY8.W*esaythatLisvpQositiv9eifthecurvqatureFtofListheKaehlerformofaHermitianmetriconY8;8thereisapGossiblesignconventionwhichplaysnoroleinourdevelopment.F*orhexample,thehypGerplanebundleъH8ëisapositivelinebundleovercomplex `pro8jectivejvspaceCP̟TnandtheassoGciatedmetricistheF;ubini-StudyWmetric.*Moregenerally*,if%Y isaholomorphicsubmanifoldofCP{TYiforsome,thentherestriction of3HW1toYisapGositivelinebundleoverYandthemetriconYistherestrictionofvtheF*ubini-StudymetrictoY8.T}Conversely, ifvY6ZadmitsapGositivelinebundleL, `thenthereexistsaholomorphicembGedding В:Y!CPnT˫forsome_andapositive o9integer֮kCmsothatL^ k*=͙ z^^H.JKSuchamanifoldissaidtobGeaHo}'dge$manifold.ThusqwemayidentifythesetofHoGdgemanifoldswiththesetofsmoothalgebraicvqarieties._P .TGilk9ey,J.Leah9y,JH.P9arkp956Y/ɧ4.2.7Theorem.L}'etƫFbethecurvatureofapositivelinebundleLoveracomplex manifoldYTsofr}'ealdimension2n.0L}'et:خSPe!Yb}'etheassociatedS^1 +principal B֍bundle.sThen o06=F9^pd2E(0;1ɍ2pYū)and[ٟ^F9^pd2E(p(n=ܫ+1p);1ɍ2pSū)for1pn.ͱPr}'oof.pLetc0LbGeapositivelinebundleoveraholomorphicmanifoldY8.YLetF bGe thecurvqatureofL.ThenFEistheKaehler)formofthemetriconYsoF9^p \isharmonicUUfor1pn:=dimqƟCY8.qW*eUUhave9iLintu=̟Y|*(F9)Fpd=p(n8+1p)F9p1ݍisUUharmonicandYextGY3c(F9)int qY(F)Fpd=p(n8+1p)F9p:TheUUTheoremnowfollowsfromTheorem4.2.4. 6|kZG!Îcю鍵4.3TPrincipalBundles ThemfollowingresultshowsthatthehypGothesisH^1Lq(G;R)=0minassertion(2) qǍofUUTheorem3.6.1wasessential.듍4.3.1:ZTheorem.L}'et乮YbeacompactRiemannianmanifold. Supposethatthere exists2E(;^2bYG)sothat6=0,sothatjj^2 *r=aisc}'onstant,andsothat dYG'=0.9L}'etBGbeacompactconnectedLiegroupwithH^1Lq(G;R)'6=0,֙andBletb}'e+projectiononthesecondfactorofP幫:=*GY8.aF;orany*>0,ther}'eexistsametricʮgP()onPYsothat":P*!YisaRiemannianprincip}'alGbundleandso ٍthat[ٟ^()2E(8+^2|sa;^2bP).Ј4.3.2Remark.LetC:֮S^3ֱ!S^2 bGetheHopf bration.yLet2bethevolume elementxdonS^2.W*eshowed[ٟ^2׫bGelongstoE(;^2vSa3 ').Since2hasconstant #LetybGepro8jectiononthesecondfactorofPy:=GfgY8. W*eSCde neametricds^2bP()byrequiringthatH = := 4( )e1϶isSCthehorizontal Aliftof andbyrequiringthatfeiTLgisanorthonormalframeforV}.Thesplitting qǍisGequivqariantandwiththismetric, 8Uɫde nesaRiemannianprincipalGbundle. Note/thatfe^1G+ʛ;e^2|s;:::g/isthecorrespGondingdualorthonormalframeforV}^ka. UTW*eUUcomputenAԧ[Fap;FbD]=l[fap;fbD]8fa((fbD))+fb((fap))#dR=l([fap;fbD]8([fa;fbD])e1|s)dP(fa;fbD)e1|s: ˽ConsequentlyE[ٟ^J1=extGP (e^1ī+mQ)[ٟ^eint.P<().]SincedYG=0andsincethe qǍ bGersUUof.areminimal,weseeJ(2፯P[ٟ-8[ٟ2፯YG)()=2|sdP(jj2)=a2|s: m4.3.3Remark:.ThetechniqueofRiemannianproGductsusedintheproofof Lemma'2.9.4cannowbGeusedtoconstructexamplesforanyp^u2'whereeigen-vqaluesUUchange.8{ZG!Îcю鍵4.4TComplexgeometry  ThereUUareanalogousexamplesforthecomplexLaplacian.'S4.4.1Theorem.L}'etO":Z~4!Y"3beOaHermitiansubmersion.`Let0<<1,let~qJ51andletpH+q52.\Ther}'e~existsaHermitiansubmersion:5V'!Uand Ì06=2E(;1ɍp;qU )sothat[ٟ^2E(;1ɍp;qV ). q W*e$bGeginbyreducingtheproofofTheorem4.4.1tothespecialcase=0$and qǍ(p;q[٫)=(1;1)UUor(p;q[٫)=(0;2):4.4.20Lemma.Supp}'osethereisaHermitiansubmersion1SI:֮Z1!Y1 andnon-zer}'o&+1L2E(0;1ɍr;sY1 %)so^[ٲl112E(͠;1ɍr;sZ1 %)forsome+>0.PcL}'et0<,Jlet r}6p,andnletsq-Gb}'egiven.R*ThenthereisaHermitiansubmersionf:6Z!Yand06=2E(;1ɍp;qY )so[ٟ^2E(;1ɍp;qZ ).Pr}'oof.pIfM!isacomplexmanifoldwithaHermitianmetric,˲letM(c)denoteM ٍwithUUthescaledmetricc^2 tds^2bM\.qSince1ɍp;q6M,(c)zW=c^2|s1ɍp;qM ,t6E(;1ɍp;qM )=E(c2|s;1ɍp;q6M,(c)?):a͟P .TGilk9ey,J.Leah9y,JH.P9arkp976Y/ɧGive8M3:=M1_⨮M2&theproGductmetricandproductholomorphicstructure.8Then%a1^E(1|s;1ɍp1 ;q1M1w۫)8^E(2|s;1ɍp2 ;q2M2w۫)E(1S+82|s;:(p1 +p2;q1+q2)"M4ث):AssumetheconditionsoftheLemmahold.PChoGosec>0sothat=c^2|s +r1.Let qǍTbGeaholomorphic attorusofcomplexdimensionatleastmaxc(pr;q)s)._By rescalingUUthemetriconT,wemaychoGose06=2C2E(;1ɍpr;q@LsȂT).qLet<-ȒY:=Y1|s(c)8T;Z~4:=Z1|s(c)T;andqǮ[٫(z1;wD)=(1(z1);wD):ThenĮisaHermitiansubmersion.*Let-|:=1H^aի2|s.Then06=2E(;1ɍp;qY ) and G[ٟ=([ٲ፬11|s)8^2C2E(c2+=;1ɍp;qZ ): UVh4.4.31F ormsoft9ypQe(1,1).Over{yY?=Y2|s,letS0=Y1RMS^1ybGethetrivialcircle qǍbundleUUandwithF0C=0.qLetS1ȫbGeacirclebundlewithwvF1C=<$1Kwfe  (֍2<(dx8^dy[٫):LetͮZ=0߮Wc(S0|s;S1)bGethe) berproductdiscussedinx1.8.0.Lete^0@ande^1bGethecorrespGondingUUdualverticalcovectors.qW*ethenhave>`theintegra-bilitycconditiononhorizontalcovectorsisimmediatesowemustonlychecktheverticalUUcompGonent;DfP@d(ejo8p 7fe X1UVek됫)=(jk8p 7fe X1kP)[ٟF9:dChapterTF our:pWhenEigen9v\raluesChange|1006Y/ɧ4.4.7TTheorem.L}'et1pm"andlet:=(j^2Gޫ+8kP^2 )p(m +1p).Then%/_ dF9pd2E(0;1ɍ2pYū)8\E(0;1ɍp;pY );>andƍd[ٟ(F9P()2E(;1ɍ2pZū)8\E(<$33133wfe (֍2fg;1ɍp;pZ ):+oPr}'oof.pW*e9haveF9^pīisaharmonicformoftypGe(p;p).\hSinceYNisKaehler," wehave qǍthat Oa E(0;1ɍ2pYū)8\C1 0p;p (Y8)=E(0;1ɍp;pY )`\soUUF9^pd2E(0;1ɍp;pY ).qW*ecompute:!1ɍ[ٯp;p1ZE[ٟF9pd=<$K1Kwfe (֍2 -(jk8p 7fe X1UVkP)p(m +81p)(ejo+p 7fe X1ek됫)^[ٟF9p1Xso,d1ɍ[ٯp;p1ZE[ٟ^F9^p 8= Ͱ1Ͱ&fes2 V[ٟ^F9^p.LThus[ٟ^F9^p 82}E( 33133&fes2bٮ;1ɍp;pZ );theproGofofthecorre- dspGondingUUassertionintherealcaseissimilar. @ NotethatthemanifoldZ6constructedabGoveisingeneralnotKaehler.F*or example,ifQListheHopflinebundleoverQthe RiemannsphereS^2Qandifwehave(jR;kP)t=(0;1),[thenЍZK=S^1S^3istheHopfmanifoldandM:S^1S^3t!S^2is qǍessentiallyUUjusttheHopf brationwherewenormalizethemetricssuitably*.7xELG!Îcю鍵4.5THermitiansubmersions 4 LethήCe:献Z!YbGeaHermitiansubmersionwithminimal bers.2Letp>0and letUq#<>c0.rNInTheorem3.5.2wesawthat[ٟ^J?preservestheeigenspacesonformsoftypGe(p;0)ifandonlyifJ9^!"=![andthat[ٟ^ګpreservestheeigenspacesofforms oftypGe(0;q[٫)ifandonlyifJ9^!H˫=!.W*enowgiveexamplestoillustratethesetwoXcases.zThecaseJ9^!'=˵!isrelativelyeasy;Y|thecaseJ9^!'=˵!requiresmorework.@4.5.1qHermitiansubmersionswithJ9^!"=![ٵ.Let8YqbGeaRiemannsurfacesodimC㋮Y&=1.BThenl~thedistributionH1;0Edisa1dimensionalcomplexfoliationand 7henceiH1;0Bisnecessarilyintegrable.ThusiJ9^!E`=釮![٫.Thesubmersionconstructed \pinUUx4.4.3UUdealingwithexamplesofformsoftypGe(1;1)givesanexample{g":Wc(S0|s;S1)!S1H8S1withUUnon-trivialcurvqaturetensor!.satisfyingJ9^!"=![٫.e0iP .TGilk9ey,J.Leah9y,JH.P9arkÅl1016Y/ɧ4.5.2/HermitiansubmersionswithJ9^!"=![ٵ.W*e|haveJQ=1on^2;0 `}^0;2 RǍand%=Jӫ=!+1on^1;1 .LetSiybGecirclebundlesover%=atorusT^k ͫwithcurvqatures ƍF9^itand"correspGondingdualverticalcovectorse^iTL.W*eassumethatJ9^F9^inѫ=LF9^i B؍orequivqalentlythatwemaydecompGoseF9^i=4uǟ^i.+\qkdyuǟ^i%Hforuǟ^i*2^2;0 .5W*ede nean qǍalmost complexstructureonZd=HWc(S0|s;S1) byrequiringthatgZ wgisHermitian, ٍthatUU[ٟ^JpreservesJ9,thatJ(e^0|s)=e^1,UUandthatJ9(e^1)=e^0.qW*eUUcomputexngd(e0S8p 7fe X1UVe1|s)=[ٟ(F908p 7fe X1F91z)퍍`=g[ٟ(uǟ0+8p 7fe X1UVuǟ1:)8+[ٟ(\qXuǟ0 p 7fe X1\q\UVuǟ1 ):rTheUUNirenbGerg-Neulanderintegrabilityconditionissatis edifandonlyifፍ(4.5.a)彮uǟ0R=p ofe X1㎮uǟ1::De neUUhorizontal2-forms![ٟ^izbytheevqaluation:s![ٟi%(fap;fbD)=<$K1Kwfe (֍2 -gZp[(eiTL;[fa;fbD]): W*eUUthenhave.k![ٟi%(fap;fbD)=<$K1Kwfe (֍2 -eiTL([fa;fbD])=<$33133wfe (֍2fgdeiTL(fap;fbD)=<$33133wfe (֍2fg[ٟF9iR(fap;fbD):"%Thus![ٟ^i#=sƱ 33133&fes2bٮ[ٟ^F9^ivandJ9^![ٟ^i=sƱ![ٟ^i%.Thusitsucestogiveanexamplewhere ˍequationUU(4.5.a)issatis ed.qLetO?uǟ0R:=<$1Kwfe  (֍4<(dx1S+8p 7fe X1UVdy[ٟ1L)8^(dx2+p 7fe X1UVdy[ٟ2L): ThenUUwehave>e^h|F90Aī=<$y1vԏwfe  (֍2(dx1S^8dx2dy[ٟ1,^dy[ٟ2L);andύ^h|F91Aī=<$y1vԏwfe  (֍2(dx1S^8dy[ٟ2,+dx2^dy[ٟ1L):}W*eUUuseLemma1.12.1toconstructbundlesLioverUUthetoruswith 卍eSVF92Aī=<$1Kwfe  (֍2<(dx1S^8dx2|s)_;F93Aī=<$1Kwfe  (֍2<(dy[ٟ1,^8dy[ٟ2L);ύTzF94Aī=<$1Kwfe  (֍2<(dx1S^8dy[ٟ2L)_;F95Aī=<$1Kwfe  (֍2<(dx2S^8dy[ٟ1L):!卍SinceUUF9(L^;Zi)=F(LiTL)UUandpuF9(Li, 8Lj6)=F(LiTL)8+F(Lj6);L0Q:=ԩL2!' L^l3andL1:=L^l4= L5sde necirclebundlesoverthetoruswiththe m2desiredUUcurvqatures.f?0ChapterTF our:pWhenEigen9v\raluesChange|1026YOn썟ZG!Îcю鍵4.6TSpingeometry W*eϢnowgeneralizeresultsconcerningdi erentialformstothespinorsetting.E6InTheoremR4.6.1,RwedealwithAbGelianstructuregroups;SinTheorem4.6.2,wedealwithnon-AbGelianstructuregroups.OBytaking0П<inTheorem4.6.1,{weseeathateigenvqaluescandecreasesoTheorem3.4.1canfailforspinors.Theorem4.6.2UUshowsTheorem3.6.1alsocanfailforspinors.!4.6.1Theorem.L}'et GbeacompactconnectedAbelianLiegroup.$ALetYS=T^2|s.F;orrany0randforany0,Uther}'erexistsaprincipalbundleM:PUN!YVwith rstructur}'eDgroupG,thereexists0 6=2E(;D^GCYrcbY 7?),andther}'eexists 2CW^c8g ksuch Ǎthat^[ٲ፯ 2E(;D^GCYrcbP 7?).GdPr}'oof.pW*ecL rstrecallsomenotationalconventionscLextablishedpreviously.IfPisaprincipalbundle,letg$iCbGetheverticalvector eldsdiscussedinxdt1.16.IThe 4curvqature4tensoristhengivenby:![٫(Fap;FbD)=gP([Fa;FbD];g!i m)g y^i =2!abi /g8y^iū.*eThe followingendomorphismde nedinequation(2.7.e)playedanimpGortantrolein qǍtheUUintertwiningformulasofLemma2.7.3: (4.6.a)KY%Espinn= K1K&fes4 )aA^CYrcbY"n =Ln"1|snq~.-ChoGosenand"1ȃ6=0so"1|sn=zPp gzPfeWv.-Theeigenvqaluesof h덍$pUW$fe 1vcgy(uǫ)UUarejj.qChoGose"2C6=0and06= ^ϫsothatԍk/(n"1S+8p 7fe X1UV"1|s"2cgy(uǫ)) В=Hp oHfe" z:ThenUUn82E(;D^GCYrcbY 7?)and^[ٲ፯ n2E(;D^GCYrcbP 7?). u Thesituationfornon-AbGelianstructuregroupsisabitdi erent.ULetG 0be [QtheUUsmallesteigenvqalueofD^GCbGګ.4.6.2rTheorem.L}'etC0andletb}'egivenwithz+G .If=0,plet Y0=T^2|s;if< >0,fletY0=S^3.L}'etGbenon-Abelian.Thereexistsaprincipal rbundleT֫:P!YQ8withstructur}'egroupG,9othereexists06=2E(;D^GCYrcbY 7?),9oand Ǎther}'eexists В2CW^c :gPsuchthat^[ٲ፯ 2E(;D^GCYrcbP 7?).ƇPr}'oof.pSuppGose rst=0.]W*echooseeid2g Iso[e1|s;e2]=<߱6=0.]LetY=T^2+with qǍthe atproGductmetricandletPΡ:=kGzwYwhereispro8jectiononthesecond factor.#pNote;8thatPǫistopGologicallytrivialincontrasttotheprincipalbundlesgivenNintheproGofofTheorem4.6.1.o[F*or%2R,OweNde neanon-trivialhorizontal ÌstructureUUbysettingF1C:=@1ɍ8yxݍ1+84%e1ȫandF2:=@1ɍ8yxݍ2+8e2|s.qThen![٫(F1|s;F2)=4%u:ލChoGoseUU06=2CW^c8R^2ȫsocYG(dy[ٟ^1L)cY(dy[ٟ^2)=$p o$fe 1.qLetCwU9(%):=ACYrc፯G )+8p 7fe X1UV%cgy(uǫ):SinceF^2 R=է1,AD^GCYrcbP 7?^[ٲ፯ =  U9(%)^2|s z.%Let(%)bGethesmallesteigenvqalueof U9(%)^2|s.Then(0)=G and(%)!1as%!1.Thuswecan nd%sothat (%)=Gګ.qChoGose荑r4l06= В2CWc8g soN#U9(%)2|s = z:úThenUU06=2E(0;D^GCYrcbY 7?)and^[ٲ፯ 2E(;D^GCYrcbP 7?).hb<ChapterTF our:pWhenEigen9v\raluesChange|1046Y/ɧ Next^suppGose6=0.JLet^֮g0IbethestandardmetriconS^3n֫andletso (3)bethe ٍLieUUalgebraofS^3.qByEquation(A.4.b),5Z~DGCYrcٔ(Sa3I;g0 )= K9K&fes4 FonbֱCWso (3):퍫Let Y:=(S^3;"^2 tg0|s).Y3F*orsuitablychosen",YD^GCYrcbY W=onCW^c8so (3).Y3LetP*:=GkY ㍫whereUU.ispro8jectiononthesecondfactor.qChoGose06=<߱2gБ.LetUU%2R.Let&ۙݍF F1|s(y[٫):="y8i;F2(y[٫):="y8jR;F3(y[٫):="y8k#F [F1|s;F2]=2"F3|s;[F2;F3]=2"F1|s;[F3;F1]=2"F1|s;F f1C:=F1S82%u;f2:=F2|s;f3:=F3|s:'iҍTheMFaнareanorthonormalframeforso J(3).SoLetthefade nethehorizontalspace Hn ofUUPc.qThen![٫(F2|s;F3)=4"%uǫ,UU!(F3|s;F1)=0,UUand!(F1|s;F2)=0.qThus(΍.UGsACYrc፯P :=b̟Vpk#VfeW pzcYG(Fc1)cY(Fc2)cY(Fc3)8 1+ ACYrc፯G$d+8"%cYG(Fc2)cY(Fc3)8 cgy(uǫ);΍FDGCYrc፯P W=b̮8+("%cYG(Fc2)cY(Fc3)8 cgy(uǫ)+1 ACYrc፯G")2|s:)ChoGoseUU2CW^c :so(3)sothatcYG(Fc^2)cY(Fc^3)=$p o$fe 1.qLet5Zu>U9(%):=ACYrc፯G )+8"%pUWfe X1vcgy(uǫ):UThenUUD^GCYrcbP 7?^[ٲ፯ =8 (+U9(%)^2|s) z.qT*ocompletetheproGof,wechoGose%soͤw.U9(%)2|s В=8Gڮ: UV֍ W*e5OnotethatconstantG )inTheorem4.6.2neednotbGeoptimal.gForexample, suppGosethatG=S^3.ULetfeiTLgfori=1;2;3betheusualbasisfortheLiealgebra ̍ofUUS^3eUasdiscussedinx1.13.qThenG = K9K&fes4 ).Let<߫=e^1|s.ThenDLsZU9(%)=3=2cgy(g y1 )cg(g y2)cg(g y3)8+"%pUWfe X1vcgy(e1|s):Sincexcgy(g y^1 )commuteswithcg(g y^1 )cg(g y^2)cg(g y^3),4forxsuitablychosenvqaluesof%, 0 isaneigenvqalueoftheself-adjointendomorphismU9(%).!Thustheconclusions ̍of%Theorem4.6.2holdforany"%andnotjust+ "9"&fes4 I".W*erefertoW. ˍKramer*&U.Semmelmann[115]andtoMoroianu[136,_137,138,139,140]*for qǍotherUUpapGersinthisarea.iru6Y1ɧuChapterFivre:pQy[\OtherTXopics*='K虚ff .UX/./././././././././././././././././././././././././././././././ ffffLG!Îcю鍵5.1TIn9troQduction Inthissection,wetakeupsomerelatedtopics.TSInxn«5.2,wediscussobstructions totheexistenceofmetricsofpGositivescalarcurvqatureonRiemannianmanifoldsin2thespinorcategory*.fTheetainvqariant,9a2spGectralinvariant,9plays2acrucialrole Kinthissettingasdogeometric bGerbundleswithHPT2) bers.Thusthespectralgeometry[ofRiemanniansubmersionsentersintothepicture.<Thisrepresentsjoint jworkofthe rstauthorwithB.BotvinnikandS.Stolz.&Thex䍑^A -roGofgenusisaspGectralinvqariant.EBLetMbGeacompactmanifoldofdimensionm5whicheitherisLsimplyconnectedorwhichhas nitecyclicfundamentalgroup.oSuppGose rstthat`Mwadmitsaspinstructures.hThenMadmitsametricofpGositivescalarcurvqature0ifandonlyifx䍑^A `1(M;s)=0.WIf0thedimensionmisdivisibleby4,then ㍍x䍑^A(M;s)vtakesvqaluesinZandisindepGendentofthespinstructureswhichischosen.If omA1orifm2moGd8,:5thenx䍑^A p(M;s)takesvqaluesinZ2andcan qǍdepGendContheparticularspinstructurechosen.JThisinvqariantvqanishesotherwise. ThuscthereisnoobstructiontoconstructingametricofpGositivescalarcurvqatureonsuchamanifoldifm׷3,ifm5,ifm6,orifm׷7moGd8.)OTherestriction.thatm ױ5.isessential;thesituationisverydi erentifm ׫=4.RThereare7analogousresultsifMOdoGesnotadmitaspinstructurebutiftheuniversalcoverUUofMlpadmitsaspinstructure. Inxr5.3,+westudytheRiccicurvqature.t IfY9isacompactmanifoldwhichad-mits\ametricofpGositiveRiccicurvqature,thenMeyerstheoremshowsthatthe qǍfundamentalgroup1|s(M)is nite.PLet-*:QP4!M -bGeaprincipalbundlewithcompactU connectedstructuregroupG.qIfPadmitsabundlemetricwithpGositivejChapterTFiv9e:pPositiveCurv\ratureM1066Y/ɧRicci͏curvqature,then1|s(Pc)is nite.tConversely*,we͏showthatif1|s(Pc)is nite qǍand׵ifthebaseYadmitsametricofpGositivescalarcurvqature,MthenP;Dadmitsa bundlemetricwithpGositivescalarcurvqature.8^ThegeometryoftheLaplacianagainplays"acentralroleintheanalysisandagainthespGectralgeometryofRiemanniansubmersionsUUiscrucial. In]xȫ5.4,wwe]discusssomeunsolvedproblems.VInprevioussections,wwehaveshownthat}eigenvqaluesdonotchangeonfunctionsandcanchangeonformsifthedegreeisatleast2.W*edonotknowwhatthesituationisregarding1forms.W*ealsodiscussUUsomeunresolvedproblemsinthespinorcontext.5CZG!Îcю鍵5.2TManifoldswithpQositiv9escalarcurv\rature LetYbGeacompactconnectedRiemannianmanifoldwithoutboundaryofdi-mensionֶm.W*eassumem5ֶforthemostparttoensurethatcertainsurgery qǍargumentswork.LetR#<:=Rijgji"bGethescalarcurvqature.LetBq(r;Pc)denotethe 7geoGdesic\XballofradiusruaboutapointPofY0[soM3=S^2[orM=RPnT2.VtConversely*,of[course,bGoththesemanifolds)admitmetricsofpGositivescalarcurvqature.SupposethatYbګadmitsa [Qspinhstructures, Y`seexͫA.2fordetails.LetA^SbYn9;s/ګbGetheDir}'acoperatorhandlet D^GSbYn9;s .:=EX(A^SbYn9;s ֫)^2 PbGeJthespin9L}'aplacian.IfthescalarcurvqatureRispositive, TTheorem 2.6.5showstherearenoharmonicspinors;bthisusedtheLichnerowiczformuladiscussedUUinx2.6.4. This(observqationcanbGeusedtoconstructanobstructiontotheexistenceofmetricsvofpGositivescalarcurvqature.+Ifthedimensionmisdivisibleby4,4>wemayYdecompGosethespinorsintopGositiveandnegativechirality*.ThisdecompGoses [QA^SbYn9;s=A^SbYn9;s;+F+8A^SbYn9;s;,whereRq%AS፯Yn9;s;.:C1 0(S)!C1(S):kP .TGilk9ey,J.Leah9y,JH.P9arkÅl1076Y/ɧW*eUUde ne x㍍x䍑R4^OAW(Y9;s):=dimnker"x(AS፯Yn9;s;+Kf)8dim6ker!(AS፯Yn9;s;g׫);thisGistheindexofthespincomplex.mTTheAtiyah-SingerGindextheoremexpresses jx䍑^A(Y9;s)zasapGolynomialinthePontrjaginznumberszofMŕsox䍑R ^A .{(M)isdetermined qǍbythedi erentiablestructureofthemanifoldandisindepGendentoftheparticular spinTstructureschosenortheparticularmetricg[٫.8LetpiarethePontrjaginclassesofUUY8.qIndimensionsm=4UUandm=8,UUwehave:mx䍑'^FA"Ɨ(Y84W)=cZUR yY4a<$4133wfe  (֍24 fh=p1|s(TcY8)UUandx䍑j^AG (Y8)=cZUR yY8<$1wfe (֍5760+9ˮ=(7p21S84p2|s)(TcY8):W*el refertoHirzebruch[98]forfurtherdetails.ThediscussiongivenabGoveshows thatUUifx䍑^A V(Y8)6=0,UUthenY9doGesnotadmitametricofpositivescalarcurvqature. m TheKummersurfaceK^49&isacomplexhypGersurfaceofrealdimension4inCPZT3`.ItUUisde nedbythehomogeneousequations͍Įzp40%+8zp41+zp42+zp43"=0:pԍThis&4dimensionalmanifoldisasimplyconnectedcompactspinmanifoldwith jx䍑^A(K^43)=2.qThusUUK^4doGesnotadmitametricofpositivescalarcurvqature. qǍ IfUUm1moGd8wereducethedimensionofthekernelmoGd2tode nex䍑r/s^oAw (Y9;s):=dimnker"x(DGS፯Yn9;s ֫)2Z2|s;EthisSisindepGendentofthemetricchosenbutdoGesdependonthespinstructure.Similarly*,UUifm2UUmoGd8,wede ne-x䍑m^k[Ar\(Y9;s):=<$K1Kwfe (֍2 'dim-}ker+(DGS፯Yn9;s ֫)2Z2|s:Again,this;isindepGendentofthemetricbutdependsonthespinstructure.zW*e jde nex䍑^UUA V(Y9;s)=0UUintheremainingdimensions. qǍ IfAY%admitsametricofpGositivescalarcurvqature,thentherearenoharmonicspinorspandx䍑^A sq(Y9;s)Κ=0.LGromovpandLawson[81,82]conjecturedtheconversemight-holdinthesimplyconnectedsetting;seealsorelatedworkofSchoGenandY*auD[172].?Notethatif1|s(Y8)istrivial,dthereisonlyonespinstructure.This conjectureUUhasbGeenestablishedbyStolz[179]whoproved:5.2.1ٳTheorem.L}'etd3Ybeaconnectedcompactspinmanifoldofdimensionmwhichisatle}'ast5. Thenx䍑J^A ~(Y8)=0ifandonlyifY7admitsametricofpositivesc}'alarcurvature.э Note7thatthereisnoobstructionifm?K3;5;6;77moGd8.If1|s(Y8)?K6=0,o|thesituationisabitmorecomplicated.W*esuppGosethefundamentalgroupis nitecyclic8tosimplifythediscussion.>W*ereferto[30,>116]fortheproGofofthefollowingresult:l"ChapterTFiv9e:pPositiveCurv\ratureM1086Y/ɧ5.2.2ٳTheorem.L}'etd3Ybeaconnectedcompactspinmanifoldofdimensionm jwhichXisatle}'ast5with1|s(Y8)=Znq~.Thenx䍑^XA ؃(Y9;s)=0Xforallspinstructuresson qǍY˺ifandonlyifYadmitsametricofp}'ositivescalarcurvature.W TherebBareanaloguesofthistheoremifY&isnotspin,e}butiftheuniversalcoverofmwY[isspin.,W*ereferto[29,s117]fortheproGofofthe rstassertionoffollowing resultUUandto[9,65]fortheproGofofthesecondassertionofthefollowingresult.A5.2.3Theorem.L}'etYbeaconnectedcompactmanifoldofdimensionmwhich &isatle}'ast5with1|s(Y8)=Zn ewhoseuniversalcoverx䍑u~Y.isspin.98(1)!pAssume8Yisorientable.\Thenx䍑3ɫ^A 9(x䍑~Y)=08ifandonlyifYadmitsametric!pofp}'ositivescalarcurvature. r8(2)!pAssumeYisnotorientablebutw2|s(Y8)U=0. Assumem2mo}'d4. Then ㍍x䍑#^!pA(q(x䍑~Y)=0ifandonlyifY˺admitsametricofp}'ositivescalarcurvature.ɍ There3aremanyrelatedresults. \TheGromov-LawsonconjecturehasbGeen provedforsphericalspaceformgroupsandforashortlistofin nitegroupsin-cluding^freegroups,freeAbGeliangroups,andfundamentalgroupsoforientable qǍsurfaces.~W*eYreferto[9,131,163,164,165,167,168,169,179,180]YforotherrelatedUUpapGers. W*e{sketchtheproGofofTheorem5.2.1brie y*. =9Weconsidertriples(Y9;s;f)wherejYNisacompactspinmanifoldofdimensionm,wheresisaspinstructureon[֮Y8,uandwheref&:|Y{7!BqZn Tgives[֮YaZnstructure;if1|s(Y8)|=Znq~,uthere isacanonicalstructuref.vLetMSpinm(BqZnq~)bGethebordismgroups;thesearede ned]byintroGducingtheequivqalencerelation(M1|s;s1;f1)(M2|s;s2;f2)]ifthere qǍexists~acompactmanifoldNwithbGoundaryM1 disjointunionM2sothatthe structures$9extendover$9N.rStandardsurgerytheoryargumentsandtheworkofGromovUUandLawson[81]andSchoGenandY*au[172]showthatA5.2.4uTheorem.L}'etM+3beaconnectedspinmanifoldofdimensionmwhichisatele}'ast5withcyclicfundamentalgroupZnq~.RSupposethereexistsamanifoldM1whichPadmitsametricofp}'ositivescalarcurvatureso[(M;s;f)]=[(M1|s;s1;f1)]PǺin MSpinm(BqZnq~).ThenMadmitsametricofp}'ositivescalarcurvature.A ThisYresultshowsthattheGromov-Lawsonconjecturereducestoaquestionin/equivqariantspinbGordism.TAllthetorsioninthecoecientringMSpinm gʫis2torsion;thusonlytheprime2enters.0W*eassumehenceforththatthefundamentalgroupUUZnӫisacyclic2group. TheIspinbGordismgroups,unfortunately*,areIstilltoolargetodealwith.Itis KatythispGointthatthegeometryofRiemanniansubmersionsenters.X4LetHP T2NbGequaternion\>pro8jectivespacewiththeusualhomogeneousmetricofpGositivescalarcurvqature.qLetMRHP0T2tm!Z~4!YbGea berbundlewherethetransitionfunctionsarethegroupofisometriesPcSp(3) mofɤHP4T2).δW*eɤassumegivenaZn ;"structureonthebase.SinceHP4T2Kissimplycon-nected,%thisbinducesacanonicalZn structureonthetotalspaceEandallZnm֟P .TGilk9ey,J.Leah9y,JH.P9arkÅl1096Y/ɧstructures(onE3ariseinthisfashion.5cLetTm(Znq~)bGethesubgroupofMSpinm(Znq~) generatedbythetotalspacesEofsuch brations.QLetkPom(Znq~)bGetheconnectiveK qtheoryUUgroups.qResultsofStolz[179,180]thenshowthat )؍oԁkPom(BqZnq~)=<$KMSpinm(BZnq~)KwfeC (֍ FTm(BZnq~)H&:̍ThefollowingLemmashowsthattheelementsofTm ~varerepresentedbymanifolds thatFadmitmetricsofpGositivescalarcurvqature.lTheyarethereforeirrelevantandthe4questionoftheexistenceofmetricsofpGositivescalarcurvqaturethenreducesto>computationsinconnectiveKʫtheory*.-ItisbGeyondthescopGeofthepresentmonographotogivethesearguments;ntheyinvolvetheetainvqariantwhichisanotherspGectral{invqariant.)DW*ewillcontentourselves,Ntherefore,withestablishingthatsuch brationsadmitmetricsofpGositivescalarcurvqatureasthisisacrucialpointinthe[fdiscussion.Infact,\itturnsoutthatitisnecessarytostudytheetainvqariantofsucha bration;PthisisanotherexampleinwhichthespGectralgeometryofaRiemannianh~submersionarises.AW*eshallomitadiscussionoftheetainvqarianth~asitUUwouldtakeusrelativelyfara eld.EF5.2.5Lemma.L}'etX&:+MZi!YmbeXageometricalHP<T2 berbundle.{ThenthereexistsametricgZ BonZKwithp}'ositivescalarcurvature.Pr}'oof.pLetgF GbGethestandardmetricofpositivescalarcurvqatureonthe berHPT2.LetXgY DΫbGeanyRiemannianmetriconthebase.{]LetFxakbethe berofZoverXany qǍpGointOxofY8. SincethestructuregroupisPcSp(3),NweuseBesse[22,N9.59]toseethatUUexistsametricgZ ŰonthetotalspaceZ qsothat: 8(1)!TheUUinducedmetriconeachFx^9isgF.̩8(2)!EachUUgF 5istotallygeoGdesic.8(3)!TheUUmap":Z~4!Y9isUUaRiemanniansubmersion.Let)VandH߫bGetheverticalandhorizontaldistributionsofthesubmersion.DW*e de nethecanonicalvqariationgZp[(t)ofthemetricbyimpGosingthethreeconditions:e8$gZp[(t)jV5=tgF;gZ(t)jH ӫ=[ٟ(gYG);andqǮgZ(t)(V};H)=0:LetUURF 5andRZp[(t)bGethescalarcurvqatureofthemetricsonFandonZ.qThenՍ}oWRZp[(t)=t1 tRF+8OG(1);seeUUBesse[22,9.70].qInparticular,RZp[(t)!1UUast#0: nȟChapterTFiv9e:pPositiveCurv\ratureM1106YOn썟ZG!Îcю鍵5.3TManifoldswithpQositiv9eRiccicurv\rature e LetBY&bGeacompactconnectedRiemannianmanifoldwithoutboundary*.KIffeiTLg qǍisO9anorthonormalbasisforthetangentspaceTPM,Prandifu;"2TPM,Prwede ne theUURiccitensor ׍w)(u;[٫):=X tׯi㉮Rǫ(;eiTL;ei;[٫);}GthisөisasymmetricbilinearformindepGendentofthechoiceoffeiTLg.Ifthisform is+pGositivede nite,=i.e.Iif(u;)Ѯ>0+forr>ѫ0,thenYHissaidtohave+pGositive qǍRicci`curvqature.yThisconditionimpGosesrestrictionsonthetopologyofY8.yIfY֫haspGositiveRiccicurvqature,ٙthenMeyer'stheoremimpliesthefundamentalgroupO1|s(Y8)is nite.BNotethattheconverseOimplicationfails;=therearesimplyconnectedqRiemannianmanifoldswhichdonotadmitmetricsofpGositiveRicci curvqature.XF*or example,asnotedinx|v5.2,theKummersurfaceK^4>VThefollowingconversewasprovedbyGilkey*, Park,gandd$T*uschmann[75];ktheproGofofthisresultformsthefocusofthissectioninvolving,UUasitdoGes,asigni cantamountofspGectralgeometry*.5.3.1rTheorem(Gilk9ey-Park-T uschmann).L}'et޸gY beametricofpositiveRic}'cicurvatureonacompactconnectedRiemannianmanifold.HLetP5beaprincipalbundleoverY(ewithc}'ompactconnectedstructuregroupG.cAssumethat1|s(Pc)is nite.J-Thenther}'eexistsaGinvariantmetricgP PonP :withpositiveRiccicurvaturesothat":(PG;gP)7!(Y9;gYG)isaRiemanniansubmersion. IfethestructuregroupGadmitsametricwithpGositiveRiccicurvqature,#thisresultfollowsfromworkofBGerardBergery[[12];the bGerscanbetakentobetotally͒geoGdesicinthisspecialcase.~BGerardBergeryalsoshowedthatundertheassumptions?ofTheorem5.3.1,z0thatP-always?admitsametricofpGositiveRiccicurvqature.mjHowever,JhisH>argumentdoGesnotyieldaGinvqariantmetricnordoGesityieldUUametricsothat.aRiemanniansubmersion. e The9;caseofacirclebundleisthecrucialoneintheworkofBGerardBergeryandFwesummarizehisargument:S,LetLbGeacomplexlinebundleoverY~with assoGciatedcirclebundleS;oƮ1|s(S)is niteifandonlyif06=c1(L)inH^2Lq(Y8;R).ByjLemma1.11.4,wemaychoGoseaconnectiononLsotheassociatedcurvqature2-form(F&isharmonic.ByLemma1.10.1,]wemayusetheconnectiontode nea}splittingofTc(S)X=VwH3into}theverticalandhorizontalsubspacesofthe ٍpro8jectionث:)S!Y8.lLetds^2bH bGethepullbackofthemetriconYūtothe horizontalh|distributionH.>&0,theRiccitensorofgS BispGositivesemi-de nite.ySince1|s(S)is nite,UUc1|s(L)6=0sothereexistsy0C2Y9sothatF9(y0|s)6=0:AtthispGoint,theRiccicurvqatureofgS mispGositivede nite.YpAubin[8]hasshown qǍthatamanifoldS.YwhichadmitsametricwithpGositivesemi-de niteRiccitensor whichCispGositivede niteatapGointadmitsametricwithpGositiveRiccicurvqatureeverywhere;UUthisresultthenshowsPadmitsametricofpGositiveRiccicurvqature. The dicultywiththisproGofisthatwhentheworkofAubinisapplied,86theGinvqarianceofthemetricislostandd«isnolongeraRiemanniansubmersion.Consequently*,rwexshallproGceedabitdi erently.ۧThecaseofaprincipalcirclebundlebSCiscrucial.W*echoGoseaconnectiononSwithharmoniccurvqatureFto)splitthetangentspaceofSvintohorizontalandverticalsubspaces.BHowever,instead?Jofconsideringaconstantrescaling,CweconsiderametriconS׫oftheformyods2፯S:=e2(C1 +%))(e1|s)2S+8ds2፲HY; i.e.webBrescalethevolumeofthecircle bGersbyavqaryingconformalfactor.Here!isasmoGothfunctiononYandC1Xand%areparameterstobespeci edlater.LetUUOrbGeanon-emptyopensetsothat؍~JjF9j2C1>0UUon8OG:OnOG,theRiccitensorispositiveforC1 ]v>>0.: OnO^cY,wewillchoGosetogivethepGositivityoftheRiccitensorfor%small.AThusacrucialfeatureofour qǍconstruction,KinIcontrasttothatgivenin[12]isthatthe bGershavenon-c}'onstantvolumesMsothe bGersarenotminimalandhencearenottotallygeodesic.This proves>Theorem5.3.1inthecasethatG=S^1.The>generalcasewillfollowbyconsidering rstG=HwhereHistheconnectedcompGonentofthecenterofGandthenUUbyconsideringatowerofcirclebundles. The-remainderofthissectionisdevotedtotheproGofofTheorem5.3.1.ULetYvadmitגametricofpGositiveRiccicurvqature.}W*ebeginwithatechnicallemmafrom0 algebraictopGologyconcerning.LetPbeaprincipletorusbundleover0 Y8. qǍLetϮLiKbGetheassociatedlinebundles.V5W*eshowthat1|s(Pc)is niteifandonly ٍiftheChernclassesc1|s(LiTL)arelinearlyindepGendentinH^2Lq(Y8;R).Thenwegiveseveral[technicalresultsregardingthegeometryofcirclebundles.TheseLemmasfollowLfromtheO'Neillformulas[150].nW*eshallgiveaself-containedproGofoftheformulas=whichweshallneedfortheconvenienceofthereader.x~Theyarequite straightforward;inQanyevent,Ϲitisnomorediculttoderivethemdirectlythantoadapt9theO'Neillformulastothesettingathand.hW*ealsorefertothediscussion qǍin5Besse[22]).hW*ewillusethesetechnicalresultstoproveTheorem5.3.1inthe ٍspGecialmcasethatG=S^1.W*emthengeneralizethisargumenttotorusbundlesofhigherrankand nallytogeneralcompactconnectedLiegroupstocompletetheproGofofTheorem5.3.1.qXW*earegratefultoProfessorTuschmannforgivingus,pGermissiontopresentthejointworkwiththe rstandthirdauthorsinthispresentUUmonograph.pzChapterTFiv9e:pPositiveCurv\ratureM1126Y/ɧ5.3.27Lemma.L}'et}(Y9;gYG)beacompactconnectedRiemannianmanifoldwith qǍp}'ositiveoRiccicurvature.LetP3beaprincipaltorusbundleoverYwithassociatedc}'omplexlinebundlesL1|s,3...,Lrm.qThen1(Pc)is niteifandonlyifthe rstChern ٍclassesc1|s(LiTL)ar}'elinearlyindependentinH^2Lq(Y8;R).RPr}'oof.pThe6fundamentalgroupofYMis nite.\Bypassingtoa nitecover,!-̮C1 s(y[٫)=h_XfC0|s(y[٫)(I«+8C0(y)1 tC1(y))g1$`:=h_XfI±8C0|s(y[٫)1 tC1(y)8+OG(jyj2|s)gC0(y)1`:=h_XC0|s(y[٫)1 T8C0(y)1 tC1(y)C0(y)1 T+8OG(jyj2):r=ChapterTFiv9e:pPositiveCurv\ratureM1146Y/ɧApplyingthistoC0C=diag[٫(e^2f +;1;:::;1)andtoC1=ds^2bS>sC0oyieldsassertion(1). AThe# rstidentitiesof(2)areimmediate.aW*euseassertion(1)toraiseindicesand deriveassertion(2)fromLemma5.3.3;'notethatabc 핫(y0|s)=0.ZT*oproveassertion(3),UUwecomputesttaƟt(4=tat g[ٯttS e+8tab ` g[ٯtbS ,=@ap(f) l1l&fes2e^2f +AbD(@aAb=$@bDAa)+OG(jy[ٱj^2|s):ZW*eKdi erentiatethisexpressionwithrespGecttoaandevqaluateaty0 1toprove qǍassertionUU(4).qTheproGofofassertion(5)issimilar:<8abZt"=abt ` g[ٯttS e+8abc 핮g[ٯctSuD= K1K&fes2 )e^2fkf@ap(e^2f +AbD)8+@b(e^2f +Aap)g+OG(jy[ٱj^2|s)溍D= K1K&fes2 )(@apAb=$+8@bDAaP+2@a(f)Ab=$+2@bD(f)Aap)+OG(jy[ٱj^2|s)Zandw%ݍ<ڮ@apbbt ޫ(y0|s)8@bDabZt (y0)#IZ= K1K&fes2 )f2@ap@bDAb=$8@b@aAb=$8@b@bAaR̼+84@bDfLo@apAb=$2@afLo@bDAb=$2@bfLo@bAap)g(y0|s):%h썫W*eUUproveassertionsthe naltwoassertionsbycomputing:3Pz,3bba&=8Ybg[ٟad [bbd +8g[ٟat7bbt T=bbaرAapbbt ƾ+OG(jy[ٱj2|s)j0D=8YbYbbaث+ l1l&fes2e^2f +f2@bD(AbAap)8@a(A^2vb|s)2Aa@bDAb)g8+OG(jy[ٱj^2|s);and~ZabZa<=8Ybg[ٟad [abdF+8g[ٟat7abt'"=abaAapabt +OG(jy[ٱj2|s)0D=8YbYabZaj+ l1l&fes2e^2f +f@bD(A^2፯ap)8Aa(@aAb=$+@bDAa)g: UVS(T;Fc)^2|s.t%ChapterTFiv9e:pPositiveCurv\ratureM1166Y/ɧ W*e rstproveTheorem5.3.1inthespGecialcasethatG=S^1.FHChooseaunitary connectionFonthecomplexlinebundleLsothecurvqatureFisharmonic.ߚSince c1|s(L)UdoGesnotvqanishinH^2Lq(Y8;R),ձFdoesnotvqanishidentically*.sSinceFis qǍharmonic,UU`FQ=0.T[8(1)!ChoGoseUU>0soS(FG;Fc)8Fc.#8(2)!ChoGoseUUanon-emptyopensetOrand1C>0sothatjF9(y[٫)j^21ȱ8y"2OG. J8(3)!ChoGoseUUasmoothfunctionsothat(y[٫)1UUfory"62O.8(4)!LetUUfڧ:=C1S+8%.qChoGoseC1ȫsoe^2f +jint q(Fc)F9j^2C< MK&fes3 F8%2(0;1).ᇍ8(5)!ChoGoseUU2C>0soe^2f +=42ȱ8%2(0;1).#8(6)!ChoGoseۮ%0 4>0soj%;FFOje+j%^2|sFc()^2j >lҟ&fes33S8F|jifۮ%2(0;%0). YThen ˍ!S(FG;Fc) MK&fes3 ).8(7)!ChoGosepKsothatS(T;T)1|s2 K%ponO,vS(T;T)%K%^2qon ٍ!OG^cY,UUandsothatS(T;Fc)^2CK%^2|s.ItEnowfollowsthatS(T;T)S(FG;Fc)EdominatesS(T;Fc)^2pforsmallvqaluesof% qǍandUUthusS \ispGositivede nite.Ⓧ NextweproveTheorem5.3.1ifGisatorusofrankrG;weuseinductiononrG. LetǮL1|s,3#...,LrtibGethecomplexlinebundlesde nedbythetorusbundlePc.Let T^rXK:=ꩮS^1WG:::S^1zbGejtherɫdimensionaltorus.Assume1|s(Pc)is nite.LetPr71bGethetorusbundleofrankrq:*1de nedbythecomplexlinebundlesL1|s,...,Lr71. GW*eUUusethelongexactsequenceofthe bration7:S1!P*!Pr71P썫toKsee1|s(Pr71)is nite.TxByinduction,wemay ndametricgr71onPr71sor71%:JPr71!YΫisaRiemanniansubmersion,sogr71NhaspGositiveRiccitensor, y.andUUsothatgr71kisinvqariantUUundertheactionofT^r71. ) W*e!workequivqariantly*.+Letx䍑H[~Lp:=^[ٲlr71Lrm;|thiscomplexlinebundleadmitsa naturalT^r71action.LLetx䍑?~r0bGeanarbitraryunitaryconnectiononx䍑~L Ϋ.W*eaverage &x䍑~r0ggovertheactionofT^r71?toconstructaT^r71invqariantconnectionx䍑@H~r1gg.2LetF1 ӍbGeOthecurvqatureofx䍑~r1!«;JthisisinvariantOundertheT^r71action.aLetFN1bGetheharmonicXrepresentativeofc1|s(x䍑.:~L:).zSinceT^r71+isaconnectedLiegroup,itacts ٍtriviallyponthecohomologyofPr71.SinceT^r71actsbyisometries,itpreserves y.theWharmonicformsandthusFisT^r71Tminvqariant.pW*edecompGoseFR=TűF1+qd![٫;byUaveraging!overT^r71,Uwemayassume!isT^r71invqariant.sFLetx䍑~rs:=x䍑r~r1+95![٫.Thenx䍑~BcrisBcaT^r71yinvqariantBcconnectionwithharmoniccurvatureF9.8ChoGoseO so,thatjF9j^2H}1>0,onOG.HKSinceFeisT^r71Binvqariant,awe,canchooseO9Itobe ٍT^r71invqariantېandhenceOG^c5isT^r71invqariant.xChoGoseېsoϱ1ېonO^cY;by averagingT^r71invqariantandhenceT^rinvqariant.jKSincethecompGositionofRiemannian submersionsn|isaRiemanniansubmersion,theremainingassertionsofTheorem5.3.1UUnowfollow. W*e)cannowcompletetheproGofofTheorem5.3.1forgeneralgroups. LetCbGe'theconnectedcomponentofthecenterofacompactconnectedLiegroupG.SinceLChisacompactconnectedAbGelianLiegroup, CQΫ=T^rAisatorusofrankr &forBsomerG.kLetP*!Y{ɫbeaprincipalGbundlewith1|s(Pc) nite.kLetx䍑~G ɫ:=G=C ԍand5letx䍑0~Y ޫ:=YPV=C.yfThenthenaturalpro8jectionfromx䍑0~YatoYde nesaprincipalx䍑C~G Ibundle.rSincel9Gisreductive,x䍑~qGissemisimplesox䍑~G>admitsametricofpGositiveRiccicurvqature.SqTheargumentinBGerardBergery[12]showsthatx䍑$~YH9admitsax䍑K~GinvqariantզmetricwithpGositiveRiccicurvaturesothat̫:(x䍑~Y;gC7~YG)!(Y9;gYG)զisa RRiemannian@submersionwithtotallygeoGdesic bers.jThenaturalpro8jectionfromP,etox䍑Ld~֮Yɫde nesaprincipalT^r6xtorusbundle.KThestructuregroupGactsonthis principal:bundlesinceCiscentral.hW*eaverageoverthegroupGateachstageinapplyingy*theconstructionusedtoprovey*Theorem5.3.1inthecaserI>ѫ1andtheremainderUUoftheargumentisthesame.  A TheproGofofTheorem5.3.1useswhatmightbecalledaKaluza-Kleinansatz.W*eUUconcludethissectionwithabriefhistoricalsummaryofsomerelatedwork. ݍ Nash]Z[149],_\PoGor[158],andBGerard-Bergery[12]usedRiemanniansubmersions qǍwithGtotallygeoGdesic berstoconstructmetricsofpositiveRiccicurvqatureonthetotalspacesofcertaincompact bGerbundlesandoncertainvectorbundlesofrankatleast2.aThesameconstructionwasusedbyD'Atri&Ziller[41],UJensen[104], andbyW*ang&Ziller[191]toconstructleftinvqariantandbiinvqariantEinsteinmetrics withpGositivescalarcurvqatureoncertaincompactLiegroupsGwhereGwasUUviewedasaprincipalH%SbundleH!G!G=H.A Severalotherauthors[36,*42,113,189]constructedEinsteinmetricswithpGosi-tivefscalarcurvqatureonthetotalspacesofprincipaltorusbundlesoverproGductsofKaehler+EinsteinmanifoldswithpGositive rstChernclass.cCheeger[38],4%Derdzin-ski)&Rigas[44],2andStrake&W*alschap[183]usedthesamemethoGdtoconstructmetricsn ofnonnegativesectionalcurvqatureoncertainmanifolds.W*einstein[198]studiedsymplecticstructuresonfatbundles.M@Aprincipalingredientinthecon- structionsnusedbyalltheseauthorsinvolvedrescalingthe bGersbyaconstantfactor. A NonconstantwarpinghasalsobGeenused.iNabonnand[145]gaveexamplesof (non-compact)_acompletemanifoldswithpGositiveRiccicurvqatureandin nitefun-damentalggroup.BGerardBergeryz[15]showedthatifMiscompleteandadmitsa rmetric*withnon-negativeRiccicurvqature,(fthenMߥȊR^kcarriesametricwithpGos- itiveRiccicurvqaturefork3.\F*orexample,!ifM+ istheK^3Gsurface,thenM]BR^3 admitsametricwithpGositiveRiccicurvqaturebutnotpositivesectionalcurvqature.Gromoll&Meyer[80]usedwarpGedproductstoconstructmetricsofpositiveRiccicurvqature=oncertainnoncompactmanifolds.*Anderson[2,w4]andSha&Y*ang[172]GusedwarpGedproductstoconstructmetricsofpositiveRiccicurvqatureontheconnected OsumofcertainsimplyconnectedmanifoldswithpGositiveRiccicurvqa-vOChapterTFiv9e:pPositiveCurv\ratureM1186Y/ɧture.1W*ei[197]showedthatifLisasimplyconnectednilpGotentLiegroup, then qǍLIzR^p admitsnN߫dim(Y8).2Thisconstruction$iscompleteanalogouswiththeconstructionofthe rstChernclass giveninx 1.11.2W*eusetensorproGducttogiveV*ect 1R R֫(Y8)thestructureofanAbGeliangroup.rLet<x>bGeinhomogeneouscoordinates.rThemapwhichsendsthe `proGductF<Y3xi>tomakesRPT1[oandCPT1intoHgroups; \ptheseaare\groups"inthehomotopycategory*,dseeSpanier[178]fordetails.QThe H-groupstructureonRPcT1뇫givesthesetofhomotopyclasses[X:;RPT1x]agroup _Jstructure1.SRealpro8jectivespaceRPQTm僫isorientableifandonlyifmisoGdd ٍ!andspinifandonlyifm3moGd4.Z^W*ehaveH^1Lq(RP UVTm;Z2|s)=Z2sothere r!areUU2inequivqalentspinstructuresonRPT4jg1#>. .8(3)!ComplexL!pro8jectivespaceCPwTm3admitsaspinmstructureifandonlyifmis :!oGdd.qTheUUspinstructureUUisuniquesinceCPTmissimplyconnected.8(4)!Thegroupofn^th GroGotsofunityactsbycomplexmultiplicationontheunit r!sphereS^2k+B1WinC^k됫.PrF*ork> t2,thelensõsp}'aceL(kP;n)isthequotient!S^2k+B1=Znq~;2qthesemanifoldswerediscussedinxZ3.9.8.,Ifk9TisoGddandifn !isceven,&L(kP;n)doGesnotadmitaspinstructure;3L(k;n)admitsaspin qǍ!structureWifnisoGddorifknandnarebotheven.Thespinstructureis !uniqueUUifnisoGdd;therearetwoUUspinstructuresifniseven. .8(5)!TheproGductofspinmanifoldsisspin';qtheproductofspin'*c ׫manifoldsis d!spin2X*c69.y;TheWconnectedsumofspinmanifoldsisspin-);Ytheconnectedsumof *+!spin2X*c:=manifoldsUUisspin**c=.<ZG!Îcю鍵A.3TTheDiracopQerator 7g LeftCli ordmultiplicationde nesarepresentationofthespinorgroupSpin(m) onXtheCli ordalgebraClifA(m).IfMohasaspinstructure,NletCWMbGetheassociated qǍvector[ubundle;ޅthisisnottheCli ordalgebrabundlede nedbyTc^sM.&Right Cli ordmultiplicationcommuteswithleftCli ordmultiplicationsoCWM9inherits [Qa rightClif(m)moGdulestructure.YThereisanaturalconnectionr^C=onCWM$3called thespinc}'onnectionwhichispreservedbytherightClifKZ(m)moGdulestructure.LeftCli ordmultiplicationinducesanaturalmapcM fromTc^sMtothebundle of9endomorphismsofCWMTsothatcM\(uǫ)^2X=ۓjj^2|sI.ctRelative9toanorthonormal{lP .TGilk9ey,J.Leah9y,JH.P9arkÅl1236Y/ɧframeBfF\gforTcM]andrelativetotheinducedframeforCWM,theconnection1 EformUUA^C3ofr^C0ޫ,andtheopGeratorsA^CbM 5andD^CbM 5arede nedby'{(A.3.a=)GvAC:= b۬1b۟&fes4gFc^0 8cM\(Fc^]X)cM(Fc^%)^Mwi82Tc^sMO End#(CWM);jCAC፯M t:=acM <8rC:C1 0(CWM)!C1(CWM);andCDGC፯M t:=a(AC፯M\)2C:C1 0(CWM)!C1(CWM);(z鍮A^CbM ;andZD^GCbMareellipticself-adjointpartialdi erentialopGeratorswhichcommute AwithUUtherightClif(m)moGdulestructureonCWM. TherepresentationofSpinF(m)onthecomplexi cationoftheCli ordalgebraClifucV(m)byleftmultiplicationisnotirreducible._Ifm=2kp]iseven,*}thisrepresen- rtation-.decompGosesasadirectsumof2^kcopiesofafundamentalrepresentationS calledthespinrepresentation._LetSMbGetheassociatedvectorbundleandlet [QA^SbM bGe*theassociated rstorderoperator.EThenA^SbM istheDiracoperatorzTand *D^GSbM t:=(A^SbM\)^2ȫisUUthespinLaplacian.W*emaydecompGose{rCWc8M3:=CWMO RC=2k$p8SM^asf2^kcopiesofthefundamentalrepresentation;thisdecompGositioninducesdecom- pGositions >rc` ACYrc፯M ::=AC፯M < 81UUandDGCYrc፯M W:=DGC፯M 81~IasOthedirectsumof2^k߫copiesoftheopGeratorsA^SbM ۫andD^SbM ۫respectively*.cThus *thespGectraltheoryoftheoperatorsA^CbM andD^CbM isessentiallythesameasthespGectral3theoryoftheoperatorsA^SbM mandD^SbM\.f[Thereisasimilardecompositionif ㍮mUUisoGdd;weomitdetailsintheinterestsofbrevity*. ThereZaresigni cantadvqantagesinsomeinstancestoworkingwiththebundle [QCWMހandetheopGeratorsA^CbM andD^CbM ratherthanwiththespinorbundlesandtheGopGeratorsA^SbM ࣫andD^SbM\.UmTheClif(m)modulestructureenablesonetode nea re nedindexoftheDiracopGeratorwhichtakesvqaluesintherealKv4theorygroups, qǍseecHitchin[99];-jthisplayscrucialroleinthestatementoftheGromov-Lawson- RosenbGerg conjecture,+seeGromov-Lawson[81,+82]andRosenbGerg[163]fordetails;seeUUalsorelatedworkofSchoGenandY*au[172]. Whenworkingwithpro8jectablespinorsinxlX2.7.3, x~2.8, andx4.6, itismorecon- venienttoworkwiththefullCli ordbundle;wthecrucialdistinctionisthattheparitynofthedimensionmdoGesnotenterinde ningthebundleCWMwhereasthecasemevenisquitedi erentfromthecasemoGddwhende ningthespinorrepre-sentations.The useofCWM%ratherthanSMwillsimplifythisdiscussion.OntheotherChand,G:instudyingtheGromov-LawsonCconjecture,weCusethebundleS(M)asQitwastheonewearemorefamilarwithinthiscontext;RtheotherviewpGointisequallyUUprevqalantintheliteratureonthissub8ject.|AppQendix%,1246YOn썟ZG!Îcю鍵A.4TSpinorsonthetorusandsphere LetAx=(x^1|s;:::;x^rm)bGetheusualperiodicparametersonther^dimensionaltorus qǍT:=T^rm.dGive.zTthe atmetric.Since=0,6?relative.ztothenaturaltrivialization 4ofUUCWT^rinducedbytheframing@id:=@8=@x^iofUUTc(T^rm)wehaveqS(A.4.a=)bͿACT=iTLcT*(dxi)@iandkDGCT?=i@82፯i :W*eVImayidentifytheunitsphereS^3fIwiththeunitquaternions.tLete1|s,Ve2,andVIe3 bGetheusualbasisfortheLiealgebrag 5ofleftinvqariantvector eldsonS^3.cLet reivCejī=ijgk vek됫.qIfUU.isapGermutationoftheindices,d9):@L(1)(2)(3)09=sig[n();܍theremainingChristo elsymbGolsvqanish.oThisframingofTcS^3«de nesanaturaltrivializationUUofCW(S^3).qW*esummarizetheequationsofstructurebGelow:2s(A.4.bv)ߔ083e1|s(x)=x8i;e2(x)=x8jR;e3(x)=x8kP;#83[e1|s;e2]=2e3|s;[e2;e3]=2e1|s;[e3;e1]=2e2|s;j83ACvSa3 y?=cS(e1|s)e1S+8cS(e2)e2S+8cS(e3)e3S+83=2cS(e1)cS(e2)cS(e3);83DGCvSa3 y?=(e21S+8e22+e23|s) 13cS(e1)cS(e2)e3Nlr83cS(e2|s)cS(e3)e1S83cS(e3)cS(e2)e1S+ l9l&fes4:TލZG!Îcю鍵A.5TComplexgeometry ThereisacomplexanalogueoftherealLaplacian.Letz"8:=(zp^1 ;:::;zp^m 2)bGea qǍsystemޅofloGcalholomorphiccoordinatesonaholomorphicmanifoldMofcomplex dimensionumwherezp^j :=wx^j+LI$p $fe 1hy[ٟ^j. I'W*ecomplexifytherealtangentandcotangentUUbundlesandde neD(A.5.a=)kASsdzpjn[:=dxjo+8p 7fe X1UVdy[ٟj;v0dɫzpj :=dxjo8p 7fe X1UVdy[ٟj;jPsSdzpID:=dzpi1 ^8:::^dzpip;v0dɫzpJ s=dɫzpj1=^8:::^dɫzpjq ߓ;Sԫp;q (M):=SpanqǟqƲjIJj=p; ۟:jJj=qȄfdzpI@ ^8dɫzpJ αg;ڸQ @8zj:= K1K&fes2 )(@^8x;Zj8$p 7$fe 1UV@1ɍ8yˍj\);v0@ 8zj:= K1K&fes2 )(@^8x;Zj+8$p 7$fe 1UV@1ɍ8yˍj\);?jJ9(@8xj)=@1ɍ8yˍj\;J(@1ɍ8yˍj)=@8xj;v0J9(dxj6)=dy[ٟj;J9(dy[ٟj)=dxjOF@8P sIJ;J(fIJ;J EdzpI@ ^8dɫzpJ s:= ۟PjT;IJ;Jcή@8zj)ɫ(fIJ;J E)dzpj#^8dzpI@ ^dɫzpJ ή;G!\qPGOF@U_P_֟IJ;JkfIJ;J EdzpI@ ^8dɫzpJ s:= ۟PjT;IJ;Jcή@ 8zj)ɫ(fIJ;J E)dɫzpj ^8dzpI@ ^dɫzpJ ή:} P .TGilk9ey,J.Leah9y,JH.P9arkÅl1256Y/ɧAUUcomplexfunctionfhisholomorphicifandonlyif\qV@ 2fڧ=0.qSinced=@+\q|8@5,+'e"@8@UP=0;\qC@ R@+8@\q9@ 2=0;and\qqǮ@\qO@&=0:OAUURiemannianmetricgM 5isHermitianifZz?gM\(X:;Y8)=gM(J9X:;JY8)foraallrealtangentvector elds.[W*eextendsuchametrictothecomplexi ed tangent&ubundletobGecomplexlinearinthe rstfactorandconjugatelinearinthesecondUUfactor.qW*ede netheDolb}'eaultLaplacianUUby+'aqtp;q :=(\qD@\q!VU@8n+\q|8@8\q  @)UUon8C1 0p;q (M):ZLetUUbGearealcotangentvector.qW*edecompGose;<߫=uǟ(1;0)Ǐ+8uǟ(0;1)into_formsofbi-degrees(0;1)and(1;0);thesetwo_formsarecomplexconjugates qǍofUUeachother.qSinceisreal, ;\q׃uǟ(0;1)K=uǟ(1;0):ZW*eUUuseequation(A.5.a)toseethat!⍍n]j?Lt(\qD@U)(uǫ)=extc((0;1))UUand$]j?Lt(\qD@8 v9)(uǫ)=L(\qD@U)(uǫ)=int q((1;0)):!;W*eUUde ne IPpwc(uǫ)=extc((0;1))8int*((1;0)):ThisUUyieldsaCli ordmoGdulestructuresince}cc(uǫ)2C= 33133&fes2bٱjj^2|s:+Thus,]moGduloasuitablenormalizingconstant,\qx4@ ӫ+\q*@8^isanopGeratorofDiractypGe ȍandUU^(p;q@L)>isanopGeratorofLaplacetype. Mp LetV":Z~4!Y:bGeaRiemanniansubmersion.3Inthecomplexsetting,weassume thatNPZlandY4areholomorphic,Othat)isholomorphic,andthatthemetricsonZandUUonY9areHermitian.qThecomplexi cationofpullbackde nesZmi[ٟի:C1 0p;q (Y8)!C1p;q (Z):~AppQendix%,1266Y/ɧ TheklinearmapJäde nedinequation(A.5.a)satis esJ9^29ī=1andisanalmost qǍc}'omplexǟstructureonM.nNoteveryalmostcomplexstructurearisesfromacom- plex:structure;,thereisanintegrability:conditiondescribGedinthetheNir}'enberg-Neulanderťthe}'orem.LetfMbGeanalmostcomplexmanifold.ThismeansthatweassumeUUgivenalinearmap_jJQ:TcM3!TMlpsoUUthat&^;J929ī=1:W*eUUextendJKtothecotangentbundlebyduality;thusw(J9X:;G):=(X;J9G):W*eUUmimictheconstructiongiveninequation(A.5.a).qWedecompGoseeTcsMO 8C=Tc(1;0)|w(M)Tc(0;1)(M)wheref cTc(1;0)|wM3:=SpanqDZf8p 7fe X1UVJ9uDZgUUandTc(0;1)M:=SpanqDZf+8p 7fe X1UVJ9uDZg:W*eUUthende ne aV(p;q@L))(M):=pR(Tc(1;0)|wM)8 qj(Tc(0;1)|wM):;qThisUUgivesadecompGositionXrO(M)8 C=p;q (p;q@L))(M):0LetUU[ٟ^(p;q@L)denotethenaturalpro8jectionof^(M)8 CUUon^(p;q@L))(M).qW*ede ne>@8(p;q@L)9:=[ٟ(p+1;q@L)Ůd:C1 0(p;q@L))(M)!C1(p+1;q@L)I(M)and\q>_=^@8(p;q@L)Wa:=[ٟ(p;q@L+1)Ůd:C1 0(p;q@L))(M)!C1(p;q@L+1)I(M):Let $ʍZTcM3=SpanqDZfX±8p 7fe X1UVJ9XgTcMO 8C;qbGeuthecomplextangentbundle.(Thisisthespanoftheholomorphictangent vectors.qSimilarlyUUletPx䍑\ZTcdM3=SpanqDZfX«+8p 7fe X1UVJ9XgTcMO 8CbGeUUthespanoftheanti-holomorphictangentvectors.qW*ehave,$y>TcMO RC=TcMx䍑+8Tc #M:W*e~osayTcMisintegrableifthec}'omplex1(1964),UU3{38.[7]MFAtiyah,3VKPatoGdi,andIMSinger,Sp}'ectralasymmetryandRiemann-ianUge}'ometryI,II,III:,Math. ProGc.Cambr.Phil.Soc.[77(1975)43{69;'͵78(1975)UU405{432;*9T(1976)71{99.[8]ToAubin,LM$etriques4jriemanniennesetc}'ourbure,J.Di .O.Geo.4(1970),383{ 424.[9]EUUBarrera-Y*anez,Ph.qD.thesis,UniversityUUofOregon(1997). 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