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SpincStructuresonManifoldsandGeometricApplicationsRogerNAKADApril10,2012MaxPlanckInstituteforMathematicsVivatsgasse7,53111BonnGermanyE-mail:nakad@mpim-bonn.mpg.deAbstractInthismini-course,wemakeuseofSpincgeometrytostudyspecialhyper-surfaces.Forthis,webeginbyselectingbasicfactsaboutSpincstructuresandtheDiracoperatoronRiemannianmanifoldsandtheirhypersurfaces.WeendbygivingaLawsontypecorrespondenceforconstantmeancurvaturesurfacesinsome3-dimensionalThurstongeometries.Contents1Introductionandmotivations2Algebraicfacts3SpincstructuresandtheDiracoperator4Examplesandremarks5TheSchro¨dinger-Lichnerowiczformula6HypersurfacesofSpincmanifolds7Geometricapplications12358012141
1IntroductionandmotivationsHavingaSpinorSpincstructureonaRiemannianmanifold(Mn,g),wecandefineanaturalfirstorderellipticdifferentialoperatorcalledtheDiracoperator.Itactsonspinorfields:sectionsofacomplexvectorbundleΣMcalledthespinorbundle.ThegeometryandtopologyofaRiemannianSpinorSpincmanifoldandtheirsubmanifoldsarestronglyrelatedtothespectralpropertiesofthisoperator.OnacompactRiemannianSpinmanifold(Mn,g)ofpositivescalarcurvature,A.Lichnerowicz[Lich63]provedthatanyeigenvalueλoftheDiracoperatorDsatisfies1λ2>infScal,4MwhereScaldenotesthescalarcurvatureof(Mn,g).Then,thekerneloftheDiracoperatoristrivialandbytheAtiyah-Singertheorem,thetopologicalindexofMniszero.Thisyieldsatopologicalobstructionfortheexistenceofpositivescalarmetrics.Th.Friedrich[Fri80]refinedtheargumentofA.Lichnerowiczandprovedthatnλ2infScal.4(n1)MTheequalitycaseischaracterizedbytheexistenceofarealKillingspinor.Theexistenceofsuchspinorsleadstorestrictionsonthemanifold.Forexample,themanifoldisEinsteinandindimension4,ithasconstantsectionalcurvature.TheclassificationofsimplyconnectedRiemannianSpinmanifoldscarryingrealKillingspinors[Ba¨r93]gives,insomedimensions,otherexamplesthanthesphere.TheseexamplesarerelevanttophysicistsingeneralrelativitywheretheDiracoperatorplaysacentralrole.Fromanextrinsicpointofview,Th.Friedrich[Fri98]characterisedsimplycon-nectedsurfacesisometricallyimmersedinR3bytheexistenceofaspinorfieldsatisfyingtheDiracequation.Indeed,M2isasimplyconnectedSpinsurface(M2,g),R3carryingaspinorfieldϕofconstantnorm.ofmeancurvatureHsatisfying=|{z}TheDiracequationThespinorfieldϕistherestrictiontothesurfaceMofaparallelspinoronR3.AsimilarresultholdsforsurfacesinS3andH3[Mor05].Asanapplication,wehaveanelementaryproofofaLawsontypecorrespondence.H.B.Lawsonprovedacorrespon-dencebetweensurfacesofconstantmeancurvatureinR3,S3andH3:everysimplyconnectedminimalsurfaceinS3(resp.inR3)isisometrictoasimplyconnectedsur-faceinR3(resp.H3)withconstantmeancurvatureequalto1.In2001,O.Hijazi,S.MontielandX.Zhang[HMZ01a,HMZ01b]provedthatthefirstpositiveeigenvalueof2
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