The Energy Momentum tensor on low dimensional Spinc manifolds

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The Energy-Momentum tensor on low dimensional Spinc manifolds Georges Habib Lebanese University, Faculty of Sciences II, Department of Mathematics P.O. Box 90656 Fanar-Matn, Lebanon Roger Nakad Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany On a compact surface endowed with any Spinc structure, we give a formula involving the Energy-Momentum tensor in terms of geometric quantities. A new proof of a Bar-type inequality for the eigenvalues of the Dirac operator is given. The round sphere S2 with its canonical Spinc structure satisfies the limiting case. Finally, we give a spinorial characterization of immersed surfaces in S2 ? R by solutions of the generalized Killing spinor equation associated with the induced Spinc structure on S2 ? R. Keywords: Spinc structures, Dirac operator, eigenvalues, Energy-Momentum tensor, compact surfaces, isometric immersions. Mathematics subject classifications (2000): 53C27, 53C40, 53C80. 1 Introduction On a compact Spin surface, Th. Friedrich and E.C. Kim proved that any eigen- value ? of the Dirac operator satisfies the equality [7, Thm. 4.5]: ?2 = pi?(M) Area(M) + 1 Area(M) ∫ M |T?|2vg, (1.1) where ?(M) is the Euler-Poincare characteristic of M and T? is the field of quadratic forms called the Energy-

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TheEnergy-MomentumtensoronlowdimensionalSpincmanifoldsGeorgesHabibLebaneseUniversity,FacultyofSciencesII,DepartmentofMathematicsP.O.Box90656Fanar-Matn,Lebanonghabib@ul.edu.lbRogerNakadMaxPlanckInstituteforMathematics,Vivatsgasse7,53111Bonn,Germanynakad@mpim-bonn.mpg.deOnacompactsurfaceendowedwithanySpincstructure,wegiveaformulainvolvingtheEnergy-Momentumtensorintermsofgeometricquantities.AnewproofofaBa¨r-typeinequalityfortheeigenvaluesoftheDiracoperatorisgiven.TheroundsphereS2withitscanonicalSpincstructuresatisﬁesthelimitingcase.Finally,wegiveaspinorialcharacterizationofimmersedsurfacesinS2×RbysolutionsofthegeneralizedKillingspinorequationassociatedwiththeinducedSpincstructureonS2×R.Keywords:Spincstructures,Diracoperator,eigenvalues,Energy-Momentumtensor,compactsurfaces,isometricimmersions.Mathematicssubjectclassiﬁcations(2000):53C27,53C40,53C80.1IntroductionOnacompactSpinsurface,Th.FriedrichandE.C.Kimprovedthatanyeigen-valueλoftheDiracoperatorsatisﬁestheequality[7,Thm.4.5]:Z2πχ(M)1ψ2λ=Area(M)+Area(M)|T|vg,(1.1)Mwhereχ(M)istheEuler-Poincare´characteristicofMandTψistheﬁeldofquadraticformscalledtheEnergy-Momentumtensor[13].Itisgivenonthecomplementsetofzeroesoftheeigenspinorψbyψ1Tψ(X,Y)=g(`ψ(X),Y)=Re(X∙rYψ+Y∙rXψ,2),|ψ|2foreveryX,Y∈Γ(TM).Here`ψistheﬁeldofsymmetricendomorphismsassociatedwiththeﬁeldofquadraticformsTψ.Weshouldpointoutthat1

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