The Energy Momentum tensor on low dimensional Spinc manifolds

Georges Habib Lebanese

Description

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-

• spin


• spinor ?


• any orthonormal local


• spinc manifold


• orthonormal tangent


• ?y ? ?


• has


• ?m


• immersion into


• hermitian scalar

Sujets

• Planck
• Weingarten
• Lebanese
• multiplication

Informations

• Publié par: pefav
• Nombre de lectures: 24
• Langue: English

Extrait

TheEnergy-Momentumtensoronlow dimensionalSpin c manifolds GeorgesHabib LebaneseUniversity,FacultyofSciencesII,DepartmentofMathematics P.O.Box90656Fanar-Matn,Lebanon [e-mail] RogerNakad MaxPlanckInstituteforMathematics,Vivatsgasse7,53111Bonn,Germany [e-mail] OnacompactsurfaceendowedwithanySpin c structure,wegiveaformula involvingtheEnergy-Momentumtensorintermsofgeometricquantities.A newproofofaBa¨r-typeinequalityfortheeigenvaluesoftheDiracoperator isgiven.Theroundsphere S 2 withitscanonicalSpin c structuresatisfiesthe limitingcase.Finally,wegiveaspinorialcharacterizationofimmersedsurfaces in S 2 × R bysolutionsofthegeneralizedKillingspinorequationassociated withtheinducedSpin c structureon S 2 × R . Keywords:Spin c structures,Diracoperator,eigenvalues,Energy-Momentum tensor,compactsurfaces,isometricimmersions. Mathematicssubjectclassifications(2000):53C27,53C40,53C80. 1Introduction OnacompactSpinsurface,Th.FriedrichandE.C.Kimprovedthatanyeigen- valueλoftheDiracoperatorsatisfiestheequality[7,Thm.4.5]: Z 2 πχ(M)1 ψ2 λ= Area(M) + Area(M) | T | v g ,(1.1) M whereχ(M)istheEuler-Poincare´characteristicofMandT ψ isthefieldof quadraticformscalledtheEnergy-Momentumtensor[13].Itisgivenonthe complementsetofzeroesoftheeigenspinorψby ψ1 T ψ (X,Y)=g(` ψ (X),Y)=Re(X ∙r Y ψ+Y ∙r X ψ, 2 ), | ψ | 2 foreveryX,Y ∈ Γ(TM).Here` ψ isthefieldofsymmetricendomorphisms associatedwiththefieldofquadraticformsT ψ .Weshouldpointoutthat 1