The Energy-Momentum Tensor on $Spin^c$ Manifolds

22 pagesInternational audienceOn Spin^c manifolds, we study the Energy-Momentum tensor associated with a spinor field. First, we give a spinorial Gauss type formula for oriented hypersurfaces of a Spin^c manifold. Using the notion of generalized cylinders, we derive the variationnal formula for the Dirac operator under metric deformation and point out that the Energy-Momentum tensor appears naturally as the second fundamental form of an isometric immersion. Finally, we show that generalized Spin^c Killing spinors for Codazzi Energy-Momentum tensor are restrictions of parallel spinors