The Energy-Momentum tensor on $Spi{n}^c$ manifolds

On $Spi{n}^c$ 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 $Spi{n}^c$ 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 $Spi{n}^c$ Killing spinors for Codazzi Energy-Momentum tensor are restrictions of parallel spinors.