  On $Spin^c$ 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.
