The energy-momentum tensor as a second fundamental form

We show that it is natural to consider the energy-momentum tensor associated with a spinor field as the second fundamental form of an isommetric immersion. In particular we give a generalization of the warped product construction over a Riemannian manifold leading to this interpretation. Special sections of the spinor bundle, generalizing the notion of Killing spinor, are studied. First applications of such a construction are then given.