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.

Simply connected 3-dimensional homogeneous manifolds E(κ, τ ), with 4-dimensional isometry group, have a canonical Spin c structure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces into E(κ, τ ). As application, we get an elementary proof of a Lawson type correspondence for constant mean curvature surfaces in E(κ, τ ). Real hypersurfaces of the complex projective space and the complex hyperbolic space are also characterized via Spin c spinors.

Abstract. We study boundary value problems for the Dirac operator on Riemannian Spin c manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by Ch. Bär and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound of Hijazi-Montiel-Zhang, involving the mean curvature of the boundary, for the spectrum of the Dirac operator on the noncompact boundary of a Spin c manifold, and the limiting case is studied.

In this paper, we extend the Hijazi inequality, involving the Energy-Momentum tensor, for the eigenvalues of the Dirac operator on Spin c manifolds without boundary. The limiting case is then studied and an example is given.

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