About the Dirac operator

We apply a result of Kim about the eigenvalue estimation of the Dirac operator on a Riemannian compact spin manifold (M,g), considering M = N x S-1, where N is a Riemannian compact spin manifold admitting a parallel vector field. We show that the lower bounds given in a theorem of Hijazi and Zhang for the eigenvalues of the so called (submanifold) twisted Dirac operator D-H in the case when H not equal 0 is true for H = 0 also. As an example, we consider every spin Kahler manifold as a totally geodesic submanifold of its twistor space and we study its twisted Killing spinors.