We apply the Cartan-Kähler theorem for the k-Dirac operator studied in Clifford analysis and to the parabolic version of this operator. We show that for the tableaux of the first prolongations of these two operators are involutive. This gives us a new characterization of the set of initial conditions for the 2-Dirac operator.
We describe all compact spin Kähler manifolds of even complex dimension and positive scalar curvature with least possible first eigenvalue of the Dirac operator.
On a pseudo-Riemannian manifold we introduce a system of partial differential Killing type equations for spinor-valued differential forms, and study their basic properties. We discuss the relationship between solutions of Killing equations on and parallel fields on the metric cone over for spinor-valued forms.