Twistor spinors and their zeroes

A generalization of the Einstein condition for Killing spinors is given for twistor spinors on Riemannian manifolds. We study the zeroes of twistor spinors on manifolds with parallel Ricci-tensor, in particular of Einstein manifolds. Furthermore, we consider the conformal deformation of the metric defined by a twistor spinor and find a condition for the completeness of this metric on the complement of the zero set. Examples are constructed for which the situation is realized. Finally, we obtain results concerning the conformal vector field defined by a twistor spinor. (orig.)