Furthermore, a Riemannian symmetric space cannot admit parallel spinors. Using de Rham's Theorem we obtain a description of Riemannian manifolds admitting parallel spinor fields in terms of the holonomy group. On the other hand the pseudo-Riemannian situation is much more complicated. Here de Rham's theorem does not reduce the study of manifolds to that of irreducibles ones. It is possible that the holonomy representation is reducible but does not decompose into irreducible holonomy representations of semi-Riemannian manifolds.
Examples are manifolds with parallel orthogonal optical structures.
We will explain the relation between these structures and the existence of parallel pure spinor fields. Other results which will be presented concern a further pseudo-Riemannian phenomenon. Riemannian symmetric spaces which admit parallel spinors. We will study double extensions of Lie groups which yield a big class of examples for such symmetric spaces.