Parallel spinors and holonomy groups

In this paper we complete the classification of spin manifolds admitting pa rallel spinors, in terms of the Riemannian holonomy groups. More precisely, we show that on a given n-dimensional Riemannian manifold, spin structures with parallel spinors are in one to one correspondence with lifts to Spin( n) of the Riemannian holonomy group, with fixed points on the spin represen tation space. In particular, we obtain the first examples of compact manifo lds with two different spin structures carrying parallel spinors. (C) 2000 American Institute of Physics. [S0022-2488(00)01302-5].