Space-times admitting a covariantly constant spinor field

It is shown that if a space-time V admits a covariantly constant spinor field psisub(A), and hence a covariantly constant null vector field lsub(μ) determined by psisub(A), its Ricci tensor is proportional to the tensor product of lsub(μ) by itself. Further, the conformal tensor of V is shown to be of Petrov-Penrose type N. That is the four index symmetric spinor product of psisub(A) with itself. The Bianchi identities are used to show that empty asymptotically flat space-times with infinitely extendible null geodesics tangent to lsub(μ) are flat