TITLE: Spin manifolds with one parallel spinor
QUESTION [7 upvotes]: Are there any examples of D-dimensional Ricci-flat Riemannian (spin) manifolds of dimension D= 2,3,4,5 with the dimension of the space of parallel spinors equal to 1? And the same question for the pseudo-Euclidean case with dimensions D= 4,6. Here the manifolds are not assumed to be compact or complete, or simply connected.

REPLY [5 votes]: In the split cases of interest to you ($D=4,6$ and of arbitrary signature), you can find a discussion of the local existence and generality in a 2000 paper of mine entitled Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor (http://arxiv.org/abs/math/0004073).  
Note that, unlike the Riemannian case, a pseudo-Riemannian metric with a parallel spinor field need not be Ricci-flat; this usually imposes extra conditions.  
In the discussion, I point out exactly what the possibilities are for holonomy groups, so that tells you when you can have the dimension of the space of parallel spinor fields equal to $1$.  For example, with $D=4$, this can only happen with metrics of type $(2,2)$ and with $D=6$, this can only happen with metrics of type $(3,3)$.  (In each case, such examples, even Ricci-flat ones, do exist.)