Holonomy rigidity for Ricci-flat metrics

On a closed connected oriented manifold $M$ we study the space ${\mathcal{M}}_{\mathrm{\parallel }}(M)$ of all Riemannian metrics which admit a non-zero parallel spinor on the universal covering. Such metrics are Ricci-flat, and all known Ricci-flat metrics are of this form. We show the following: The space ${\mathcal{M}}_{\mathrm{\parallel }}(M)$ is a smooth submanifold of the space of all metrics, and its premoduli space is a smooth finite-dimensional manifold. The holonomy group is locally constant on ${\mathcal{M}}_{\mathrm{\parallel }}(M)$. If $M$ is spin, then the dimension of the space of parallel spinors is a locally constant function on ${\mathcal{M}}_{\mathrm{\parallel }}(M)$.