Abstract

On a closed connected oriented manifold M we study the space $$\mathcal {M}_\Vert (M)$$ 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 $$\mathcal {M}_\Vert (M)$$ 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 $$\mathcal {M}_\Vert (M)$$ . If M is spin, then the dimension of the space of parallel spinors is a locally constant function on $$\mathcal {M}_\Vert (M)$$ .

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