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arXiv · 1512.07390

Holonomy rigidity for Ricci-flat metrics

Bernd Ammann·Klaus Kroencke·Hartmut Weiss·Frederik Witt
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Abstract

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

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Bernd Ammann, Klaus Kroencke, Hartmut Weiss, Frederik Witt. 2016-05-10. Holonomy rigidity for Ricci-flat metrics. https://arxiv.org/abs/1512.07390

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