Holonomy rigidity for Ricci-flat metrics

TL;DR

This paper investigates the structure of Ricci-flat metrics with non-zero parallel spinors on closed manifolds, showing their moduli space is smooth and the holonomy group remains locally constant within this space.

Contribution

It proves that the space of such Ricci-flat metrics forms a smooth submanifold with a finite-dimensional premoduli space and establishes the local constancy of holonomy groups and parallel spinor dimensions.

Findings

The space of metrics with non-zero parallel spinors is a smooth submanifold.

The premoduli space of these metrics is finite-dimensional and smooth.

Holonomy groups are locally constant on this space.

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 $\mathcal{M}_\|(M)$.