Holonomy rigidity for Ricci-flat metrics

On a closed connected oriented manifold M we study the space 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 M ‖ ( M ) is a smooth su...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2019-02, Vol.291 (1-2), p.303-311
Main Authors: Ammann, Bernd, Kröncke, Klaus, Weiss, Hartmut, Witt, Frederik
Format: Article
Language:English
Subjects:
Manifolds (mathematics)
Mathematics
Mathematics and Statistics
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Summary:On a closed connected oriented manifold M we study the space 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 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 M ‖ ( M ) . If M is spin, then the dimension of the space of parallel spinors is a locally constant function on  M ‖ ( M ) .
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-018-2084-3