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Holonomy rigidity for Ricci-flat metrics

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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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Acknowledgements

We thank X. Dai for discussions about the history of the subject and H.-J. Hein for some discussions related to Tian-Todorov theory.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, Universitätsstraße 31, 93040, Regensburg, Germany

    Bernd Ammann

  2. Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, 20146, Hamburg, Germany

    Klaus Kröncke

  3. Mathematisches Seminar, Universität Kiel, Ludewig-Meyn Straße 4, 24098, Kiel, Germany

    Hartmut Weiss

  4. Institut für Geometrie und Topologie, Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Frederik Witt

Authors
  1. Bernd Ammann
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  2. Klaus Kröncke
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  3. Hartmut Weiss
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  4. Frederik Witt
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Corresponding author

Correspondence to Bernd Ammann.

Additional information

Bernd Ammann has been partially supported by SFB 1085 Higher Invariants, Regensburg, funded by the DFG.

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Ammann, B., Kröncke, K., Weiss, H. et al. Holonomy rigidity for Ricci-flat metrics. Math. Z. 291, 303–311 (2019). https://doi.org/10.1007/s00209-018-2084-3

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  • DOI: https://doi.org/10.1007/s00209-018-2084-3

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