Math. Z. https://doi.org/10.1007/s00209-018-2084-3 Mathematische Zeitschrift Holonomy rigidity for Ricci-ﬂat metrics 1 2 Bernd Ammann · Klaus Kröncke · 3 4 Hartmut Weiss · Frederik Witt Received: 7 October 2016 / Accepted: 13 April 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract 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 ). 1 Overview over the results Let M be a compact connected oriented manifold without boundary, and let π : M → M be its universal covering. We assume throughout the article that M is spin. We deﬁne M (M ) Bernd Ammann has been partially supported by SFB 1085 Higher Invariants, Regensburg, funded by the
Mathematische Zeitschrift – Springer Journals
Published: Jun 5, 2018
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