Holonomy rigidity for Ricci-flat metrics

Holonomy rigidity for Ricci-flat metrics 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Holonomy rigidity for Ricci-flat metrics

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
D.O.I.
10.1007/s00209-018-2084-3
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Mathematische ZeitschriftSpringer Journals

Published: Jun 5, 2018

References

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