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Locally 2-fold symmetric manifolds are locally symmetric

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Abstract

A manifold is locally k-fold symmetric if for any point and any k-dimensional vector subspace tangent to this point, there exists a local isometry such that this point is a fixed point and the differential of the isometry restricted to that k-dimensional vector subspace is minus the identity. We show that for \(k \ge 2\), Riemannian, pseudoriemannian, and Finslerian locally k-fold symmetric manifolds are locally symmetric.

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References

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China

    Shaoqiang Deng

  2. Institut für Mathematik, Friedrich-Schiller Universität Jena, 07737, Jena, Germany

    Vladimir S. Matveev

Authors
  1. Shaoqiang Deng
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  2. Vladimir S. Matveev
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Corresponding author

Correspondence to Vladimir S. Matveev.

Additional information

The result was obtain during the visit of V.M. to Nankai University related to the 2016 International Conference on Riemann–Finsler Geometry; he thanks the Nankai University and the conference organizers for their hospitality and financial support and also acknowledges the financial support of DFG. We are grateful to O. Yakimova for useful discussions. The first author was supported by NSFC (nos. 11671212, 51535008).

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Deng, S., Matveev, V.S. Locally 2-fold symmetric manifolds are locally symmetric. Arch. Math. 108, 521–525 (2017). https://doi.org/10.1007/s00013-017-1036-1

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  • DOI: https://doi.org/10.1007/s00013-017-1036-1

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