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Non-degenerate para-complex structures in 6D with large symmetry groups

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Abstract

For an almost product structure J on a manifold M of dimension 6 with non-degenerate Nijenhuis tensor \(N_J\), we show that the automorphism group \(G=\mathrm{Aut}(M,J)\) has dimension at most 14. In the case of equality G is the exceptional Lie group \(G_2^*\). The next possible symmetry dimension is proved to be equal to 10, and G has Lie algebra \(\mathfrak {sp}(4,{\mathbb R})\). Both maximal and submaximal symmetric structures are globally homogeneous and strictly nearly para-Kähler. We also demonstrate that whenever the symmetry dimension is at least 9, then the automorphism algebra acts locally transitively.

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Notes

  1. If \(o\in M\) belongs to a singular orbit, this fact is usually false.

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Acknowledgements

Both authors were partially supported by the Norwegian Research Council and DAAD project of Germany.

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  1. Department of Mathematics and Statistics, Faculty of Science and Technology, UiT the Arctic University of Norway, 90-37, Tromsø, Norway

    B. S. Kruglikov & H. Winther

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  1. B. S. Kruglikov
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  2. H. Winther
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Correspondence to B. S. Kruglikov.

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Kruglikov, B.S., Winther, H. Non-degenerate para-complex structures in 6D with large symmetry groups. Ann Glob Anal Geom 52, 341–362 (2017). https://doi.org/10.1007/s10455-017-9561-5

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