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Almost complex structures in 6D with non-degenerate Nijenhuis tensors and large symmetry groups

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An Erratum to this article was published on 23 February 2017

Abstract

For an almost complex structure J in dimension 6 with non-degenerate Nijenhuis tensor \(N_J\), the automorphism group \(G=\mathop {\mathrm{Aut}}\nolimits (J)\) of maximal dimension is the exceptional Lie group \(G_2\). In this paper, we establish that the sub-maximal dimension of automorphism groups of almost complex structures with non-degenerate \(N_J\), i.e. the largest realizable dimension that is less than 14, is \(\dim G=10\). Next, we prove that only three spaces realize this, and all of them are strictly nearly (pseudo-) Kähler and globally homogeneous. Moreover, we show that all examples with \(\dim \mathop {\mathrm{Aut}}\nolimits (J)=9\) have semi-simple isotropy.

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Notes

  1. The statement about the Lie algebra of symmetries is stronger than that about the Lie group, and so we give only the local version.

  2. This form differs in [9] and [10], and we use the expression from the latter (the former form, though it looks differently, is equivalent).

  3. This is different from “right” or extensions by derivations [4].

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Acknowledgments

Henrik Winther is grateful to Ilka Agricola for her hospitality during his DGF-funded research stay at the University of Marburg. Both authors were partially supported by the Norwegian Research Council and the DAAD project of Germany.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, NT-Faculty, University of Tromsø, 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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An erratum to this article is available at http://dx.doi.org/10.1007/s10455-017-9546-4.

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Kruglikov, B.S., Winther, H. Almost complex structures in 6D with non-degenerate Nijenhuis tensors and large symmetry groups. Ann Glob Anal Geom 50, 297–314 (2016). https://doi.org/10.1007/s10455-016-9513-5

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  • DOI: https://doi.org/10.1007/s10455-016-9513-5

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