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Half-flat structures on \( S^3\times S^3\)

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Abstract

We describe left-invariant half-flat \( \mathrm{SU }(3) \)-structures on \( S^3\times S^3\) using the representation theory of \( \mathrm SO (4) \) and matrix algebra. This leads to a systematic study of the associated cohomogeneity one Ricci-flat metrics with holonomy \( \mathrm G _2\) obtained on \( 7 \)-manifolds with equidistant \( S^3\times S^3\) hypersurfaces. The generic case is analysed numerically.

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Acknowledgments

Both authors thank Mark Haskins for discussions that helped initiate this research and, in particular, for bringing [31] to their attention. The first author gratefully acknowledge financial support from the Danish Council for Independent Research, Natural Sciences.

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  1. Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UK

    Thomas Bruun Madsen & Simon Salamon

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  1. Thomas Bruun Madsen
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  2. Simon Salamon
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Correspondence to Simon Salamon.

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Madsen, T.B., Salamon, S. Half-flat structures on \( S^3\times S^3\) . Ann Glob Anal Geom 44, 369–390 (2013). https://doi.org/10.1007/s10455-013-9371-3

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