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Half-flat structures inducing Einstein metrics on homogeneous spaces

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Abstract

In this paper, we consider half-flat \(\mathrm{SU}(3)\)-structures and the subclasses of coupled and double structures. In the general case, we show that the intrinsic torsion form \(w_1^-\) is constant in each of the two subclasses. We then consider the problem of finding half-flat structures inducing Einstein metrics on homogeneous spaces. We give an example of a left-invariant half-flat (non-coupled and non-double) structure inducing an Einstein metric on \(S^3\times S^3\) and we show there does not exist any left-invariant coupled structure inducing an \(\mathrm{Ad}(S^1)\)-invariant Einstein metric on it. Finally, we show that there are no coupled structures inducing the Einstein metric on Einstein solvmanifolds and on homogeneous Einstein manifolds of nonpositive sectional curvature.

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Acknowledgments

The author would like to thank Anna Fino for suggesting the problem and for the fruitful conversations on the work and Thomas Madsen and Paolo Lella for the useful comments.

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  1. Dipartimento di Matematica “G. Peano”, Università di Torino, via Carlo Alberto 10, 10123, Torino, Italy

    Alberto Raffero

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  1. Alberto Raffero
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Raffero, A. Half-flat structures inducing Einstein metrics on homogeneous spaces. Ann Glob Anal Geom 48, 57–73 (2015). https://doi.org/10.1007/s10455-015-9457-1

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