Half-flat structures inducing Einstein metrics on homogeneous spaces

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In this paper, we consider half-flat SU(3)-structures and the subclasses of coupled and double structures. In the general case we show that the intrinsic torsion form w₁- 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 S³× S³ and we show there does not exist any left invariant coupled structure inducing an Ad(S¹)-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.