Half-flat structures inducing Einstein metrics on homogeneous spaces

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.