Every homogeneous hypercomplex manifold with transitive compact Lie group action admits a unique invariant HKT-Einstein metric up to scaling, and invariant strong HKT metrics have Bismut-parallel torsion and curvature.
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On homogeneous HKT manifolds and the Einstein condition
Every homogeneous hypercomplex manifold with transitive compact Lie group action admits a unique invariant HKT-Einstein metric up to scaling, and invariant strong HKT metrics have Bismut-parallel torsion and curvature.