Compact homogeneous Einstein 6-manifolds. (Q1417345)

The goal of the authors is to give a classification of all compact simply connected homogeneous Einstein 6-dimensional manifolds \((M= G/H,g)\), that is to classify all invariant Einstein metrics \(g\) on 6-dimensional homogeneous simply connected spaces of a compact Lie group \(G\). Without loss of generality, one can assume that \(G\) is a compact connected semisimple Lie group. The authors prove that in this case, \((M = G/H,g )\) is either a symmetric space or one of the following manifolds : (1) \(\mathbb{C} P^3 = Sp_2/Sp_1 \times U_1\) with the sqashed metric, (2) the Wallach space \(SU_3/T^2\) with the standard metric or with the Kähler invariant metric, (3) the Lie group \(SU_2 \times SU_2\) with some left invariant Einstein metric. This reduces the problem to the classification of left invariant Einstein metrics on the group \(G= SU_2 \times SU_2\). The authors get the following partial result in this direction: If the Einstein metric \(g\) is right invariant with respect to a subgroup \(T^1\) of \(G\), then it is homothetic to the standard biinvariant metric or to the Jensen metric.