Coindex and Rigidity of Einstein Metrics on Homogeneous Gray Manifolds

Abstract Any 6-dimensional strict nearly Kähler manifold is Einstein with positive scalar curvature. We compute the coindex of metric respect to Einstein–Hilbert functional on each compact homogeneous examples. Moreover, we show that infinitesimal deformations $$F_{1,2}={\text {SU}}(3)/T^2$$ F 1 , 2 = SU ( 3 ) / T are not integrable into a curve metrics.