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 the metric with respect to the Einstein–Hilbert functional on each of the compact homogeneous examples. Moreover, we show that the infinitesimal Einstein deformations on $$F_{1,2}={\text {SU}}(3)/T^2$$ F 1 , 2 = SU ( 3 ) / T 2 are not integrable into a curve of Einstein metrics.