We proved that the normalized Ricci flow does not preserve the positivity of the Ricci curvature of invariant Riemannian metrics on every generalized Wallach space with \(a_1+a_2+a_3\le 1/2\) , in particular, such a property takes place on the homogeneous spaces \(\operatorname {SU}(k+l+m)/\operatorname {S}(\operatorname {U}(k)\times \operatorname {U}(l) \times \operatorname {U}(m))\) and \(\operatorname {Sp}(k+l+m)/\operatorname {Sp}(k)\times \operatorname {Sp}(l) \times \operatorname {Sp}(m)\) independently on their parameters k, l and m. We proved that the positivity of the Ricci curvature is preserved under the normalized Ricci flow on generalized Wallach spaces with \(a_1+a_2+a_3> 1/2\) if the conditions \(4\left( a_j+a_k\right) ^2\ge (1-2a_i)(1+2a_i)^{-1}\) are satisfied for all \(\{i,j,k\}=\{1,2,3\}\) . We also established that the homogeneous spaces \(\operatorname {SO}(k+l+m)/\operatorname {SO}(k)\times \operatorname {SO}(l)\times \operatorname {SO}(m)\) satisfy the above conditions if \(\max \{k,l,m\}\le 11\) , moreover, additional conditions were found to keep \(\operatorname {Ric}>0\) in cases when \(\max \{k,l,m\}\le 11\) is violated. Answers have also been found to similar questions about maintaining or non-maintaining the positivity of the Ricci curvature on all other generalized Wallach spaces given in the classification of Yu. G. Nikonorov.
