Prescribing Ricci curvature on a product of spheres

We prove an existence result for the prescribed Ricci curvature equation certain doubly warped product metrics on $$\mathbb {S}^{d_1+1}\times \mathbb {S}^{d_2}$$ , where $$d_i \ge 2$$ . If T is a metric satisfying assumptions, we show that can be scaled independently two factors so as to itself tensor of some metric. This first global problem closed cohomogeneity one manifolds since work Hamilton in 1984.