Prescribing Ricci curvature on a product of spheres

Abstract

We prove an existence result for the prescribed Ricci curvature equation for 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 certain curvature assumptions, we show that T can be scaled independently on the two factors so as to itself be the Ricci tensor of some metric. This is the first global existence result for the prescribed Ricci curvature problem on closed cohomogeneity one manifolds since work of Hamilton in 1984.