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 S d 1 +1 × S d 2 , where di ≥ 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.

On the variational properties of the prescribed Ricci curvature functional

“…More precisely, suppose that the metric g and the tensor T are invariant under a Lie group G acting on M . In the case where the quotient M/G is onedimensional, the problem was addressed by Hamilton [21], Cao-DeTurck [13], Pulemotov [29,30] and Buttsworth-Krishnan [10]. The case where M is a homogeneous space G/H has been studied extensively; see the survey [11] and the more recent references [12,25,26,4,3].…”