Prescribing Ricci curvature on a Product of Spheres

TL;DR

This paper establishes conditions under which specific warped product metrics on products of spheres can be prescribed to have a given Ricci curvature, expanding understanding of curvature equations on complex manifolds.

Contribution

It proves an existence theorem for the prescribed Ricci curvature equation on doubly warped product metrics on spheres, with independent scaling of factors.

Findings

Existence of metrics with prescribed Ricci curvature on product of spheres.

Conditions under which the metric can be scaled to match a given curvature.

Extension of Ricci curvature prescription to doubly warped product geometries.

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 $d_i \geq 2$. 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.