Conformal product structures on compact Einstein manifolds

Andrei Moroianu; Mihaela Pilca

paper

Conformal product structures on compact Einstein manifolds

arXiv:2504.07886

Abstract

In this note we generalize our previous result, stating that if $(M_{1}, g_{1})$ and $(M_{2}, g_{2})$ are compact Riemannian manifolds, then any Einstein metric on the product $M := M_{1} \times M_{2}$ of the form $g = e^{2 f_{1}} g_{1} + e^{2 f_{2}} g_{2}$ , with $f_{1} \in C^{\infty} (M_{2})$ and $f_{2} \in C^{\infty} (M_{1} \times M_{2})$ , is a warped product metric. Namely, we show that the same conclusion holds if we replace the assumption that the manifold $M$ is globally the product of two compact manifolds by the weaker assumption that $M$ is compact and carries a conformal product structure.

11 pages