Conformal product structures on compact Einstein manifolds
Andrei Moroianu, Mihaela Pilca
TL;DR
The paper addresses when a compact Einstein metric lies in a conformal product class by weakening the global product hypothesis to a special Lee form condition. It analyzes the Lee form decomposition $ heta= heta_1+ heta_2$ under rank-based cases and uses local product arguments, Bochner formulas, and Myers-type results to show ${ m d} heta_1=0$, hence the Weyl structure is closed. Consequently, on a compact Einstein LCP manifold, the Weyl connection must be exact, so the metric is globally conformal to one with reducible holonomy, with the universal cover yielding a warped product structure. This extends previous global-product results and clarifies the structure of Einstein metrics in broader conformal-product settings.
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 $g=e^{2f_1}g_1+e^{2f_2}g_2$, 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.