Reducible Riemannian manifolds with conformal product structures

Abstract We study conformal product structures on compact reducible Riemannian manifolds, and show that under a suitable technical assumption, the underlying Riemannian manifolds are either conformally flat or local triple products, i.e., locally isometric to Riemannian manifolds of the form $(M,g)$ with $M=M_1\times M_2\times M_3$ and $g=e^{2f}g_1+g_2+g_3$ , where $g_i$ is a Riemannian metric on $M_i$ , for $i\in \{1,2,3\}$ , and $f\in C^\infty (M_1\times M_2)$ .