Reducible Riemannian manifolds with conformal product structures

We study conformal product structures on compact reducible Riemannian manifolds, and show that under a suitable technical assumption, the underlying Riemannian mani\-folds are either conformally flat, or triple products, i.e. locally isometric to Riemannian manifolds of the form $(M,g)$ with M=M₁× M₂× M₃ and g=e²ᶠg₁+g₂+g₃, where gᵢ is a Riemannian metric on Mᵢ, for i∈\1,2,3\, and f∈ C^∞(M₁× M₂).