Complete Riemannian manifolds admitting a pair of Einstein-Weyl structures

summary:We prove that a connected Riemannian manifold admitting a pair of non-trivial Einstein-Weyl structures $(g, \pm \omega )$ with constant scalar curvature is either Einstein, or the dual field of $\omega $ is Killing. Next, let $(M^n, g)$ be a complete and connected Riemannian manifold of dimension at least $3$ admitting a pair of Einstein-Weyl structures $(g, \pm \omega )$. Then the Einstein-Weyl vector field $E$ (dual to the $1$-form $\omega $) generates an infinitesimal harmonic transformation if and only if $E$ is Killing