Complete Riemannian manifolds admitting a pair of Einstein-Weyl structures. (Q2828793)

A Weyl structure on a manifold \(M\) is a pair consisting of a conformal class \([g]\) and a traceless connection \(\nabla\) preserving it. It is called Einstein-Weyl if the symmetrized Ricci tensor of \(\nabla\) is proportional to \(g\). Weyl connections \(\nabla\) are bijective with 1-forms \(\omega\) given by \(\nabla g=\omega\otimes g\). A change of representative \(g\mapsto e^fg\) results in the change \(\omega\mapsto\omega+df\). A conformal structure represented by \(g\) can admit several Weyl potentials \(\omega\) such that \((g,\omega)\) is Einstein-Weyl. The paper studies the case, when a connected Riemannian manifold \(M\) admits a pair of Einstein-Weyl structures \((g,\pm\omega)\). It is proved that if in such a case the scalar curvature of \(g\) is constant, then either \((M,g)\) is Einstein or the \(g\)-dual vector field \(e\) of \(\omega\) is Killing. If \((M,g)\) is, in addition, complete of \(\dim\geq3\), then the flow of \(e\) generates an infinitesimal harmonic transformation if and only if \(e\) is Killing.