Geodesic
Complete Riemannian manifolds admitting a pair of Einstein-Weyl structures
Mathematica Bohemica, Tome 141 (2016) no. 3, pp. 315-325.

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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.
DOI : 10.21136/MB.2016.0072-14
Classification : 53C15, 53C20, 53C25
Keywords: Weyl manifold; Einstein-Weyl structure; infinitesimal harmonic transformation 
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Ghosh, Amalendu. Complete Riemannian manifolds admitting a pair of Einstein-Weyl structures. Mathematica Bohemica, Tome 141 (2016) no. 3, pp. 315-325. doi : 10.21136/MB.2016.0072-14. http://geodesic.mathdoc.fr/articles/10.21136/MB.2016.0072-14/

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