We prove that a connected Riemannian manifold admitting a pair of nontrivial Einstein-Weyl structures (g, ±ω) with constant scalar curvature is either Einstein, or the dual field of ω is Killing. Next, let (Mn , g) be a complete and connected Riemannian manifold of dimension at least 3 admitting a pair of Einstein-Weyl structures (g,±ω). Then the Einstein-Weyl vector field E (dual to the 1-form ω) generates an infinitesimal harmonic transformation if and only if E is Killing.
The concept of the Ricci soliton was introduced by R. S. Hamilton. The Ricci soliton is defined by a vector field and it is a natural generalization of the Einstein metric. We have shown earlier that the vector field of the Ricci soliton is an infinitesimal harmonic transformation. In our paper, we survey Ricci solitons geometry as an application of the theory of infinitesimal harmonic transformations.