Infinitesimal Harmonic Transformations and Ricci Solitons

A Ricci soliton on a smooth manifold $M$ is a triple $(g_0,\xi,\lambda)$, where $g_0$ is a complete Riemannian metric, $\xi$ a vector field, and $\lambda$ a constant such that the Ricci tensor $\mathrm{Ric}_0$ of $g_0$ satisfies the equation $-2\mathrm{Ric}_0=L_\xi g_0+2\lambda g_0$. In the paper, we study the geometry of Ricci solitons in dependence of the properties of the vector field $\xi$. In particular, we prove that this vector field is a harmonic transformation.