Infinitesimal harmonic transformations and Ricci solitons on complete Riemannian manifolds

The definition of a Ricci soliton was introduced by R. Hamilton; it naturally generalizes the Einstein metric. A Ricci soliton on a smooth manifold $M$ is the triplet $(g_0,\xi,\lambda)$, where $g_0$ is a complete Riemannian metric, $\xi$ is a vector field, and $\lambda$ is a constant value such that the Ricci tensor $\mathrm{Ric}_0$ of the metric $g_0$ satisfies the equation $-2\mathrm{Ric}_0=L_\xi g_0+2\lambda g_0$. The following assertion is one of the main results of this paper. Assume that $(g_0,\xi,\lambda)$ is a Ricci soliton such that $(M,g_0)$ is a compete noncompact oriented Riemannian manifold, $\int_M\|\xi\|\,dv<\infty$, and the scalar curvature $s_0$ of the metric $g_0$ has a constant sign on $M$. Then $(M,g_0)$ is an Einstein manifold.