New Characteristics of Infinitesimal Isometry and Ricci Solitons

We prove that a vector field $X$ on a compact Riemannian manifold $(M,g)$ with Levi-Cività connection $\nabla$ is an infinitesimal isometry if and only if it satisfies the system of differential equations: $\operatorname{trace}_g(L_X\nabla)=0$, $\operatorname{trace}_g(L_X\operatorname{Ric})=0$, where $L_X$ is the Lie derivative in the direction of $X$ and $\operatorname{Ric}$ is the Ricci tensor. It follows from the second assertion that the Ricci soliton on a compact manifold $M$ is trivial if its vector field $X$ satisfies one of the following two conditions: $\operatorname{trace}_g(L_X\operatorname{Ric})\le 0$ or $\operatorname{trace}_g(L_X \operatorname{Ric})\ge 0$.