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 ? is an infinitesimal isometry if and only if it satisfies the system of differential equations: trace _g (L _X ?) = 0, trace _g (L _X Ric) = 0, where L _X is the Lie derivative in the direction of X and 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: trace _g (L _X Ric) ≤ 0 or trace _g (L _X Ric) ≥ 0.