Sufficient conditions for triviality of Ricci solitons

We found conditions on an $n$-dimensional Ricci soliton $\left(M,g,\mathbf{u},\lambda \right)$ to be trivial. First, we showed that under an appropriate upper bound on the squared length of the covariant derivative of the potential field $\mathbf{u}$, the Ricci soliton $\left(M,g,\mathbf{u},\lambda \right)$ reduces to a trivial soliton. We also showed that appropriate upper and lower bounds on the Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$ of a compact Ricci soliton $\left(M,g,\mathbf{u},\lambda \right)$ with potential field $\mathbf{u}$ geodesic vector field makes it a trivial soliton. We showed that if the Ricci operator $S$ of the Ricci soliton $\left(M,g,\mathbf{u},\lambda \right)$ is invariant under the potential field $\mathbf{u}$, then $\left(M,g,\mathbf{u},\lambda \right)$ is trivial and the converse is also true. Finally, it was shown that if the potential field $\mathbf{u}$ of a connected Ricci soliton $\left(M,g,\mathbf{u},\lambda \right)$ is a concurrent vector field, then the Ricci soliton is shrinking.