A Note on Ricci Solitons

In this paper, we characterize trivial Ricci solitons. We observe the important role of the energy function f of a Ricci soliton (half the squared length of the potential vector field) in the charectrization of trivial Ricci solitons. We find three characterizations of connected trivial Ricci solitons by imposing different restrictions on the energy function. We also use Hessian of the potential function to characterize compact trivial Ricci solitons. Finally, we show that a solution of a Poisson equation is the energy function f of a compact Ricci soliton if and only if the Ricci soliton is trivial.