Abstract
In this paper we consider a complete connected Ricci soliton (M, g, xi, lambda) of positive Ricci curvature and assign the Ricci tensor Ric = (g) over bar a role of another Riemannian metric on M. It is shown that the identity map i : (M, g) -> (M, (g) over bar) is a harmonic map. In addition, we also study compact shrinking gradient Ricci soliton (M, g, del f, lambda) of positive Ricci curvature and obtain a lower bound for the average value of the potential function f and show that if the lower bound is attended then the gradient Ricci soliton is an Einstein manifold.