Abstract
In this paper, we introduce the notion of Jacobi-type vector fields on Riemannian manifolds, which is a generalization of the Jacobi field along a geodesic. We study Ricci solitons with positive Ricci curvature whose potential vector field is a Jacobi-type vector field and show that if the metric on Ricci soliton is replaced by the Ricci tensor, then we get a Riemannian manifold that is an Einstein manifold. As a by-product, we get a criterion for compactness of a complete Ricci soliton using a Jacobi-type vector field. Finally it is shown that a Ricci soliton of positive Ricci curvature whose potential field is Jacobi-type vector field is necessarily an Einstein manifold.