Ricci Solitons and Generalized Ricci Solitons Whose Potential Vector Fields Are Jacobi Type

Abstract

‎This paper is devoted to Ricci solitons admitting a Jacobi-type vector field‎. ‎First‎, ‎we present some rigidity results for Ricci solitons $(M^n‎, ‎g‎, ‎V‎, ‎\lambda)$ admitting a Jacobi-type vector field $\xi$ and provide conditions under which $\xi$ is Killing‎. ‎We also present conditions under which the Ricci soliton $(M^n‎, ‎g‎, ‎\xi‎, ‎\lambda)$ is isometric to $\mathbb{R}^n$‎. ‎Next‎, ‎we demonstrate that Jacobi-type vector fields which are the potential vector fields of a quasi-Einstein manifold are Killing and of constant length‎. ‎Finally‎, ‎we prove that quasi-Einstein manifolds whose potential vector fields are Jacobi-type‎, ‎are necessarily of constant scalar curvature‎.