Certain Results on Ricci Solitons in -Sasakian Manifolds

We study Ricci solitons in -Sasakian manifolds and show that it is a shrinking or expanding soliton and the manifold is Einstein with Killing vector field. Further, we prove that if is conformal Killilng vector field, then the Ricci soliton in 3-dimensional -Sasakian manifolds is shrinking or expanding but cannot be steady. 1. Introduction A Ricci soliton is a generalization of an Einstein metric and is defined on a Riemannian manifold by where is a complete vector field on and is a constant. The Ricci soliton is said to be shrinking, steady, or expanding according as is negative, zero, and positive, respectively. Long-existing solutions, that is, solutions which exist on an infinite time interval, are the self-similar solutions, which in Ricci flow are called Ricci soliton. Compact Ricci solitons are the fixed points of the Ricci flow projected from the space of metrics onto its quotient modulo diffeomorphisms and scalings and often arise as blow-up limits for the Ricci flow on compact manifolds. If the vector field is the gradient of a potential function , then is called a gradient Ricci soliton and (1) assumes the form A Ricci soliton on a compact manifold is a gradient Ricci soliton. A Ricci soliton on a compact manifold has constant curvature in dimension [1] and also in dimension [2]. In [3], Perelman proved that a Ricci soliton on a compact -manifold is a gradient Ricci soliton. In [4], Sharma studied Ricci solitons in -contact manifolds, where the structure field is Killing, and he proved that a complete -contact gradient soliton is compact Einstein and Sasakian. In [5], Tripathi studied Ricci solitons in -contact metric and manifolds. In [6], Ghosh and Sharma studied -contact metrics as Ricci solitons. In [7], Nagaraja and Premalatha studied Ricci solitons in -Kenmotsu manifolds and 3-dimensional trans-Sasakian manifolds. Recently, Bagewadi and Ingalahalli [8] studied Ricci solitons in Lorentzian -Sasakian manifolds. Motivated by the previous studies on Ricci solitons, in this paper, we study Ricci solitons in an -Sasakian manifolds, where is some constant. 2. Preliminaries Let be an almost contact metric manifold of dimension equipped with an almost contact metric structure consisting of a tensor field , a vector field , a 1-form , and a Riemannian metric , which satisfy for all . An almost contact metric manifold is said to be -Sasakian manifold if the following conditions hold: for some nonzero constant on . In an -Sasakian manifold, we have the following relations: for all , where is the Riemannian curvature tensor, is the Ricci tensor, and