A Study on Ricci Solitons in Kenmotsu Manifolds

We study and obtain results on Ricci solitons in Kenmotsu manifolds satisfying , , , and , where and are C-Bochner and pseudo-projective curvature tensor. 1. Introduction A Ricci soliton is a natural generalization of an Einstein metric and is defined on a Riemannian manifold . A Ricci soliton is a triple with a Riemannian metric, a vector field, and a real scalar such that where is a Ricci tensor of and denotes the Lie derivative operator along the vector field . The Ricci soliton is said to be shrinking, steady, and expanding accordingly as is negative, zero, and positive, respectively [1]. In this paper, we prove conditions for Ricci solitons in Kenmotsu manifolds to be shrinking, steady, and expanding. In 1972, Kenmotsu [2] studied a class of contact Riemannian manifolds satisfying some special conditions and this manifold is known as Kenmotsu manifolds. Kenmotsu proved that a locally Kenmotsu manifold is a warped product of an interval and a Kaehler manifold with warping function , where is a nonzero constant. Kenmotsu proved that if in a Kenmotsu manifold the condition holds, then the manifold is of negative curvature , where is the curvature tensor of type and denotes the derivation of the tensor algebra at each point of the tangent space. The authors in [3–7] have studied Ricci solitons in contact and Lorentzian manifolds. The authors in [8] have obtained some results on Ricci solitons satisfying , , and and now we extend the work to , , , and . 2. Preliminaries An -dimensional differential manifold is said to be an almost contact metric manifold [9] if it admits an almost contact metric structure consisting of a tensor field of type , a vector field , a -form , and a Riemannian metric compatible with satisfying for all vector fields , on . An almost contact metric manifold is said to be Kenmotsu manifold [2] if From (3), we have where denotes the Riemannian connection of . In an -dimensional Kenmotsu manifold, we have where is the Riemannian curvature tensor. Let be a Ricci soliton in an -dimensional Kenmotsu manifold . From (4) we have From (1) and (9) we get The above equation yields that where is the Ricci tensor, is the Ricci operator, and is the scalar curvature on . 2.1. Example for 3-Dimensional Kenmotsu Manifolds We consider -dimensional manifold , where are the standard coordinates in . Let be linearly independent given by Let be the Riemannian metric defined by , , where is given by The structure is given by The linearity property of and yields that , , , for any vector fields on . By definition of Lie bracket, we have Let be the