Ricci solitons and gradient Ricci solitons in three-dimensional trans-Sasakian manifolds

Abstract

The object of the present paper is to study 3-dimensional trans-Sasakian manifolds admitting Ricci solitons and gradient Ricci solitons. We prove that if (1,V, ?) is a Ricci soliton where V is collinear with the characteristic vector field ?, then V is a constant multiple of ? and the manifold is of constant scalar curvature provided ?, ? =constant. Next we prove that in a 3-dimensional trans-Sasakian manifold with constant scalar curvature if 1 is a gradient Ricci soliton, then the manifold is either a ?-Kenmotsu manifold or an Einstein manifold. As a consequence of this result we obtain several corollaries.

Gradient Ricci solitons on almost Kenmotsu manifolds

“…Following Perelman [15], we know that a Ricci soliton on a compact manifold is a gradient Ricci soliton. Ricci solitons on contact metric manifods, three-dimensional trans-Sasakian manifolds and N (k)-quasi-Einstein manifolds were studied by Ghosh [8], Turan, De, and Yildiz [18] and Crasmareanu [3], respectively. With regard to the studies of Ricci solitons on Kenmotsu manifolds, we refer the reader to De and Matsuyama [4] and Ghosh [7], respectively.…”

Gradient Ricci solitons on multiply warped product manifolds

“…In [2], De studied gradient Ricci solitons in para-Sasakian manifolds. In [3], [4], [16] and [19], gradient Ricci solitons in almost contact metric manifolds were studied. Motivated by the above studies, in the present paper, we consider gradient Ricci solitons on multiply warped product manifolds.…”

Ricci solitons on η-einstein contact manifolds

“…Also Ricci solitons and gradient Ricci solitons on some kinds of almost contact metric manifolds of dimension three were studied by many authors. For instances, De et al [18] and Turan et al [33] investigated Ricci solitons and gradient Ricci solitons on threedimensional normal almost contact metric manifolds and three-dimensional trans-Sasakian manifolds respectively. Moreover, A. Ghosh [23] and J. T. Cho [10] classified Ricci solitons on three-dimensional Kenmotsu manifolds respectively.…”