Three dimensional Sasakian manifolds admitting η-Ricci solitons

Abstract

In this paper we characterize the three dimensional Sasakian manifolds admitting η-almost Ricci solitons. After the introduction, in section 2, we study three dimensional Sasakian manifolds. In section 3, we prove that an η-Ricci soliton in Sasakian manifolds satisfying the curvature property R.Q = 0 is shrinking and reduces to Ricci soliton. In section 4, we show that the necessary and suficient condition for a Sasakian manifold not admitting a proper η-Ricci soliton is that it is Ricci symmetric. In sections 5 and 6, we study projectively at and concircularly at Sasakian manifold of dimension 3 respectively and find the type of an η-Ricci soliton on such manifold. The next section is devoted to the study of such a manifold admitting η-Ricci soliton and we prove some equivalent conditions. Finally, in section 8, we prove that in a three dimensional Sasakian manifold an η-Ricci soliton becomes Ricci soliton if and only if it is Ricci pseudo-symmetric.