Abstract. In this paper we study η-Ricci solitons on Sasakian 3-manifolds. Among others we prove that an η-Ricci soliton on a Sasakian 3-manifold is an η-Einstien manifold. Moreover we consider η-Ricci solitons on Sasakian 3-manifolds with Ricci tensor of Codazzi type and cyclic parallel Ricci tensor. Beside these we study conformally flat and φ-Ricci symmetric η-Ricci soliton on Sasakian 3-manifolds. Also η-Ricci soliton on Sasakian 3-manifolds with the curvature condition Q.R = 0 have been considered. Finally, we construct an example to prove the non-existence of proper η-Ricci solitons on Sasakian 3-manifolds and verify some results.

We consider η-Ricci solitons on Lorentzian para-Sasakian manifolds with Codazzi type of the Ricci tensor. Then we study η-Ricci solitons on ϕ-conformally semi-symmetric, ϕ-Ricci symmetric, and conformally Ricci semi-symmetric Lorentzian para-Sasakian manifolds. Finally, we construct an example of a three dimensional Lorentzian para-Sasakian manifold which admits η-Ricci solitons with non-constant scalar curvature. 2010 Mathematics Subject Classification. 53C15, 53C25. Key words and phrases. Ricci solitons; η-Ricci solitons; Lorentzian para-Sasakian manifolds; Codazzi type of the Ricci tensor. Debabrata Kar is supported by the Council of Scientific and Industrial Research, India (File no. 09/028(1007)/2017-EMR-1).

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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 sufficient 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 flat and concircularly flat 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.