Abstract. In this study we consider φ−conformally flat, φ−conharmonically flat, φ−projectively flat and φ−concircularly flat Lorentzian α−Sasakian manifolds. In all cases, we get the manifold will be an η−Einstein manifold.

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.

ON 3-DIMENSIONAL f-KENMOTSU MANIFOLDS AND RICCI SOLITONS ПРО 3-ВИМIРНI f-МНОГОВИДИ КЕНМОЦУ ТА СОЛIТОНИ РIЧЧI The object of the present paper is to study 3-dimensional f-Kenmotsu manifolds and Ricci solitons. Firstly we give an example of a 3-dimensional f-Kenmotsu manifold. Then we consider Ricci-semisymmetric 3-dimensional f-Kenmotsu manifold and prove that a 3-dimensional f-Kenmotsu manifold is Ricci semisymmetric if and only if it is an Einstein manifold. Also η-parallel Ricci tensor in a 3-dimensional f-Kenmotsu manifold have been studied. Finally we study Ricci solitons in a 3-dimensional f-Kenmotsu manifold. Метою даної статтi є вивчення 3-вимiрних f-многовидiв Кенмоцу та солiтонiв Рiччi. Спочатку наведено приклад 3-вимiрного f-многовиду Кенмоцу. Потiм розглянуто напiвсиметричний за Рiччi 3-вимiрний f-многовид Кенмоцу i доведено, що 3-вимiрний f-многовид Кенмоцу є напiвсиметричним за Рiччi тодi i тiльки тодi, коли вiн є многовидом Ейнштейна. Також дослiджено η-паралельний тензор Рiччi у 3-вимiрному f-многовидi Кенмоцу. Насамкiнець, дослiджено солiтони Рiччi у 3-вимiрному f-многовидi Кенмоцу.

In this paper we study h-projectively semisymmetric, ϕ-projectively semisymmetric, h-Weyl semisymmetric and ϕ-Weyl semisymmetric non-Sasakian (k, µ)-contact metric manifolds. In all the cases the manifold becomes an η-Einstein manifold. As a consequence of these results we obtain that if a 3-dimensional non-Sasakian (k, µ)-contact metric manifold satisfies such curvature conditions, then the manifold reduces to an N (k)-contact metric manifold.

Abstract. In this paper we prove that a ϕ-recurrent (k, µ)-contact metric manifold is an η-Einstein manifold with constant coefficients. Next, we prove that a three-dimensional locally ϕ-recurrent (k, µ)-contact metric manifold is the space of constant curvature. The existence of ϕ-recurrent (k, µ)-manifold is proved by a non-trivial example.