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The purpose of the paper is to prove that if the metric of a 3-dimensional α-Sasakian structure on a complete Riemannian manifold is a Yamabe soliton then it is of constant curvature. We also derive some properties of the flow vector field U of the Yamabe soliton together with an example of an α-Sasakian manifold admitting Yamabe soliton.

α-SASAKIAN 3-METRIC AS A RICCI SOLITON * α-САСАКIЄВА 3-МЕТРИКА ЯК СОЛIТОН РIЧЧI We prove that if the metric of a 3-dimensional α-Sasakian manifold is a Ricci soliton, then it is either of constant curvature or of constant scalar curvature. We also establish some properties of the potential vector field U of the Ricci soliton. Finally, we give an example of an α-Sasakian 3-metric as a nontrivial Ricci soliton. Доведено, що якщо метрика тривимiрного α-сасакiєвого многовиду є солiтоном Рiччi, то вiн має або сталу кривину, або сталу скалярну кривину. Встановлено деякi властивостi потенцiального векторного поля U солiтона Рiччi. Наведено приклад α-сасакiєвої 3-метрики як нетривiального солiтона Рiччi.

Abstract. In the present paper, we prove three fundamental results concerning almost ∗-Ricci soliton in the framework of para-Sasakian manifold. The paper is organised as follows:• If a para-Sasakian metric g represents an almost ∗-Ricci soliton with potential vector field V is Jacobi along Reeb vector field ξ, then g becomes a ∗-Ricci soliton.• If a para-Sasakian metric g represents an almost ∗-Ricci soliton with potential vector field V as infinitesimal paracontact transformation, then V is killing and g is η-Einstein.• If a para-Sasakian metric g represents an almost ∗-Ricci soliton with potential vector field V is collinear with the Reeb vector field ξ, then λ = 0, V is strict and g is η-Einstein.

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