The present paper aims to deliberate the geometric composition of a perfect fluid spacetime with torse-forming vector field $\xi$ in connection with conformal Ricci–Yamabe metric and conformal $\eta$-Ricci–Yamabe metric. We delineate the conditions for conformal Ricci–Yamabe soliton to be expanding, steady or shrinking. We also discuss conformal Ricci–Yamabe soliton on some special types of perfect fluid spacetime such as dust fluid, dark fluid and radiation era. Furthermore, we design conformal $\eta$-Ricci–Yamabe soliton to find its characteristics in a perfect fluid spacetime and lastly acquired Laplace equation from conformal $\eta$-Ricci–Yamabe soliton equation when the potential vector field $\xi$ of the soliton is of gradient type. Overall, the main novelty of the paper is to study the geometrical phenomena and characteristics of our newly introduced conformal Ricci–Yamabe and conformal $\eta$-Ricci–Yamabe solitons to apply their existence in a perfect fluid spacetime.

The object of the present paper is to study some properties of (LCS) _n -manifolds whose metric is Yamabe soliton. We establish some characterization of (LCS) _n -manifolds when the soliton becomes steady. Next we have studied some certain curvature conditions of (LCS) _n -manifolds admitting Yamabe solitons. Lastly we construct a 3-dimensional (LCS) _n -manifold satisfying the results.

The present paper is to deliberate the class of $3$-dimensional trans-Sasakian manifolds which admits $\eta$-Einstein solitons. We have studied $\eta$-Einstein solitons on $3$-dimensional trans-Sasakian manifolds where the Ricci tensors are Codazzi type and cyclic parallel. We have also discussed some curvature conditions admitting $\eta$-Einstein solitons on $3$-dimensional trans-Sasakian manifolds and the vector field is torse-forming. We have also shown an example of $3$-dimensional trans-Sasakian manifold with respect to $\eta$-Einstein soliton to verify our results. Ця стаття має на меті обговорення класу $3$-вимірних транс-Сасакянових многовидів, що допускають $\eta$-солітони Айнштайна. Ми вивчили $\eta$-солітони Айнштайна на $3$-вимірних транс-Сасакянових многовидах, де тензори Річчі мають тип Кодацці та є циклічно паралельними. Ми також обговорили деякі умови кривизни, що допускають $\eta$-солітони Айнштайна на $3$-вимірних транс-Сасакянових многовидах, а векторне поле торцеутворююче. Ми також показали приклад $3$-вимірного транс-Сасакянового многовиду відносно $\eta$-солітону Айнштайна для перевірки наших результатів.

We study conformal $$\eta$$η -Einstein solitons on the framework of trans-Sasakian manifold in dimension three. Existence of conformal $$\eta$$η -Einstein solitons on trans-Sasakian manifold is discussed. Then we find some results on trans-Sasakian manifold which are conformal $$\eta$$η -Einstein solitons where the Ricci tensor is cyclic parallel and Codazzi type. We also consider some curvature conditions with addition to conformal $$\eta$$η -Einstein solitons on trans-Sasakian manifold. We also use torse-forming vector fields in addition to conformal $$\eta$$η -Einstein solitons on trans-Sasakian manifold. Finally, an example of conformal $$\eta$$η -Einstein solitons on trans-Sasakian manifold is constructed.