In this paper we study ∗-η-Ricci soliton on Sasakian manifolds. Here, we have discussed some curvature properties on Sasakian manifold admitting ∗-η-Ricci soliton. We have obtained some significant results on ∗-η-Ricci soliton in Sasakian manifolds satisfying R(ξ, X) · S = 0, S(ξ, X) · R = 0, P̄ (ξ, X) · S = 0, where P̄ is Pseudo-projective curvature tensor.The conditions for ∗-η-Ricci soliton on ϕ-conharmonically flat and ϕ-projectively flat Sasakian manifolds have been obtained in this article. Lastly, we have given an example of 5-dimensional Sasakian manifolds satisfying ∗-η-Ricci soliton.

The goal of the paper is to deliberate conformal Ricci soliton and *-conformal Ricci soliton within the framework of paracontact geometry. Here we prove that if an ?-Einstein para-Kenmotsu manifold admits conformal Ricci soliton and *-conformal Ricci soliton, then it is Einstein. Further we have shown that 3-dimensional para-cosymplectic manifold is Ricci flat if the manifold satisfies conformal Ricci soliton where the soliton vector field is conformal. We have also constructed some examples of para-Kenmotsu manifold that admits conformal and *-conformal Ricci soliton and verify our results.

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.