Geometry of para-Sasakian metric as an almost conformal $\eta$-Ricci soliton

Abstract

In this paper, we initiate the study of conformal $\eta$-Ricci soliton and almost conformal $\eta$-Ricci soliton within the framework of para-Sasakian manifold. We prove that if para-Sasakian metric admits conformal $\eta$-Ricci soliton, then the manifold is $\eta$-Einstein and either the soliton vector field $V$ is Killing or it leaves $\phi$ invariant. Here, we have shown the characteristics of the soliton vector field $V$ and scalar curvature when the manifold admitting conformal $\eta$-Ricci soliton and vector field is pointwise collinear with the characteristic vector field $\xi$. Next, we show that a para-Sasakian metric endowed an almost conformal $\eta$-Ricci soliton is $\eta$-Einstein metric if the soliton vector field $V$ is an infnitesimal contact transformation. We have also displayed that the manifold is Einstein if it represents a gradient almost conformal $\eta$-Ricci soliton. We have developed an example to display the alive of conformal $\eta$-Ricci soliton on 3-dimensional para-Sasakian manifold.