$*$-conformal $\eta$-Ricci soliton on Lorentzian para-Sasakian manifolds

Abstract

The objective of this paper is to characterize LP-Sasakian manifolds admitting $*$-conformal $\eta$-Ricci soliton. In this way, we let $*$-conformal $\eta$-Ricci soliton LP-Sasakian manifolds satisfying $\mathfrak{R}\cdot\mathcal{S}^\flat=0$. Further, we study Codazzi type of Ricci tensor and cyclic parallel Ricci tensor endowed with LP-Sasakian manifolds admitting $*$-conformal $\eta$-Ricci soliton. Furthermore, we investigate the Ricci-recurrent, pseudo Ricci symmetric and $\eta$-parallel $\varphi$-tensor in LP-Sasakian manifolds admitting $*$-conformal $\eta$-Ricci soliton. Moreover, we study the conformal curvature tensor on LP-Sasakian manifolds admitting $*$-conformal $\eta$-Ricci soliton. Finally, we give a $3$-dimensional example which justify the presence of $*$-conformal $\eta$-Ricci soliton in a LP-Sasakian manifolds.