Some Solitons on Lorentzian Para-Kenmotsu Manifolds

In this paper we study the nature of the Einstein soliton and $\eta $-Einstein soliton in the framework of Lorentzian para-Kenmotsu manifolds (briefly, LP-Kenmotsu manifolds). We find an expression for scalar curvature of LP-Kenmotsu manifolds admitting the Einstein soliton and $\eta $-Einstein soliton in various cases. We prove that if an LP-Kenmotsu manifold containsan $\eta $-Einstein soliton with a parallel Reeb vector field then the manifold is $\eta $-Einstein. We study the nature of the $\eta $-Einstein soliton on these manifolds with conformal, collinear and torse forming potential vector fields. We also study the $\eta $-Einstein soliton on these manifolds satisfying the curvature conditions: $\left( \xi .\right) _{R}.S=0, $ $\left( \xi .\right) _{W_{2}}.S=0$ and $\left( \xi .\right) _{S}.W_{2}=0,$ where $R,$ $S$ and $W_{2}$ are, respectively, the Riemannian curvature tensor, Ricci curvature tensor and $W_{2}$-curvature tensor.