Study of $\eta$-Ricci-Bourguignon solitons on $\xi$-conformally flat Lorentzian para-Kenmotsu manifolds
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The aim of this study is to characterize the $\eta$-Ricci-\linebreak Bourguignon soliton (briefly, $\eta$-$\mathcal{RBS}$) in the context of $\xi$-conformally flat Lorentzian para-Kenmotsu manifold. Further, we establish that the scalar curvature $\kappa$ of a $p$-dimensional Lorentzian para-Kenmotsu (briefly, $\mathcal{(LPK)}_{p})$ manifold admitting an $\eta$-$\mathcal{RBS}$ satisfies the Poisson's equation. Next, the condition for which the scalar curvature of $\mathcal{(LPK)}_{p}$ manifold admitting an $\eta$-$\mathcal{RBS}$ establishes Laplace equation. The presence of gradient $\eta$-$\mathcal{RBS}$ on $\mathcal{(LPK)}_{p}$ manifold has been established. At the end, an illustration of an $\mathcal{(LPK)}_{4}$ manifold is constructed to confirm the results of this manuscript.
Abstract
The aim of this study is to characterize the $\eta$-Ricci-\linebreak Bourguignon soliton (briefly, $\eta$-$\mathcal{RBS}$) in the context of $\xi$-conformally flat Lorentzian para-Kenmotsu manifold. Further, we establish that the scalar curvature $\kappa$ of a $p$-dimensional Lorentzian para-Kenmotsu (briefly, $\mathcal{(LPK)}_{p})$ manifold admitting an $\eta$-$\mathcal{RBS}$ satisfies the Poisson's equation. Next, the condition for which the scalar curvature of $\mathcal{(LPK)}_{p}$ manifold admitting an $\eta$-$\mathcal{RBS}$ establishes Laplace equation. The presence of gradient $\eta$-$\mathcal{RBS}$ on $\mathcal{(LPK)}_{p}$ manifold has been established. At the end, an illustration of an $\mathcal{(LPK)}_{4}$ manifold is constructed to confirm the results of this manuscript.