Almost \(\eta\)-Ricci solitons in \((LCS)_n\)-manifolds (Q668956)

In [Kyungpook Math. J. 43, No. 2, 305--314 (2003; Zbl 1054.53056)] \textit{A. A. Shaikh} introduced the notion of \textit{Lorentzian concircular structure} (briefly, $(\mathrm{LCS})_n$ Structure) as follows. Let $(M,g)$ be an $n$-dimensional Lorentzian manifold and $\zeta$ a unit time-like concircular vector field, $\nabla\zeta=\alpha(I+\eta\otimes\zeta)$, with $\alpha$ a nowhere zero smooth function on $M$ such that $d\alpha=\rho\eta$, for a smooth function $\rho$, where $\nabla$ is the Levi-Civita connection of $g$ and $\eta=i_{\zeta}g$ and $\varphi=I+\eta\otimes\zeta$. Another key definition is an almost $\eta$-Ricci soliton on $M$ which is a data $(g,\zeta,\lambda,\mu)$ satisfying the equation: \[ \mathfrak{L}_{\zeta}g+2S+2\lambda g+2\mu \eta\otimes \eta=0, \] where $\mathfrak{L}_{\zeta}$ is the Lie derivative operator along $\eta$, $S$ is the Ricci curvature tensor of $g$, and $\lambda$ and $\mu$ are smooth functions on $M$. In this paper, the author investigates the question whether $\zeta$ is of potential type, i.e., $\zeta=\mathrm{grad}(f)$, and provides lower and upper bounds for the Ricci curvature norm and a Bochner-type formula for the gradient almost $\eta$-Ricci soliton case.