A note on Almost Riemann Soliton and gradient almost Riemann soliton

Abstract

The quest of the offering article is to investigate \emph{almost Riemann soliton} and \emph{gradient almost Riemann soliton} in a non-cosymplectic normal almost contact metric manifold $M^3$. Before all else, it is proved that if the metric of $M^3$ is Riemann soliton with divergence-free potential vector field $Z$, then the manifold is quasi-Sasakian and is of constant sectional curvature -$λ$, provided $α,β=$ constant. Other than this, it is shown that if the metric of $M^3$ is \emph{ARS} and $Z$ is pointwise collinear with $ξ$ and has constant divergence, then $Z$ is a constant multiple of $ξ$ and the \emph{ARS} reduces to a Riemann soliton, provided $α,\;β=$constant. Additionally, it is established that if $M^3$ with $α,\; β=$ constant admits a gradient \emph{ARS} $(γ,ξ,λ)$, then the manifold is either quasi-Sasakian or is of constant sectional curvature $-(α^2-β^2)$. At long last, we develop an example of $M^3$ conceding a Riemann soliton.