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 -$\lambda$, provided $\alpha ,\beta =$ constant. Other than this, it is shown that if the metric of $M^3$ is \emph{ARS} and $Z$ is pointwise collinear with $\xi$ and has constant divergence, then $Z$ is a constant multiple of $\xi$ and the \emph{ARS} reduces to a Riemann soliton, provided $\alpha ,\beta =$constant. Additionally, it is established that if $M^3$ with $\alpha ,\beta =$ constant admits a gradient \emph{ARS} $(\gamma ,\xi ,\lambda )$, then the manifold is either quasi-Sasakian or is of constant sectional curvature $-({\alpha }^2-{\beta }^2)$. At long last, we develop an example of $M^3$ conceding a Riemann soliton.

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