A note on almost Riemann Solitons and gradient almost Riemann Solitons

Abstract

The object of the offering article is to investigate an almost Riemann soliton and a 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 a 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. Also, it is shown that if the metric of $M^3$ is an almost Riemann Soliton and Z is pointwise collinear with $\xi$ and has constant divergence, then Z is a constant multiple of $\xi$ and the almost Riemann Soliton reduces to a Riemann soliton, provided $\alpha ,\;\beta =$constant. Additionally, it is established that if $M^3$ with $\alpha ,\; \beta =$ constant admits a gradient almost Riemann soliton $(\gamma ,\xi ,\lambda )$, then the manifold is either quasi-Sasakian or is of constant sectional curvature $-(\alpha ^2-\beta ^2)$. Finally, we develop an example of $M^3$ admitting a Riemann soliton.