A note on almost Ricci soliton and gradient almost Ricci soliton on para-Sasakian manifolds

TLDR: In this article, it was shown that if a 3D para-Sasakian manifold admits gradient almost Ricci soliton under certain conditions then either the manifold is of constant sectional curvature $-1$ or it reduces to a gradient Ricci Soliton.

Abstract: The object of the offering exposition is to study almost Ricci soliton and gradient almost Ricci soliton in 3-dimensional para-Sasakian manifolds. At first, it is shown that if $(g, V,\lambda)$ be an almost Ricci soliton on a 3-dimensional para-Sasakian manifold $M$, then it reduces to a Ricci soliton and the soliton is expanding for $\lambda$=-2. Besides these, in this section, we prove that if $V$ is pointwise collinear with $\xi$, then $V$ is a constant multiple of $\xi$ and the manifold is of constant sectional curvature $-1$. Moreover, it is proved that if a 3-dimensional para-Sasakian manifold admits gradient almost Ricci soliton under certain conditions then either the manifold is of constant sectional curvature $-1$ or it reduces to a gradient Ricci soliton. Finally, we consider an example to justify some results of our paper.