Almost conformal Ricci solituons on $3$-dimensional trans-Sasakian manifold

Abstract

In this paper we have shown that if a $3$-dimensional trans-Sasakian manifold M admits conformal Ricci soliton $(g,V,\lambda )$ and if the vector field $V$ is point wise collinear with the unit vector field $\xi$, then $V$ is a constant multiple of $\xi$. Similarly we have proved that under the same condition an almost conformal Ricci soliton becomes conformal Ricci soliton. We have also shown that if a $3$-dimensional trans-Sasakian manifold admits conformal gradient shrinking Ricci soliton, then the manifold is an Einstein manifold.