Ricci Soliton and Certain Related Metrics on a Three-Dimensional Trans-Sasakian Manifold

In this article, a Ricci soliton and *-conformal Ricci soliton are examined in the framework of trans-Sasakian three-manifold. In the beginning of the paper, it is shown that a three-dimensional trans-Sasakian manifold of type $(\alpha ,\beta )$ admits a Ricci soliton where the covariant derivative of potential vector field V in the direction of unit vector field $\xi$ is orthogonal to $\xi$. It is also demonstrated that if the structure functions meet ${\alpha }^2={\beta }^2$, then the covariant derivative of V in the direction of $\xi$ is a constant multiple of $\xi$. Furthermore, the nature of scalar curvature is evolved when the manifold of type $(\alpha ,\beta )$ satisfies *-conformal Ricci soliton, provided $\alpha \ne 0$. Finally, an example is presented to verify the findings.