Ricci solitons and certain related metrics on 3-dimensional trans-Sasakian manifold

Abstract

In this paper we study certain types of metrics such as Ricci soliton, $\ast$-conformal Ricci soliton in 3-dimensional trans-Sasakian manifold. First we have shown that a 3-dimensional trans-Sasakian manifold of type $(\alpha ,\beta )$ admits a Ricci soliton where the covariant derivative of potential vector field in the direction of unit vector field $\xi$ is orthogonal to $\xi$. It is also shown that if the structure functions satisfy ${\alpha }^2={\beta }^2$ then the covariant derivative of the potential vector field in the direction of $\xi$ is a constant multiple of $\xi$. Further, we have evolved the nature of scalar curvature when the manifold satisfies $\ast$-conformal Ricci soliton of type $(\alpha ,\beta )$, provided $\alpha \ne 0$. Finally, we present an example to verify our findings.

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• Geometry of para-Sasakian metric as an almost conformal $\eta$-Ricci soliton