Communications of the
Korean Mathematical Society CKMS

We characterize a three-dimensional Riemannian manifold endowed with a type of semi-symmetric metric $P$-connection. At first, it is proven that if the metric of such a manifold is a gradient $m$-quasi-Einstein metric, then either the gradient of the potential function $psi$ is collinear with the vector field $P$ or, $lambda=-(m+2)$ and the manifold is of constant sectional curvature $-1$, provided $Ppsi
eq m$. Next, it is shown that if the metric of the manifold under consideration is a gradient $ho$-Einstein soliton, then the gradient of the potential function is collinear with the vector field $P$. Also, we prove that if the metric of a 3-dimensional manifold with semi-symmetric metric $P$-connection is a gradient $omega$-Ricci soliton, then the manifold is of constant sectional curvature $-1$ and $lambda+mu=-2$. Finally, we consider an example to verify our results.

Keywords: Ricci soliton, gradient Ricci soliton, gradient $m$-quasi-Einstein metric, gradient $ho$-Einstein soliton, $eta$-Ricci soliton, gradient $eta$-Ricci soliton, semi-symmetric metric connection, semi-symmetric metric $P$-connection

MSC numbers: Primary 53D10, 53C25, 53C15

Supported by: Gour Gopal Biswas is financially supported by UGC, Ref. ID 423044.