Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2022; 37(3): 825-837

Online first article May 13, 2022      Printed July 31, 2022

https://doi.org/10.4134/CKMS.c210046

Copyright © The Korean Mathematical Society.

Study of gradient solitons in three dimensional Riemannian manifolds

Gour Gopal Biswas, Uday Chand De

University of Kalyani, Kalyani-741235; 35, Ballygaunge Circular Road, Kolkata -700019

Abstract

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.