Communications of the
Korean Mathematical Society
CKMS

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

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Commun. Korean Math. Soc. 2020; 35(2): 625-637

Online first article January 28, 2020      Printed April 30, 2020

https://doi.org/10.4134/CKMS.c190121

Copyright © The Korean Mathematical Society.

$\ast$-Ricci solitons and $\ast$-gradient Ricci solitons on 3-dimensional trans-Sasakian manifolds

Dibakar Dey, Pradip Majhi

Kolkata - 700019; Kolkata - 700019

Abstract

The object of the present paper is to characterize $3$-dimen\-sional trans-Sasakian manifolds of type $(\alpha,\beta)$ admitting $\ast$-Ricci solitons and $\ast$-gradient Ricci solitons. Under certain restrictions on the smooth functions $\alpha$ and $\beta$, we have proved that a trans-Sasakian $3$-manifold of type $(\alpha,\beta)$ admitting a $\ast$-Ricci soliton reduces to a $\beta$-Kenmotsu manifold and admitting a $\ast$-gradient Ricci soliton is either flat or $\ast$-Einstein or it becomes a $\beta$-Kenmotsu manifold. Also an illustrative example is presented to verify our results.

Keywords: Trans-Sasakian manifolds, $\ast$-Ricci soliton, $\ast$-gradient Ricci soliton, $\ast$-Einstein manifold

MSC numbers: Primary 53D15, 53C25

Supported by: The author Dibakar Dey is thankful to the Council of Scienti c
and Industrial Research, India (File no: 09/028(1010)/2017-EMR-1) for their
assistance

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