Geometry of conformal <i>η</i>-Ricci solitons and conformal <i>η</i>-Ricci almost solitons on paracontact geometry

Abstract

Abstract We prove that if an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>η</m:mi> </m:math> \eta -Einstein para-Kenmotsu manifold admits a conformal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>η</m:mi> </m:math> \eta -Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a conformal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>η</m:mi> </m:math> \eta -Ricci soliton is Einstein if its potential vector field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> </m:math> V is infinitesimal paracontact transformation or collinear with the Reeb vector field. Furthermore, we prove that if a para-Kenmotsu manifold admits a gradient conformal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>η</m:mi> </m:math> \eta -Ricci almost soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits conformal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>η</m:mi> </m:math> \eta -Ricci soliton and satisfy our results. We also have studied conformal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>η</m:mi> </m:math> \eta -Ricci soliton in three-dimensional para-cosymplectic manifolds.