m-quasi-Einstein metric and contact geometry
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We study m-quasi-Einstein metric in the framework of contact metric manifolds. The existence of such metric has been confirmed on the class of \( \eta \)-Einstein K-contact manifold, in which the potential vector field V is a constant multiple of the Reeb vector field \(\xi \). Next, we consider closed m-quasi-Einstein metric on a complete K-contact manifold and prove that it is Sasakian and Einstein provided \(m\ne 1\). We also proved that, if a K-contact manifold M admits an m-quasi-Einstein metric such that the potential vector field V is conformal, then V becomes Killing and M is \( \eta \)-Einstein. Finally, we obtain a couple of results on a contact metric manifold M admitting an m-quasi-Einstein metric, whose potential vector field is a point wise collinear with the Reeb vector field.
Keywordsm-quasi-Einstein metric Ricci soliton Gradient Ricci soliton Contact metric manifold K-contact manifold
Mathematics Subject Classification53C25 53C15 53D10
The author expresses his sincere thanks to the referee for offering many valuable suggestions towards the improvement of the paper.
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