Ricci solitons and Ricci almost solitons on para-Kenmotsu manifold
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The purpose of this article is to study the Ricci solitons and Ricci almost solitons on para-Kenmotsu manifold. First, we prove that if a para-Kenmotsu metric represents a Ricci soliton with the soliton vector field $V$ is contact, then it is Einstein and the soliton is shrinking. Next, we prove that if a $\eta$-Einstein para-Kenmotsu metric represents a Ricci soliton, then it is Einstein with constant scalar curvature and the soliton is shrinking. Further, we prove that if a para-Kenmotsu metric represents a gradient Ricci almost soliton, then it is $\eta$-Einstein. This result is also hold for Ricci almost soliton if the potential vector field $V$ is pointwise collinear with the Reeb vector field $\xi$.
Abstract
The purpose of this article is to study the Ricci solitons and Ricci almost solitons on para-Kenmotsu manifold. First, we prove that if a para-Kenmotsu metric represents a Ricci soliton with the soliton vector field $V$ is contact, then it is Einstein and the soliton is shrinking. Next, we prove that if a $\eta$-Einstein para-Kenmotsu metric represents a Ricci soliton, then it is Einstein with constant scalar curvature and the soliton is shrinking. Further, we prove that if a para-Kenmotsu metric represents a gradient Ricci almost soliton, then it is $\eta$-Einstein. This result is also hold for Ricci almost soliton if the potential vector field $V$ is pointwise collinear with the Reeb vector field $\xi$.
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