Generalized Ricci soliton and paracontact geometry

Abstract

In the present paper, we study generalized Ricci soliton in the framework of paracontact metric manifolds. First, we prove that if the metric of a paracontact metric manifold M with $Q\varphi =\varphi Q$ is a generalized Ricci soliton (g, X) and if $X\ne 0$ is pointwise collinear to $\xi$, then M is K-paracontact and $\eta$-Einstein. Next, we consider closed generalized Ricci soliton on K-paracontact manifold and prove that it is Einstein provided $\beta (\lambda +2n\alpha )\ne 1$. Next, we study K-paracontact metric as gradient generalized almost Ricci soliton and in this case we prove that (i) the scalar curvature r is constant and is equal to $-2n(2n+1)$; (ii) the squared norm of Ricci operator is constant and is equal to $4n^2(2n+1)$, provided $\alpha \beta \ne -1$.