Generalized Ricci soliton and paracontact geometry

TLDR: In this article, the authors studied generalized Ricci soliton in the framework of paracontact metric manifolds and proved that the scalar curvature r is constant and the squared norm of Ricci operator is constant.

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 $$\alpha \beta \ne -1$$ .