Certain paracontact metrics satisfying gradient ϱ-Ricci–Bourguignon almost solitons

In this paper, we thoroughly study Formula: see text-Ricci–Bourguignon almost soliton and gradient Formula: see text-Ricci–Bourguignon almost soliton in the paracontact geometry, precisely, on Formula: see text-paracontact and para-Sasakian manifolds. Here, we prove that if the metric Formula: see text of the Formula: see text-paracontact manifold endows a Formula: see text-Ricci–Bourguignon almost soliton with the nonzero potential vector field Formula: see text parallel to Formula: see text, then the manifold is an Einstein with Einstein constant Formula: see text. Next, we show that if a para-Sasakian manifold represents a gradient Formula: see text-Ricci–Bourguignon almost soliton, then the manifold is an Einstein with constant scalar curvature Formula: see text. We also discuss Formula: see text-Ricci–Bourguignon almost soliton on Formula: see text-paracontact manifold.