Ricci–Bourguignon Soliton on Three-Dimensional Contact Metric Manifolds

This paper aims to classify a certain type of three-dimensional complete non-Sasakian contact manifold with specific properties, namely Qξ=σξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q\xi =\sigma \xi $$\end{document} and admitting Ricci–Bourguignon solitons. In the case of constant σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}, the paper proves that if the potential vector field of the Ricci–Bourguignon soliton is orthogonal to the Reeb vector field, then the manifold is either Einstein or locally isometric to E(1, 1). Under a similar hypothesis, the paper shows that a (κ,μ,ϑ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\kappa ,\mu ,\vartheta )$$\end{document}-contact metric manifold is locally isometric to E(1, 1). Finally, the paper considers the scenario where the potential vector is pointwise collinear with the Reeb vector field and presents some results.