Research PaperNo Access

On $K$ $K$ -contact metric manifolds satisfying an almost gradient Ricci–Bourguignon soliton

Santu Dey

https://orcid.org/0000-0002-2601-3788

Department of Mathematics, Bidhan Chandra College, Asansol — 4, West Bengal 713304, India

E-mail Address: santu.mathju@gmail.com

E-mail Address: santu@bccollegeasansol.ac.in

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and

Young Jin Suh

https://orcid.org/0000-0003-0319-0738

Department of Mathematics, Kyungpook National University, Daegu 41566, South Korea

Research Institute of Real and Complex Manifolds (RIRCM), Kyungpook National University, Daegu 41566, South Korea

E-mail Address: yjsuh@knu.ac.kr

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https://doi.org/10.1142/S0129055X24500326Cited by:0 (Source: Crossref)

Abstract

The main purpose of the paper is to study an almost gradient Ricci–Bourguignon soliton (RB soliton) within the framework of $K$ $K$ -contact manifolds and $(κ, ψ)$ $(κ, ψ)$ -contact manifolds. First, we prove that if complete $K$ $K$ -contact manifold endows a gradient RB soliton, then the manifold is compact Sasakian and isometric to unit sphere $S^{2 n + 1}$ $S^{2 n + 1}$ . Next, we show that if a complete contact metric satisfies an almost RB soliton with a non-zero potential vector field is collinear with the Reeb vector field $ξ$ $ξ$ and the Reeb vector field $ξ$ $ξ$ acting as an eigenvector of the Ricci operator, then it is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field $ξ$ $ξ$ . Lastly, we prove that if the metric of a non-Sasakian $(κ, ψ)$ $(κ, ψ)$ -contact manifold is an almost gradient RB soliton, then it is flat in dimension 3 and in higher dimensions it is locally isometric to $E^{n + 1} \times S^{n} (4)$ $E^{n + 1} \times S^{n} (4)$ .

Keywords:

AMSC: 53D15, 53C15, 53C25

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