Characterization of contact structures for the dual form of the vector field of an h-Ricci-Bourguignon soliton on the product manifold D2 and R

Abstract

In this paper, we study h-Ricci-Bourguignon solitons on the product manifold D2 x R, where D2 is the Poincare disk equipped with its standard hyperbolic metric. We first establish that this manifold admits a gradient Ricci-Bourguignon h-soliton only if the non-zero function h depends solely on the manifold's last coordinate. Next, in the case where h:=1, we explicitly determine the vector field that makes D2 x R a Ricci-Bourguignon soliton. Finally, we provide the necessary and sufficient condition for the dual 1-form of this vector field to define a contact structure, which corresponds to the existence of an explicitly determined function independent of the manifold's second local coordinate, thereby allowing us to compute its Reeb vector field.