Gradient Ricci-Bourguignon solitons and applications

In this article, we discuss the geometry of gradient Ricci-Bourguignon solitons (GRB) and then we characterize the general relativistic space-time with Ricci-Bourguignon (RB) and GRB solitons. First, we obtain the expression for the scalar curvature of a compact GRB soliton. Then, we prove that the gradient of the potential function of GRB soliton is bounded if its scalar curvature satisfies a boundedness condition. Then the Riemannian curvature of a 4-dimensional GRB soliton and its derivative are investigated whenever the Weyl tensor is harmonic. It is proven that if the potential vector field is torse-forming, then a compact RB soliton becomes perfect fluid space-time. We also discuss the bounds of the first eigenvalue of Laplacian. A volume formula for GRB soliton is obtained. Further, we study the application of conformal vector field on a RB soliton and obtain the expression for the Ricci curvature. Next, we find when RB soliton is expanding, shrinking and steady, if relativistic perfect fluid space-time admits a RB soliton with conformal vector field. We also construct a non-trivial example of GRB soliton equipped with a conformal potential vector field.