In this investigation, we examine the geometric character of almost Riemann solitons and gradient almost Riemann solitons in the context of perfect fluid solutions of the Einstein equations that admit a torse-forming vector field ζ. We first examine the conditions on the scalar curvature, which are necessary for the existence of an almost Riemann soliton or a gradient almost Riemann soliton in such solutions. We then examine the case of several physically reasonable types of perfect fluids, such as dark fluids, dust-filled universes, and the radiation-dominated epoch. We also show that any spacetime bearing an almost Riemann soliton with a conformal potential vector field must necessarily have an Einstein geometry. In addition, in the case of a perfect fluid spacetime with a torse-forming vector field, given the fulfillment of the almost Riemann soliton compatibility equation and Q·P=0, the scalar curvature of the spacetime must be constant. Finally, a rigidity theorem states that any parallel symmetric (0,2)-tensor defined on the spacetime must be a constant multiple of the metric tensor.
Jafari et al. (Wed,) studied this question.