Riemann solitons on imperfect fluid spacetimes

Abstract This research paper seeks to investigate the characteristics of almost Riemann solitons and almost gradient Riemann solitons within the framework of generalized Robertson–Walker (GRW) spacetimes that incorporate imperfect fluids. Our study begins by defining specific properties of the potential vector field linked to these solitons. We examine the potential vector field of an almost Riemann soliton on GRW imperfect fluid spacetimes, establishing that it aligns collinearly with a unit timelike torse-forming vector field. This leads us to express the scalar curvature in relation to the structures of soliton and spacetime. Furthermore, we explore the characteristics of an almost gradient Riemann soliton with a potential function ψ across a range of GRW imperfect fluid spacetimes, deriving a formula for the Laplacian of ψ . We also categorize almost Riemann solitons on GRW imperfect fluid spacetimes into three types: shrinking, steady, and expanding, when the potential vector field of the soliton is Killing. We prove that a GRW imperfect fluid spacetime with constant scalar curvature and a Killing vector field admits an almost Riemann soliton. Additionally, we demonstrate that if the potential vector field of the almost Riemann soliton is a ν ( Ric )-vector, or if the GRW imperfect fluid spacetime is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">W</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> -flat or pseudo-projectively flat, the resulting spacetime is classified as a dark fluid.