Riemann Solitons on Relativistic Bulk-Viscous Fluid String Spacetime

In this article, we undertake a comprehensive study of almost Riemann solitons and their gradient counterparts within the framework of relativistic bulk-viscous fluid string (BVFS) geometries. We begin by analyzing the potential vector field. It is shown that, on BVFS backgrounds, this potential vector must coincide with a torse-forming field which is unit timelike. This correspondence allows the scalar curvature to be reformulated explicitly through both the soliton data and the intrinsic geometric invariants of the spacetime. Subsequently, we turn to almost gradient Riemann solitons generated by a potential function ψ, obtaining a relation that links the Laplacian of ψ with the curvature properties of BVFS metrics. A classification result is derived as well, asserting that whenever the soliton vector satisfies the Killing condition, the almost Riemann soliton falls into the shrinking, steady, or expanding categories. Moreover, we establish that any BVFS spacetime admitting a Killing field automatically accommodates an almost Riemann soliton. Finally, we prove that the model behaves like a dark fluid provided the potential vector field is of νRic-type, or if the BVFS structure is W2-flat or pseudoprojectively flat.