Riemann Solitons on Relativistic Space-Times

We examine almost Riemann solitons and almost gradient Riemann solitons in generalized Robertson–Walker space-times and perfect fluid space-times. At first, we prove that if a generalized Robertson–Walker space-time admits an almost Riemann soliton or an almost gradient Riemann soliton, then it becomes a perfect fluid space-time. Next, we observe that a space-time with an almost Riemann soliton whose potential vector field, is a conformal vector field, is an Einstein manifold, and if the potential vector field is a nonhomothetic conformal vector field, then space-time is of Petrov type O or N. In final, we prove that if a generalized Robertson–Walker space-time satisfies the definition of an almost Riemann soliton, and Q. P=0 then it is an Einstein manifold, and consequently it is a perfect fluid space-time.