Existence and Physical Properties of Gradient Ricci-Yamabe Solitons

We first prove the existence of the gradient Ricci-Yamabe soliton (briefly GRYS) by constructing an explicit example endowed with the Robertson-Walker metric. Then we focus on the physical properties of the gradient Ricci-Yamabe solitons satisying Einstein's field equations, under the assumptions of different subspaces of Gray's decompositions. For instance, we prove that if a GRYS space-time satisfying Einstein's field equations, in which the gradient of the potential function psi is a unit-timelike torse-forming vector field, belongs to the subspaces B and B', then it is a Robertson-Walker space-time with vanishing shear and vorticity. Moreover, its possible local cosmological structures are of Petrov types I, D, or O. Finally, we obtain the equations of state of a perfect-fluid space-time admitting the GRYS whose velocity field is a unit-timelike Killing vector field.