Gradient almost Ricci soliton warped product

In this paper we present a necessary and sufficient condition for constructing gradient almost Ricci solitons that are realized as warped products. This will be done by means of Bishop-O'Neill's formulas and a particular study of Riemannian manifolds satisfying a Ricci-Hessian type equation. We prove existence results and give an example of particular solutions of the PDEs that arise from our hypothesis of construction. We also prove rigidity results for gradient Ricci soliton Riemannian product and Ricci-Hessian type equation.