Ricci-Yamabe Soliton on a Class of $4$-Dimensional Walker Manifolds

This article explores Ricci-Yamabe solitons on a specific class of 4-dimensional Walker manifolds. Walker manifolds, characterized by the existence of a parallel null distribution, find applications in general relativity and are fundamental objects of geometric study. We consider a particular pseudo-Riemannian metric, which depends on the smooth functions $f_1$, $f_2$, $f_3$. The main objective is to determine the conditions under which this manifold admits a Ricci-Yamabe soliton. We will explicitly calculate the components of the Ricci tensor, the scalar curvature, and the components of the Hessian Perelman potential. Solving the resulting system of partial differential equations, we will identify the constraints on the functions f1, f2, f3 and the vector field X for the existence of such solitons. Specific examples and their geometric properties will also be discussed.