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

TL;DR

This paper investigates conditions under which a specific class of 4-dimensional Walker manifolds admits Ricci-Yamabe solitons, involving explicit calculations of geometric tensors and solving PDEs for functions defining the metric.

Contribution

It provides explicit criteria and examples for Ricci-Yamabe solitons on 4D Walker manifolds with a particular pseudo-Riemannian metric.

Findings

Derived conditions on functions f1, f2, f3 for soliton existence

Explicit solutions to PDEs defining Ricci-Yamabe solitons

Examples illustrating geometric properties of solutions

Abstract

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.