Ricci Geometry of Twin Metric on Almost Product Lorentzian Walker Three-Manifold

In this study, several theorems and results concerning the geometry of the Ricci tensor associated with a twin metric G_f on an almost product Lorentzian Walker three-manifold (M^3,J,g_f ) are presented. Firstly, we constitute the necessary and sufficient condition under which the quadruple (M^3,J,g_f,G_f) is Ricci-flat, and subsequently examine the relationship between the Ricci-flatness and flatness of the manifold. Furthermore, we establish a collection of theorems describing the conditions under which the quadruple (M^3,J,g_f,G_f) is locally Ricci symmetric and cyclic parallel Ricci. As a consequence of these theorems, the intrinsic relationship between these two geometric classes of manifolds is explicitly characterized.