On Ricci Vector Fields and Almost Product Structures in Three-Dimensional Walker Manifolds

Motivated by the vector fields φ(Ric) defined through the condition ∇φ=μRic, where μ is a real constant and φ denotes a vector field on the underlying manifold, this paper investigates the existence problem of Ricci vector fields (μ=1) on three-dimensional Lorentzian Walker manifolds (M3,gf). A theorem is established providing the necessary and sufficient conditions for their existence, followed by several propositions offering a local classification of the component functions of Ricci vector fields and of the functions f determining the Walker metric under specific geometric constraints. Further results characterize the existence of spacelike and timelike Ricci vector fields. Introducing a product structure J on the Walker manifold, we show that the metric gf is pure with respect to J, and we construct the associated twin metric Gf, whose index exhibits a particularly noteworthy behavior when f>0. Finally, a theorem is presented giving the necessary and sufficient conditions for the existence of Ricci vector fields with respect to the twin metric Gf on the almost product Walker manifold (M3,J,gf).