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

Abstract

Motivated by the vector fields \(\varphi (\textrm{Ric}) \) defined through the condition \(\nabla \varphi = \mu \textrm{Ric} \), where \(\mu \) is a real constant and \(\varphi \) denotes a vector field on the underlying manifold, this paper investigates the existence problem of Ricci vector fields \(( \mu = 1 )\) on three-dimensional Lorentzian Walker manifolds \((M^3, g_f) \). 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 \(g_f \) is pure with respect to J, and we construct the associated twin metric \(G_f \), 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 \(G_f \) on the almost product Walker manifold \((M^3, J, g_f) \).