Properties of Three-Dimensional Almost Pure Twin Walker Manifolds

In this paper, it has been investigated three-dimensional almost pure twin Walker manifolds equipped with a Walker metric gεf, an almost product structure F satisfying the purity condition, and the associated twin metric Gεf. First, the curvature properties of the manifold are studied, and explicit expressions for the Ricci curvature tensor are obtained. Later, the necessary and sufficient conditions for the manifold to be Ricci-flat are derived. Furthermore, the symmetry structure of the manifold is analyzed by determining its Killing and conformal vector fields through the solution of the corresponding systems of partial differential equations. In addition, the geodesic equations are derived and solved for certain special cases. The obtained results show that the qualitative behavior of geodesics strongly depends on the defining function fx,y,z. Finally, graphical representations of selected geodesics are presented using Python-based visualizations, illustrating the geometric behavior of the manifold.