Geometric properties of almost pure metric plastic pseudo-Riemannian manifolds

This paper investigates the geometric and structural properties of almost plastic pseudo-Riemannian manifolds, with a specific focus on three-dimensional cases. We explore the interplay between an almost plastic structure and a pseudo-Riemannian metric, providing a comprehensive analysis of the conditions that define pure metric plastic P-Kählerian manifolds. In this context, the fundamental tensor field is symmetric and also represents another pure metric. A key finding is the necessary and sufficient condition for the integrability of the plastic structure by means of partial differential equations, which is characterized by a specific function depending solely on one variable. Furthermore, the necessary and sufficient condition for an almost pure metric plastic pseudo-Riemannian manifold to be pure metric plastic P-Kählerian is obtained. The study also reveals that the Riemannian curvature tensor vanishes under specific conditions, while the scalar curvature vanishes if a particular polynomial form is satisfied. The analysis extends to the properties of vector fields, identifying conditions under which they become Killing vector fields or form Ricci soliton structures. Additionally, we examine three-dimensional Walker manifolds, detailing the conditions for vanishing scalar curvature, Killing vector fields, and Ricci soliton structures. Our findings provide a detailed framework for understanding almost plastic manifolds, contributing to the broader field of differential geometry by clarifying the relationship between plastic structures and pseudo-Riemannian metrics.