TURK 2025

This study searches the geometric and structural characteristics of almost plastic pseudo-Riemannian manifolds, placing particular emphasis on the three-dimensional scenario. It investigates the interaction between an almost plastic structure and a pseudo-Riemannian metric, aiming to characterize thoroughly the conditions that define pure metric plastic P-Kählerian manifolds. In this framework, the associated fundamental tensor field not only exhibits symmetry but also serves as an alternative pure metric, enriching the underlying geometry.
One of the main results involves establishing the necessary and sufficient conditions for the integrability of the plastic structure, formulated through a system of partial differential equations. These conditions are governed by a specific scalar function that depends on a single variable. Additionally, the paper derives criteria for when an almost pure metric plastic pseudo-Riemannian manifold can be regarded as a pure metric plastic P-Kählerian manifold.
The research further examines the behavior of the Riemannian curvature tensor, showing that it becomes null under certain constraints, while the scalar curvature is shown to vanish when a particular polynomial condition is fulfilled. The study also extends to the analysis of vector fields, identifying the precise circumstances in which such fields constitute Killing vector fields or exhibit Ricci soliton behavior.
Special attention is given to three-dimensional Walker manifolds, with a detailed discussion on the necessary conditions for vanishing scalar curvature, the existence of Killing vector fields, and Ricci soliton configurations. Overall, this work contributes to the broader understanding of differential geometry by offering new insights into the intricate relationship between almost plastic structures and pseudo-Riemannian metrics.