Geometry of Almost Pure Metric Pseudo-F-Manifolds: Insights from Walker 4-Manifolds

This article investigates the geometric and structural properties of almost pure metric pseudo-F-manifolds, with a focus on Walker 4-manifolds. We analyze integrability conditions and characterize pure metric pseudo-F-Kählerian Walker manifolds, identifying criteria under which the Riemann curvature tensor vanishes. The study also examines the existence of Killing vector fields and Ricci soliton structures. In particular, we show that integrability conditions for almost pure metric pseudo-F-structures are governed by partial differential equations and highlight the role of constant functions in defining pure metric pseudo-F-Kählerian properties. Additionally, we investigate special connections with torsion that preserve certain tensor structures, exploring their relationship to Codazzi pairs and the conditions necessary for torsion-freeness.