Proposed results concerning Codazzi and torsion coupleds on pure metallic metric geometries

The paper explores the intricacies of metallic pseudo-Riemannian manifolds represented as (M, J, g), where M is a smooth manifold with a metallic structure denoted as J, a pseudo-Riemannian metric denoted as g and a linear connection with torsion denoted as ?. The focus is on linear connections with torsion, introducing novel conditions and classifications. The paper introduces a new integrability condition for the metallic structure J, including the torsion-coupling condition, and provides specific outcomes for Codazzi-coupled and torsion-coupled scenarios. It explores Codazzi-couplings involving a torsion tensor and investigates properties of associated tensor fields. Additionally, it discusses conditions for the purity of connections and their operators, deriving significant results through torsion or Codazzi coupling. It establishes conditions for a metallic pseudo-Riemannian manifold to become locally metallic pseudo-Riemannian manifold and quasi-metallic pseudo-Riemannian manifold. Overall, the paper contributes to understanding integrable structures with torsion in metallic pseudo-Riemannian manifolds.