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

Abstract

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 $\nabla $. 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.