Investigating the Interplay of Codazzi Couplings and Connections in Metallic-like Pseudo-Riemannian Manifolds

In this paper, we explore the characteristics of metallic-like pseudo-Riemannian manifolds that employ various linear connections. We establish a series of relationships regarding these connections, specifically examining the links between distinct types and their counterparts. Our findings reveal that a particular type of connection is associated with another if and only if its counterpart exhibits a corresponding relationship, and vice versa. Furthermore, we present significant equalities pertaining to the intrinsic properties of the manifolds, illustrating how these characteristics interact within a metallic-like context. We discuss the conditions required for a particular type of coupling, revealing significant connections between the properties of different associated connections. Additionally, we derive crucial equivalences that emphasise the relationship between torsion, statistical structures, and Codazzi coupling across various connections. Collectively, our results offer a cohesive framework that clarifies the geometric and algebraic foundations of metallic-like pseudo-Riemannian manifolds, enhancing our understanding of their structure and properties.