Extending Information Geometry: Quasi-Statistical Structures on Metallic-Like Pseudo-Riemannian Manifolds

In this paper, we investigate the conditions under which quasi-statistical structures can be realized on metallic-like pseudo-Riemannian manifolds. By combining the flexibility of quasi-statistical geometry with the algebraic richness of metallic-like structures, we provide a unified framework for analyzing compatibility conditions among metrics, conjugate connections and structure tensors. We demonstrate that distinct conjugate connections such as h,˜h, J and J∗−conjugates, may yield quasi-statistical manifolds under appropriate compatibility assumptions. In particular, we establish a number of geometric results under the assumptions of Codazzi coupling and d∇-closedness. The novelty of our approach lies in combining the framework of metallic-like manifolds with quasi-statistical structures in the presence of torsion, thereby extending existing results in the literature and opening new directions for further research. Finally, we also present a theorem concerning the Tachibana operator, which highlights additional structural properties of the manifolds under consideration.