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 andstructure tensors. We demonstrate that distinct conjugate connections such as $h,\widetilde{h},J$ and $J^{\ast }-$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 $d^{\nabla }$-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 extendingexisting 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.