Meta-metallic Riemannian manifolds

In this study, motivated by the Meta-Golden-Chi ratio, we develop essentially Meta-Metallic manifolds by using Meta-Metallic-Chi ratio and Metallic manifolds, provide an example and explore certain features of its Meta-Metallic structure. We give the conditions for integrability of the almost Meta-Metallic structure and examine its relation to the curvature tensor field. We further demonstrate that the Meta-Metallic Riemannian manifold is flat if and only if its curvature is constant. As a result, we show that a different notion of sectional curvature is needed in Meta-Metallic Riemannian manifolds.