Metallic Riemannian Structures on the Tangent Bundles of Riemannian Manifolds with $g-$Natural Metrics

Let $(M,g)$ be a Riemannian manifold and $(TM,\tilde{g})$ be its tangent bundle with the $g-$natural metric. In this paper, a family of metallic Riemannian structures $J$ is constructed on $TM,$ found conditions under which these structures are integrable. It is proved that $(TM,\tilde{g},J)$ is decomposable if and only if $(M,g)$ is flat.