Liftings of metallic structures to tangent bundles of order $r$

It is well known that the prolongation of an almost complex structure from a manifold $M$ to the tangent bundle of order $r$ on $M$ is also an almost complex structure if it is integrable. The general quadratic structure $F^2=\alpha F+\beta I$ is a generalization of an almost complex structure where $\alpha =0,\beta =-1.$ The purpose of this paper is to characterize a metallic structure defined by the general quadratic structure $F^2=\alpha F+\beta I,\alpha ,\beta \in \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers. We show that the $r$-lift of the metallic structure $F$ in the tangent bundle of order $r$ is also a metallic structure. Furthermore, we deduce a theorem on the projection tensor in the tangent bundle of order $r$. Moreover, prolongations of $G$-structures immersed in the metallic structure to the tangent bundle of order $r$ and 2 are discussed. Finally, we construct examples of metallic structures that admit an almost para contact structure on the tangent bundle of order 3 and 4.