Characterizations of the Frame Bundle Admitting Metallic Structures on Almost Quadratic ϕ-Manifolds

In this work, we have characterized the frame bundle (Formula presented.) admitting metallic structures on almost quadratic (Formula presented.) -manifolds (Formula presented.), where p is an arbitrary constant and q is a nonzero constant. The complete lifts of an almost quadratic (Formula presented.) -structure to the metallic structure on (Formula presented.) are constructed. We also prove the existence of a metallic structure on (Formula presented.) with the aid of the (Formula presented.) tensor field, which we define. Results for the 2-Form and its derivative are then obtained. Additionally, we derive the expressions of the Nijenhuis tensor of a tensor field (Formula presented.) on (Formula presented.). Finally, we construct an example of it to finish.