Metallic Structures for Tangent Bundles over Almost Quadratic φ-Manifolds

This paper aims to explore the metallic structure J(2 )= pJ + qI, where p and q are natural numbers, using complete and horizontal lifts on the tangent bundle TM over almost quadratic phi-structures (briefly, (phi,xi,eta)). Tensor fields F and F* are defined on TM, and it is shown that they are metallic structures over (phi,xi,eta). Next, the fundamental 2-form Omega and its derivative d Omega, with the help of complete lift on TM over (phi,xi,eta), are evaluated. Furthermore, the integrability conditions and expressions of the Lie derivative of metallic structures F and F* are determined using complete and horizontal lifts on TM over (phi,xi,eta), respectively. Finally, we prove the existence of almost quadratic phi-structures on TM with non-trivial examples.