Abstract
It is well known that 'an almost complex structure' J that is J2 = -I on the manifold M is called 'an almost Hermitian manifold' (M, J, G) if G(JX, JY) = G(X, Y) and proved that (F2M, JD, GD) is 'an almost Hermitian manifold' on the frame bundle of the second order F2M. The term 'an almost complex structure' refers to the general quadratic structure J2 = pJ + qI, where p = 0, q = -1. However, this paper aims to study the general quadratic equation J2 = pJ + qI, where p, q are positive integers, it is named as a metallic structure. The diagonal lift of the metallic structure J on the frame bundle of the second order F2M is studied and shows that it is also a metallic structure. The proposed theorem proves that the diagonal lift GD of a Riemannian metric G is a metallic Riemannian metric on F2M. Also, a new tensor field J similar to of type (1,1) is defined on F2M and proves that it is a metallic structure. The 2-form and its derivative dF of a tensor field J similar to are determined. Furthermore, the Nijenhuis tensor NJ similar to of a metallic structure J similar to and the Nijenhuis tensor NJD of a tensor field JD of type (1,1) on the frame bundle of the second order F2M are calculated.