### Novel theorems and integrability for metallic structures on the frame bundle of the second order

#### 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 (F 2M; JD; GD) is ‘an almost

Hermitian manifold’ on the frame bundle of the second order F 2M. The

term ‘an almost complex structure’ refers to the general quadratic structure J2 = pJ + qI; where p = -1; q = 0: 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 F 2M

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 F 2M. Also, a new tensor field J~ of type

(1,1) is defined on F 2M and proves that it is a metallic structure. The

2-form and its derivative dF of a tensor field J~ are determined. Furthermore, the Nijenhuis tensor of a metallic structure J~ on the frame

bundle of the second order F 2M is calculated. Finally, a study is done

on the Nijenhuis tensor NJD of a tensor field JD of type (1,1) on F 2M

is integrable

on the manifold M is called ‘an almost Hermitian manifold’ (M; J; G) if

G(JX; JY ) = G(X; Y ) and proved that (F 2M; JD; GD) is ‘an almost

Hermitian manifold’ on the frame bundle of the second order F 2M. The

term ‘an almost complex structure’ refers to the general quadratic structure J2 = pJ + qI; where p = -1; q = 0: 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 F 2M

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 F 2M. Also, a new tensor field J~ of type

(1,1) is defined on F 2M and proves that it is a metallic structure. The

2-form and its derivative dF of a tensor field J~ are determined. Furthermore, the Nijenhuis tensor of a metallic structure J~ on the frame

bundle of the second order F 2M is calculated. Finally, a study is done

on the Nijenhuis tensor NJD of a tensor field JD of type (1,1) on F 2M

is integrable

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