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

: In this work, we have characterized the frame bundle FM admitting metallic structures on almost quadratic φ -manifolds φ 2 = p φ + qI − q η ⊗ ζ , where p is an arbitrary constant and q is a nonzero constant. The complete lifts of an almost quadratic φ -structure to the metallic structure on FM are constructed. We also prove the existence of a metallic structure on FM with the aid of the ˜ J 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 ˜ J on FM . Finally, we construct an example of it to ﬁnish.


Introduction
Numerous types of f -structures on a differentiable manifold M have been studied by Yano [1], Ishihara and Yano [2], Blair [3], Nakagawa [4] and others. Yano proposed the notion of an f -structure obeying f 3 + f = 0, f is a tensor field of type (1,1), which is the generalization of an almost complex structure and an almost contact structure [5] and investigated some basic results of it. Later, Goldberg and Yano [6] and Goldberg and Perridis [7] defined a polynomial structure P(J) = J n + a n J n−1 + ... + a 2 J + a 1 I, where a 1 , a 2 , . . . , a n are real numbers, J is a tensor field of type (1,1) and I is an identity tensor field of type (1,1) on M. Moreover, some important polynomial structures such as an f (3, ε)-structure [8], a general quadratic structure [9], an almost complex structure and an almost product structure [1], φ(4, ±2)-structures [10] and an almost r-contact structure [11] are studied and the fundamental results are established in these papers.
Recently, the polynomial structure J 2 = pJ + qI, p, q ∈ N, where N is the set of natural numbers, of degree 2 is known as a metallic structure on M [12][13][14]. For specific values of p and q, metallic structures become prominent structures given below: 1 an almost product structure [15] 0 −1 an almost complex structure [16,17] 1 1 a golden structure [18,19] 2 1 a silver structure [20] Hretceanu and Crasmareanu [21] initiated the study of golden and metallic structures on a Riemannian manifold and interpreted the geometry of submanifolds admitting both structures on M. The various geometric properties of such structures in a metallic (and golden) Riemannian manifold and a metallic (and golden) warped product Riemannian manifold were studied in [22][23][24][25][26]. Debnath and Konar [27] defined a new type of structure named as an almost quadratic φ-structure (φ, ζ, η) on M and studied some geometric properties of such structures. Next, Gonul et al. [28] established the relationship between an almost quadratic metric φ-structure and a metallic structure on M. Most recently, Gok et. al. [29] defined a generalized structure namely f (a,b) (3, 2, 1)-structures on manifolds and construct a framed f (a,b) (3, 2, 1)-structures on M.
On the other hand, let M be an m-dimensional differentiable manifold, TM its tangent bundle and FM its frame bundle. The notion of the mappings, namely vertical, complete and horzontal lifts from the manifold M to its tangent bundle TM were introduced by Sasaki [30], Yano and Ishihara [31] and Yano and Davis [32]. Kabayashi and Nomizu [33], Mok [34] and Okubo [35] have studied the complete lift of a vector field A to FM. The geometric structures such as an almost contact metric structure (φ, ζ, η, g), and almost complex structures J on FM have been studied by Bonome et al. [16], who established the integrability and normality of such structures on FM.
The main objective of this paper can be summarized as follows: • We study the complete lifts of an almost quadratic φ-structure to the metallic structure on FM.

•
We establish the existence of a metallic structure on FM in the tensor fieldJ, which we define. • We obtain results on the 2-Form and its derivative on FM.

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We derive the expressions of the Nijenhuis tensor of a tensor fieldJ on FM.

•
We construct an example related to it.
Remark: b a (M) and b a (FM) are symbolized as the set of all (a, b)-type tensor fields in M and FM respectively [17].

Preliminaries
Let F, A, f and η be a tensor field of type (1,1), a vector field, a function and a 1form, respectively, on M. The horizontal, vertical and α-vertical lifts of F, A, f and η are represented by F H , A H , A (α) , f H , η V and η H α on FM and they are expressed in terms of partial differential equations as [16,17] where Γ h ij , A i , F h j and η i are the local components of a linear connection ∇, A, F and η, respectively on M. Proposition 1. ∀A, B ∈ 1 0 (M), by using mathematical operators, we have the following where α, β = 1, . . . , m and δ α β denotes the Kronecker delta.
, R is the curvature tensor of ∇.
Let g be a Riemannian metric on a Riemannian manifold M and g D its diagonal metric on FM, then ∀Ã,B ∈ 1 0 (FM), where ∇ and∇ represent the Levi-Civita connection of (M, g) and (FM, g D ), respectively. Proposition 3. ∀A, B ∈ 1 0 (M) , by using mathematical operators, we have the following

Metallic Structure
If a (1, 1) tensor field J obeying where N is the set of natural numbers and I is an identity operator, determines a polynomial structure on a manifold M, the structure is referred to as metallic. A metallic manifold is defined as (M, J) when a manifold M possesses a metallic structure (MS) J.
The Nijenhuis tensor N J of J is expressed as ∀A, B ∈ 1 0 (M).
In addition, the 1-form η is associated with g such that and the fundamental 2-Form Φ is given by [3] Φ(A, B) = g(A, φB).

Proposed Theorems on FM Admitting Metallic Structures on Almost Quadratic φ-Manifolds
In this section, we construct the complete lifts of an almost quadratic φ-structure to the metallic structure on FM.
Next, we obtain the results on the 2-Form and its derivative on FM.
Boname et al. [16] proposed and gave the definition ofJ on FM as Recently, Khan [36] proposed and gave the definition of the tensor fieldJ on FM as where Motivated by the above definitions, let us introduce a tensor fieldJ of type (1,1) on FM asJ where

Theorem 1. LetÃ be a vector field on FM. ThenJ given by (22) is a metallic structure on FM.
Proof. To prove thatJ defined in (22) is a metallic structure, we have to prove that Taking the horizontal lift A H and β th -vertical lift A (β) for each β = 1, . . . 2n + 1 on both sides of (22), we infer where In view of (22), we providẽ Definition 1. The 2-Form Ω ofJ is given by ∀Ã,B ∈ 1 0 (FM).
Proof. Using (9) and (29), we infer Theorem 3. The differential dΩ on FM is expressed as Proof. The differential dΩ is given by Formulas (iii) and (iv) can be easily obtained.