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 φ -structures (brieﬂy, ( φ , ξ , η ) ). Tensor ﬁelds (cid:101) F and F ∗ are deﬁned on TM , and it is shown that they are metallic structures over ( φ , ξ , η ) . Next, the fundamental 2-form Ω and its derivative d Ω , with the help of complete lift on TM over ( φ , ξ , η ) , are evaluated. Furthermore, the integrability conditions and expressions of the Lie derivative of metallic structures (cid:101) F and F ∗ are determined using complete and horizontal lifts on TM over ( φ , ξ , η ) , respectively. Finally, we prove the existence of almost quadratic φ -structures on TM with non-trivial examples.


Introduction
Many ancient societies have made extensive use of the golden mean as a foundation for proportions, whether for creating music, sculptures, paintings, or buildings, such as temples and palaces [1].Fractal geometry has been explained using the silver mean [2].Some uses of a class of polynomial structures have been constructed on Riemannian manifolds for the metallic means family (a generalization of the golden mean) and generalized Fibonacci sequences in differential geometry.The geometric properties (such as totally geodesic, totally umbilical hypersurfaces, etc.) in metallic Riemannian manifolds have been explored in [3].This manuscript is focused on studying the properties of metallic structures for tangent bundles over a class of metallic Riemannian manifolds.
A quadratic equation of type where p and q are natural numbers, whose positive solutions are given by σ q p = p + p 2 + 4q 2 is known as a metallic means family [4].The most notable member is the well-known "Golden Mean" for p = q = 1.The metallic means family includes the silver mean for p = 2, q = 1, the bronze mean for p = 3, q = 1, the copper mean for p = 1, q = 2, and many others.
Let M be an n-dimensional differentiable manifold and TM be its tangent bundle.Let (M) and (TM) be the algebra of tensor fields of M and TM, respectively.The differential geometry of tangent bundle has been broadly studied by Davis [5], Sasaki [6], Tachibana and Okumura [7], Yano and Ishihara [8], and others.Yano and Kabayashi [9] defined the natural mapping (say complete lift) of (M) into (TM) and studied complete lifts of an almost complex structure and the symplectic structure on TM.Tanno [10] studied complete and vertical lifts of an almost contact structure on TM and defined a tensor field J of type (1,1) and proved that it is an almost complex structure on TM.Numerous investigators have studied various geometric structures on TM-an almost complex structure by Yano [11], paracomplex structures by Tekkoyun [12], almost r-contact structures by Das and Khan [13], and many others [14][15][16][17][18][19].
In [20], Azami explored complete and horizontal lifts of metallic structures and analyzed the geometric properties of these structures.Salimov et al. [19] studied complete lifts of symplectic vector fields on tangent and cotangent bundles.Recently, Khan [21] introduced a new tensor field J of type (1,1) and demonstrated that J is a metallic structure (MS) on the frame bundle FM.Furthermore, the derivative and the coderivative of fundamental 2-form and the Nijenhuis tensor of J on FM are discussed.
On the other hand, Sasaki [6] defined a structure named as an almost contact structure and demonstrated its basic algebraic properties such as a Riemannian metric, the fundamental 2-form, etc., on M. Later on, Sato [22] defined the notion of an almost paracontact structure and analyzed its geometrical properties.
The main aim of this paper is summarized as: • Tensor fields F and F * are defined on TM over the structure (φ, ξ, η) and we prove that they are metallic structures, which generalize the notion of almost complex structure J introduced by Tanno [10].

•
The basic geometrical properties of fundamental 2-Form and its derivative on TM over the structure (φ, ξ, η) are studied.

•
The integrability conditions and expressions of the Lie derivative of metallic structures F and F * with the help of complete and horizontal lifts, respectively, on TM over the structure (φ, ξ, η) are investigated.

•
The existence of almost quadratic φ-manifolds on TM with non-trivial examples are proved.

Preliminaries
Let M be an n-dimensional differentiable manifold of class C ∞ and TM be the tangent bundle over a manifold M such that TM = x∈M T x M with the projection map π : TM → M, where T x M represents the tangent space at a point x of M. Let (U, x h ) be a local chart in M and (x h , y h ) be a local coordinate in π −1 (U) ⊂ TM and be called the induced coordinate in π −1 (U).
Let f , η, Υ 1 , and F be a function, a 1-form, a vector field, and a tensor field of type (1,1) of M, respectively.The vertical lifts f V , η V , Υ 1 V , and F V on TM in terms of partial differential equations are given by [8,25] where η i , x h , and F h i , i, h = 1, 2, . . ., n are local components of η, Υ 1 , and F on M, respectively.The complete lifts f C , η C , Υ 1 C , and F C on TM in the term of partial differential equations are given by By the definition of the lift, we have By the definition of the Lie product of the lift, we have Let f be a function and ∇ is an affine connection on M. The horizontal lift is where ∇ f is a gradient of f on M, γ is an operator, and [8], p. 86).
Let Υ 1 , η, and S be a vector field, a 1-form, and a tensor field of arbitrary type on M, respectively.The horizontal lifts Υ 1 H , η H , and S H on TM are given by By the definitions of the lifts, we have By the definitions of the Lie product of the lifts, we have where £ Υ 1 represents the Lie derivative with respect to Υ 1 and R represents the curvature tensor of ∇ given by ∇Υ In addition, let P and Q be arbitrary elements of (M), then Let g C be the complete lift on TM of a Riemannian metric g on M. Then [20] where Υ 1 and Υ 2 are vector fields on M.

Metallic Structure
The quadratic structure J on M satisfying where J denotes a tensor field of type (1,1), I is the identity vector field, and p, q are natural numbers, named as a metallic structure.The structure (M, J) is called a metallic manifold [26][27][28][29][30][31].
Let g be a Riemannian metric on M such that or equally, where Υ 1 and Υ 2 are vector fields on M. The structure (M, J, g) is said to be a metallic Riemannian manifold [32,33].The Nijenhuis tensor of J is denoted by N J and given by

Almost Quadratic φ-Structure
Debnath et al. [23] introduced the notion of structure (φ, ξ, η) and discussed some geometric properties of such structures.Next, Gonul et al. [24] investigated the connection between MS and almost quadratic φ-structures.Consider a non-null tensor fields φ of type (1,1), a 1-form η and a vector field ξ on M satisfying where p and q are constants and I is the identity vector field.The structure (φ, ξ, η) is called an almost quadratic φ-structure on M and the manifold (M, φ, ξ, η) is called an almost quadratic φ-manifold [23,24,34].Furthermore, or equally, The structure (φ, ξ, η, g) is termed as an almost quadratic metric φ-structure and the manifold (M, φ, ξ, η, g) is called an almost quadratic metric φ-manifold.
In addition, the 1-form η associated with g such that and the 2-Form Φ is given by [35] is said to be the fundamental form of an almost quadratic metric φ-manifold.
The Nijenhuis tensor of (φ, ξ, η) is denoted by N φ and given by where Υ 1 and Υ 2 are vector fields on M.

Proposed Theorems for the Complete Lifts of Metallic Structures on the Tangent Bundle Over (φ, ξ, η)
In this section, we study the structure (φ, ξ, η) geometrically using complete lift on TM.A tensor field F on the tangent bundle is defined and show that it is an MS by using the complete lift on TM over (φ, ξ, η).Next, mathematical operators, namely fundamental 2-Form Ω and the derivative dΩ using the complete lift on TM over (φ, ξ, η), are calculated.Furthermore, the integrability condition and the Lie derivative of an MS( F) by using the complete lift on TM over (φ, ξ, η) are established.
Let M be an n dimensional differentiable manifold and φ, η, and ξ be a tensor field of type (1,1), a 1-form and a vector field on M, respectively.
From Azami [20] and Khan [21], let us introduce a new (1, 1)-type tensor field F on TM as where A = 2σ q p −p 2 √ pφ C +q .Since p, q are natural numbers and φ is non-singular, therefore pφ C + q > 0 and A = 0. Theorem 1.Let TM be a tangent bundle of M immersed with structure (φ, ξ, η).Then F given by ( 12) is a metallic structure on TM.
Proof.Let Υ 1 be a vector field on M and Υ 1 C and Υ 1 V be complete and vertical lifts of Υ 1 , respectively, on TM.Applying ξ V , ξ C , and φ C on (12), we obtain where Υ 1 is a vector field on TM.
In the view of ( 12)-( 15), we obtain This shows that F is an MS on TM.

Corollary 1.
Let Υ 1 and Υ 2 be vector fields on M and F be an MS on TM given by ( 12) such that η(Υ 1 ) = 0, then Proof.The proof is obtained by applying Υ C 1 and Υ V 1 on F given by ( 12) and using η(Υ 1 ) = 0. Let g C be the complete lift of the metric g on TM.The 2-form on TM defined by where Υ 1 and Υ 2 are vector fields and F is an MS given by (12) on TM.
Theorem 2. Let TM be the tangent bundle of M, g C be the complete lift of g and F be an MS given by ( 12) on TM, then the 2-form Ω is given by where Υ 1 and Υ 2 are vector fields on TM.
Proof.(18) and using ( 1) and ( 12), we have (18) and using ( 1) and ( 12), we have (18) and using (1) and ( 12), we have (18) and using ( 1) and ( 12), we have Theorem 3. Let TM be the tangent bundle of M, g C be the complete lift of g, and F be an MS given by ( 12), then the derivative dΩ is given by Proof.We have Here Υ1 , Υ2 , Υ3 are arbitrary vector fields on TM.
Proof.Let N F stand for the Nijenhuis tensor of F. Then where Υ 1 and Υ 2 are vector fields on TM.

Proposed Theorems for the Horizontal Lift of Metallic Structures on the Tangent Bundle Over (φ, ξ, η)
In this section, we study (φ, ξ, η) geometrically using a horizontal lift on TM.A tensor field F * on the tangent bundle is defined and shows that it is an MS by using the horizontal lift on TM over (φ, ξ, η).Furthermore, the integrability condition and Lie derivative of an MS F * by using the horizontal lift on TM over (φ, ξ, η) are established.
Let M be an n dimensional differentiable manifold and φ, η, and ξ be the tensor field of type (1,1), a 1-form, and a vector field on M. Let φ H , η H , and ξ H be horizontal lifts of φ, η, and ξ, respectively, on TM.Applying horizontal lifts on ( 9), (10), and using (1), we obtain From Azami [20] and Khan [21], let us introduced a new tensor field F * of type (1,1) on TM as where B = 2σ q p −p 2 √ pφ H +q .Since p, q are natural numbers and φ is non-singular, therefore pφ H + q > 0 and A = 0. Theorem 6.Let the tangent bundle TM of M be immersed with (φ, ξ, η).Then the metallic structure F * , given by (21), is an MS on TM.
Proof.Let Υ 1 be a vector field on M and Υ H 1 , Υ C 1 , and Υ V 1 be horizontal, complete, and vertical lifts of Υ 1 , respectively, on TM.Applying ξ H , ξ V , ξ C , and φ H on (21), we obtain (iii) In the view of ( 21) and ( 22), we obtain .
This shows that F * is an MS.

Examples of Almost Quadratic φ-Manifolds
In this section, we prove the existence of almost quadratic φ-manifolds on the tangent bundle with non-trivial examples.
Example 2. Let M = {(x, y, z) : x, y, z ∈ , z = 0} be a differentiable manifold of dimension 3, is a set of real numbers.We suppose that e C i and e V i ; i = 1, 2, 3 be complete and vertical lifts on TM of independent vector fields e i ; i = 1, 2, 3 on M, then they form a basis {e C i , e V i ; i = 1, 2, 3} for TM of M. Let g C be the complete lift of a Riemannian metric g such that g ij = δ ij , where δ ij is Kronecker delta.That is, 1 , e C 3 ) = (g(Υ 1 , e 3 )) C = (η(e 3 )) C , g C (e C 3 , e C 3 ) = 1, g V (Υ V 1 , e C 3 ) = 0, g V (e V 3 , e V 3 ) = 0, where Υ 1 is a vector field on M. If φ represents the (1,1) symmetric tensor on M such that 3 ) = 0, Then we can easily verify that where p = 2e z , q = e 2z ⇒ p 2 + 4q = 8e 2z = 0.This shows that M is an almost quadratic φ-manifold and the structure (φ, ξ, η) is an almost quadratic φ-structure on M.
The manifold M is an almost quadratic metric φ-manifold and the structure (φ, ξ, η, g) is an almost quadratic metric φ-structure on M. Example 3. A paracontact structure (φ, η, ξ) on M such that [22] φ 2 = I − η ⊗ ξ is an almost quadratic φ-structure when p = 0, q = 1 in (9).The new tensor F of type (1,1) given by (12) becomes It can be easily proved that F is almost a product structure.
Remark 1.For the horizontal lift, we can obtain the similar examples of almost quadratic φ-manifolds.

Conclusions
In this work, we have characterized a metallic structure by using the complete and horizontal lifts over an almost quadratic φ-structure (φ, ξ, η).Tensor fields F and F * are defined on TM over the structure (φ, ξ, η) and we proved that they are metallic structures, which generalizes the notion of an almost complex structure J introduced by Tanno [10].The fundamental geometrical properties of fundamental 2-Form and its derivative on TM over the structure (φ, ξ, η) were calculated.The integrability conditions and expressions of the Lie derivative of metallic structures F and F * on TM over the structure (φ, ξ, η) were determined.Finally, we demonstrated that almost quadratic φ-manifolds exist on TM with non-trivial examples.Future studies could fruitfully explore this issue further by considering the polynomial structure Q(F) = F n + a n F n−1 + • • • + a 2 F + a 1 I, where F is the tensor field of type (1,1).