Liftings of metallic structures to tangent bundles of order r

: It is well known that the prolongation of an almost complex structure from a manifold M to the tangent bundle of order r on M is also an almost complex structure if it is integrable. The general quadratic structure F 2 = α F + β I is a generalization of an almost complex structure where α = 0 , β = − 1 . The purpose of this paper is to characterize a metallic structure deﬁned by the general quadratic structure F 2 = α F + β I , α, β ∈ N , where N is the set of natural numbers. We show that the r - lift of the metallic structure F in the tangent bundle of order r is also a metallic structure. Furthermore, we deduce a theorem on the projection tensor in the tangent bundle of order r . Moreover, prolongations of G -structures immersed in the metallic structure to the tangent bundle of order r and 2 are discussed. Finally, we construct examples of metallic structures that admit an almost para contact structure on the tangent bundle of order 3 and 4.

The name MMF was clearly explained by De Spinadel in [19].In [19], the author declared that "besides carrying the name of a metals, they have common mathematical properties that attach a fundamental importance to them in modern investigations about the search of universal roads to chaos".These metallic numbers have seen several interesting modern applications in researches that "analyze the behavior of non linear dynamical systems when they proceed from a periodic regime to a chaotic one" [17,20].
Goldberg, Yano and Petridis [5,6] defined and studied polynomial structures on the differentiable manifold.The general quadratic structure satisfying J 2 = αJ + βI, α, β ∈ N, where J is a tensor field of type (1,1) and N is the set of natural numbers, is named metallic structure on the differentiable manifold.Crasmareanu and Hretcanu [7] discussed metallic structures in the Riemannian manifolds.Azmi introduced metallic structures on the tangent bundle of P-Sasakian manifolds and investigated the integrability and parallelity of such metallic structures [1].Recently, the geometry of metallic structures have been studied in [2,3,8,10].
On the other hand, liftings of tensor fields and connections to tangent bundle were defined and studied by Yano and Davis [22], Ledger and Yano [12] and Yano [21].Yano and Ishiharo [24] developed the theory of prolongations of these geometric objects to the tangent bundle of order 2 and investigated integrability conditions.Morimoto [13] has studied prolongations of tensor fields, connections and G-structures to the tangent bundle of higher order.The first author has studied the prolongation of G-structure immersed in generalized almost r-contact structure on M to its tangent bundle T M of order 2 [11].
The prolongation of some classical G-structure tensor fields and connections immersed in an almost complex structure, f -structure, generalized almost r-contact structure, etc. to the tangent bundles of order r have been investigated [11,16,23].Inspired by the above mentioned studies, we define and study liftings of metallic structures to tangent bundle of order r.
The main contributions of the paper can be listed as follows: • The r-lift is applied to the metallic structure F and we show that it is also a metallic structure in T r (M).• We show that the metallic structure F (r) in T r (M) is integrable if and only if F is an integrable metallic structure.• The projection tensors are defined for the metallic structure and a theorem is proved about them.
• Some classical G-structures defined by tensor fields immersed in metallic structures in T r (M) and T 2 (M) have been investigated.• Examples on metallic structures that admits an almost para contact structure in T 3 (M) and T 4 (M) are constructed.

The tangent bundle of order r and lifts of tensor fields
In an n-dimensional differentiable manifold, let r ≥ 1 be a fixed integer and R be the real line.Consider the following equivalence relation ∼ among all differentiable mappings: "If the mappings F : R → M and G : R → M meet the following criteria where F and G are characterized respectively by x h = F h (t) and x h = G h (t), t is an element of R) with respect to local coordinates (x h ) in a coordinate neighborhood of {U, x h } containing the point P = F(0) = G(0), then we state that F ∼ G.Each equivalence relation is called r-jet of M and denoted by j r P (F), if this class contains a mapping F : R → M such that F(0) = P.The point P is called the target of the r-jet j r P (F).The set of all r-jets of M is called the tangent bundle of order r and denoted by T r (M)" [16].
Let π r be the bundle projection such that π r = T r (M) → M, i.e. j r P (F) = P.We define π sr : T r (M) → T s (M) for r > s by π sr ( j r P (F)) = j s P (F).Then we have π r = π s • π sr .Consider an r-jet j r P (F) belonging to π −1 (U) and the set where x h = F h (t), t is an element of R) of F in U and P = F(0).The r-jet j r P (F) is represented by the set (y (ν)h , ν = 0, 1, ...., r), where (y (0)h ) = (x h ) are coordinates of P in U. Therefore, the system of coordinates (y (ν) [15,23].

Almost para contact structure
Let M be an n-dimensional differentiable manifold of class C ∞ .Suppose that there is given a tensor field F of type (1,1) satisfying where U ∈ 1 0 (M), ω ∈ 0 1 (M).The structure (F, U, ω) of such fields F, U, ω is said to be an almost para contact structure ( [23], p. 66).

Lifts of metallic structures
Let M be an n-dimensional differentiable manifold and where I is the unit vector field.
Let P(t) be a polynomial of t and F (r) its r-lift in T r (M).Then ( [23], p. 393) Theorem 3.1.Let F ∈ 1 0 (M).Then F (r) is a metallic structure in T r (M) if and only if F is a metallic structure in M.
Proof.By operating the r-lift of Eq (3.1) and by using Eq (3.2), we obtain Hence, F (r) is a metallic structure in T r (M).Theorem 3.2.Let F and F (r) be metallic structures in M and T r (M), respectively.Then F (r) is integrable in T r (M) if and only if F is integrable in M.
Proof.Let N F and N F (r) denote the Nijenhuis tensors of F and F (r) respectively.Then ( [23], p. 393) since F is integrable if and only if N F = 0 [9].So, from (3.3), we obtained N F (r) = 0. Hence, the proof is completed.
Let l and m be the projection tensors defined in [8]: and where I denotes the identity operator in M.
Theorem 3.3.Let l and m be the projection tensors.Then l + m = 0, ) Proof.In the view of Eqs (3.4) and (3.5), then Similarly, other identities can be easily proved.
Let D l and D m be the complementary distributions corresponding to l and m, respectively in M. Let rank(F) = s, therefore dimension of D l is s and the dimension of D m is (n − s), where the dimension of M is n.Theorem 3.4.Let m be a projection tensor in M. Then the r-lift m (r) of m is a projection tensor in T r (M) and the distribution D determined by m (r) in T r (M) is integrable if and only if the distribution D determined by m is so in M.
Proof.Let D be a distribution in M. D is determined by a projection tensor m, i.e. m is an element of where r) .Thus, conditions (3.7) and (3.8) are equivalent to each other.This finishes the proof.

Prolongation of G-structures immersed in metallic structures to the tangent bundle of order r
In this section, we study the tangent bundle of order r on some classical G-structures, which are defined by tensor fields immersed in metallic structures.
Let P(M, π * , G) be a G-structure over a manifold M, where G is a Lie subgroup of GL(n, R). "A G-structure on M is a G-subbundle P(M, π * , G) of the frame bundle F M over M" [14].Let u = U be an open covering of M such that in each U there exists an n-frame {X i } which is adapted to the G-structure P(M, π * , G).The structure group T r (GL(n, R)) of T (T r (M)) is reducible to the tangent group T r (G) of order r, that is, the tangent bundle T r (M) of order r admits a T r (G)-structure P, which is called the prolongation of G-structure P in M to T (T r (M)) [23].
Let o F be a tensor field of type (1,1) in R n , which is invariant by G.We consider that M admits a G-structure P. Consider a coordinate neighborhood U, X h of M and an n-frame X (i) in U. Thus, if we set The local tensor field F is defined by Eq (4.1) in M. Hence F defines a global tensor field, which is called the tensor field induced in M from ( o F, P) [23].Now, we state the following proposition for later use ( [23], p. 406).Proposition 4.1."The prolongation P of a G-structure P given in M is integrable in the tangent bundle T (M) if and only if the G-structure P is integrable in M".Theorem 4.1.Let M denote a manifold that admits a metallic structure P (as a G-structure) defined by a tensor field F of type (1,1) such that F 2 = αF + βI.Let the tangent bundle T r (M) of order r be P of P is the metallic structure defined by the r-lift F (r) of F to T r (M).Then the metallic structure P is integrable in M if and only if P of P to T r (M) is integrable.invariant.Thus, we obtain T r (G) = GL(3n, C).By Proposition 4.1, the metallic structure P is integrable in M if and only if P of P to T r (M) is integrable.
This finishes the proof.By operating 2-lift of both sides of (4.2) and (4.3), we get This finishes the proof.

Conclusions
In this work, we have characterized a metallic structure given by the general quadratic structure F 2 = αF + βI, α, β ∈ N, where N is the set of natural numbers.We have proved that the r-lift of the metallic structure F in the tangent bundle of order r is also a metallic structure.Furthermore, the projection tensor in the tangent bundle of order r is studied.Moreover, we have discussed prolongations of G-structures immersed in the metallic structure to the tangent bundle of order r and 2. Finally, the examples are given to validate obtained results.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) on the differentiable manifold M.

1 1 (
M) such that m 2 = m.From Eq (3.2) and m 2 = m, we have (m (r) ) 2 = m (r) , that is the r-lift m (r) of m is a projection tensor in T r (M).The distribution D is integrable if and only if l[mX, mY] = 0, (3.7)where l = I − m denotes a projection tensor complementary to m and [, ] is the Lie bracket.By applying r-lift on (3.7), we obtain

F 2 = α o FFF
be a tensor of type (1,1) in R 2n such that o F +βI and denote by GL(n, C) the group of all elements of G = GL(2n, C) which leave o F invariant.Then the r-lift o to T r (R 2n ) is a tensor of type (1,1) satisfying (

Theorem 4 . 2 ..F
If a manifold M of 2n-dimension admits almost para contact structure P (as a Gstructure) determined by (F, U, ω) given in Eq (2.8), then, on the tangent bundle T 2 (M), P of P is the metallic structure determined by the tensor field Proof.G = GL(n, C) × I. Let the rank of o be 2s.Let o ξ denote a contravariant vector field and o ω a 1-form in R 2n such that

1 .η
If we denote by G, the group of all the elements of GL(2n, C), which leave invariant, then it is obvious thatG = GL(n, C) × I ⊂ GL(2n, R),where I denotes the trivial group.We set is a metallic structure in T 2 (r).Hence, T 2 (R 2n ) leaves ( .Thus, we obtain T (G) ⊂ GL(3n, C) × I ⊂ GL(6n, R).