Vertical and complete lifts of sections of a (dual) vector bundle and Legendre duality

Supplementary comments about generalized Lie algebroids are presented and a new point of view over the construction of the Lie algebroid generalized tangent bundle of a (dual) vector bundle is introduced. Using the general theory of exterior differential calculus for generalized Lie algebroids, a covariant derivative for exterior forms of a (dual) vector bundle is introduced. Using this covariant derivative, the complete lift of an arbitrary section of a (dual) vector bundle is discovered. A theory of Legendre type and Legendre duality between vertical and complete lifts is presented. Finally, a duality between Lie algebroids structures is developed.


Introduction
It is well-known that the lift of geometrical objects such as functions, vector fields and 1-forms defined on the base of the usual Lie algebroid Using these lifts, it is possible to introduce the lift of a (pseudo) Riemannian metric structure. The Sasaki lift of a Riemannian metric structure on M is an important example of metric structures on T M used in differential geometry with many applications in physics [16]. Lift  were introduced and studied by several authors [8,9,17]. In many papers such as [5,11,14,15], the authors studied the lifts to the second order tangent bundle, tensor bundle and jet bundle. The Lie algebroids are important issues in physics and mechanics since the extension of Lagrangian and Hamiltonian systems to their entity [6,7,12] and catching the Poisson structure [13]. Several authors presented and studied the lift of geometrical objects of a Lie algebroid ((F, ν, N ), [, ] F , (ρ, Id N )) to the Lie algebroid prolongation. Using the vertical and complete lifts of sections of a Lie algebroid, the first author presented important results about Lie simetry and horizontal lifts in the general framework of prolongation Lie algebroid [10].
Extending the notion of Lie algebroid from one base manifold to a pair of diffeomorphic base manifolds, the second author introduced the generalized Lie algebroid [1,2]. Using the lift of a differentiable curve defined on the base of a generalized Lie algebroid, he developed a new theory of mechanical systems with many applications in physics [4]. The space used for developing this theory of mechanical systems is the Lie algebroid generalized tangent bundle (((ρ, η)T F, (ρ, η)τ F , F ), [, ] (ρ,η)T F , (ρ, Id F )), of a generalized Lie algebroid ((F, ν, N ), [, ] F,h , (ρ, η)).
This paper is arranged as follows. Some notions and results about exterior differential algebra of a vector bundle and information about generalized Lie algebroids are presented in Section 1. Using a vector bundle (E, π, M ) anchored by a generalized Lie algebroid ((F, ν, N ), [, ] F,F , (ρ, η)) and a vector bundle morphism (g, h) we obtain a new point of view over construction of the Lie algebroid generalized tangent bundle (((ρ, η)T E, (ρ, η)τ E , E), [, ] (ρ,η)T E , (ρ, Id E )), in Section 2. Using the exterior differential calculus of the exterior algebra of the generalized Lie algebroid ((F, ν, N ), [, ] F,h , (ρ, η)) presented in [3], in Section 3, we introduce a Lie covariant derivative for the exterior algebra of the vector bundle (E, π, M ). Using this Lie covarinat derivative, we introduce in Theorem 11 the complete (g, h)-lift u c ∈ Γ(T E, τ E , E) of an arbitrary section u ∈ Γ(E, π, M ). Using the complete (g, h)-lift of a function f ∈ F (N ) we obtain new results for vertical and complete (g, h)-lifts. In the final of Section 3, we introduced the complete and vertical (g, h)-lifts u C , u V ∈ Γ((ρ, η)T E, (ρ, η)τ E , E), of a section u ∈ Γ(E, π, M ) and important results are presented in Theorem 18. Also, using the dual vector bundle ( A dual theory for the vertical and complete lifts is presented in Section 5 and similar results are obtained. A general presentation of Lagrange (Finsler) and Hamilton (Cartan) fundamental functions and a theory of Legendre type are presented in Section 6. Using the tangent (ρ, η)-application of the Legendre bundle morphism associated to a Lagrange respectively Hamilton fundamental function, we obtain new results about duality between vertical and complete (g, h)-lifts in Section 7. New results about duality between Lie algebroids structures and the Legendre (ρ, η)-equivalence between the vector bundle (E, π, M ) and its dual ( * E, * π, M ) are presented in Section 8.

Preliminaries
Let (E, π, M ) be an arbitrary vector bundle. If Γ(E, π, M ) is the set of the sections of the vector bundle (E, π, M ) and F (M ) is the set of differentiable real-valued functions on M , then (Γ(E, π, M ), +, ·) is a F (M )-module.
Using the previous definition, we obtain We set Then it is easy to see that (Λ(E, π, M ), +, ·, ∧) is a F (M )-algebra. This algebra will be called the exterior differential algebra of the vector bundle (E, π, M ). Now let (F, ν, N ) be an another vector bundle and (ϕ, ϕ 0 ) is a vector bundles morphism from (E, π, M ) to (F, ν, N ) such that ϕ 0 is a isomorphism from M to N . Then using the operation it results that (Γ(F, ν, N ), +, ·) is a F (M )-module and we obtain the modules morphism Γ(E, π, M ) Definition 3 Let (ϕ, ϕ 0 ) be a vector bundles morphism from (E, π, M ) to (F, ν, N ) such that ϕ 0 is a isomorphism from M to N . Then we define the pull-back application for any u 1 , ..., u q ∈ Γ(E, π, M ).
Remark 4 If (ρ, η) and (T h, h) are two vector bundles morphisms given by the diagrams F then we obtain the modules morphism Definition 5 A generalized Lie algebroid is a vector bundle (F, ν, N ) given by the diagrams: where h and η are arbitrary isomorphisms, (ρ, η) is a vector bundles morphism from (F, ν, N ) to (T M, τ M , M ) and the operation We denote by ((F, ν, N ), [, ] F,h , (ρ, η)) the generalized Lie algebroid defined in the above. Moreover, the couple ([ , ] F,h , (ρ, η)) is called the generalized Lie algebroid structure. In particular, if η = Id M = h, then we obtain the definition of Lie algebroid. So, any Lie algebroid can be regarded as a generalized Lie algebroid.
So, for any vector local (m + r)-chart (V, t V ) of (E, π, M ), there exist the real functions Vg b α − −−−−− → R, α ∈ {1, · · · , n}, b ∈ {1, · · · , r}, such thatg b α (κ) · g α a (κ) = δ b a andg a α (κ) · g β a (κ) = δ β α , for any κ ∈ V . Thus, we can discuss about vector bundles morphism g −1 , h −1 from (F, ν, N ) to (E, π, M ) with components It is remarkable that since Definition 8 If u = u a s a is a section of (E, π, M ), then we introduce the vertical lift of u as section of Γ(T E, τ E , E) given by If {s a } be a basis of sections of Γ(E, π, M ), then using the above equation we have s ∨ a =∂ a . Using the locally expression of u ∨ we can deduce Lemma 9 If u and v are sections of E and f ∈ F (M ), then for any θ ∈ Λ q (F, ν, N ) and z 1 , ..., z q ∈ Γ(F, ν, N ), will be called the covariant Lie derivative with respect to the section z. Also for any u ∈ Γ(E, π, M ), the F (M )-multilinear application for any ω ∈ Λ q (E, π, M ) and u 1 , ..., u q ∈ Γ(E, π, M ), will be called the covariant Lie (g, h)-derivative with respect to the section u.
Definition 10 For any a = 1, · · · , r, we consider the real function U a on E such that where the real numbers y 1 , · · · , y r are the fibre components of the point u x in the arbitrary vector local (m + r)-chart (V, s V ).
Theorem 11 Let u be a section of (E, π, M ). Then there exists a unique vector field u c ∈ Γ(T E, τ E , E), the complete (g, h)-lift of u, satisfying the following conditions: Proof. At first we let that there exists u c such that satisfies in (i) and (ii).
Since u c is a vector field on E, then we can write it as follows: From two above equations we obtain On the other hand we have

Condition (i) give us
On the other hand we have Thus we have But condition (ii) gives us Since ω is arbitrary, then we suppose that ω = s b . Thus we have ω b = 1 and ω a = 0, for any a = b. Therefore we obtain So, for u c we can obtain the following locally expression: The above relation prove the existence and uniqueness of the complete lift.
where (V, s V ) is an arbitrary vector local (m + r)-chart.

Definition 15
The complete (g, h)-lift u C of a section u ∈ Γ(E, π, M ) is the section of ((ρ, η)T E, (ρ, η)τ E , E) given by Using the above definition, we can obtain In the particular case of Lie algebroids, (g, η, h) = (Id E , Id M , Id M ), the complete lifts are given by the equality: • π}∂ a , and in the classical case, ρ = Id T M , the complete lifts are given by the equality: Definition 16 If u = u a s a is a section of (E, π, M ), then we introduce the vertical (g, h)-lift of u as section of ((ρ, η)T E, (ρ, η)τ E , E) given by If u = u a e a ∈ Γ(E, π, M ), then in the locally expressions we get In particular, we have s V a =∂ a . Remark 17 Using the almost tangent (g, h)-structure J (g,h) given by Theorem 18 The Lie brackets of vertical and complete (g, h)-lifts satisfy the following equalities: Proof. Direct calculation gives us On the other hand, we have The above equation gives us From (9) and (10) we get (ii). Now we prove (iii). we have where and Using (7) and direct calculation we get where On the other hand, we have g γ d =g a α g γ a g α d . Derivative of the above expression with respect to j, we get ∂ j (g γ d ) = −∂ j (g a α )g γ a g α d . Using the above equation, we obtain Setting (17)-(21) in (15) we deduce that C r = 0. This equation together with (11) and (14) give us (iii).

The generalized tangent bundle of a dual vector bundle
We consider the following diagrams: * Setting (x i , p a ) as the canonical local coordinates on ( * E, * π, M ), where i ∈ 1, · · · , m, a ∈ 1, · · · , r, and is a change of coordinates on ( * E, * π, M ), then the coordinates p a change to p a ′ according to the rule p a ′ = M a a ′ p a . If v = v α t α is a section of (F, ν, N ), then we define its corresponding section If we define The base sections ( * ∂ α ,∂ a ) are called the natural (ρ, η)-base. Now consider the vector bundles morphism ( *
Definition 19 If f ∈ F (N ) (respectively f ∈ F (M )), then the real function is called the vertical lift of the function f.

Remark 20 Since
then using the above definition we obtain
Definition 24 For any a = 1, · · · , r, we consider the real function U a on * where the real numbers p 1 , · · · , p r are the fibre components of the point * u x in the arbitrary vector local (m + r)-chart (V, s V ).
Proof. Similar to the proof of Theorem 11, we obtain the following locally expression for * u c that show the existence and uniqueness of it.
where (V, s V ) is an arbitrary vector local (m + r)-chart.

Similar to the Lemmas 13 and 14, we have
Lemma 29 If * u is a section of ( * E, * π, M ) and f, f 1 , f 2 ∈ F (N ), then It is easy to check that Γ( * Definition 31 If * u = u a s a be a section of ( * E, * π, M ), then we introduce the vertical lift of * u as section of ((ρ, η) T * If u = u a e a ∈ Γ(E, π, M ), then in the locally expressions we get * which gives us (s a ) V =∂ a .
Remark 32 Using the almost tangent ( * g, h)-structure * J ( * g,h) given by Similar to Theorem 18 we can deduce the following Theorem 33 The Lie brackets of generalized vertical lifts and generalized complete ( * g, h)-lifts satisfy the following equalities:

Legendre transformation
Definition 34 A Lagrange fundamental function on the vector bundle (E, π, M ) is a function E L − −− → R which satisfies the following conditions: where 0 means the null section of (E, π, M ) .

Remark 35
If (U, s U ) is a local vector (m + r)-chart, then we obtain the following real functions defined on π −1 (U ): L a = ∂L ∂y a , L ab = ∂ 2 L ∂y a ∂y b . Definition 36 If L is a Lagrange fundamental function such that rank L ab (u x ) = r, for any u x ∈ π −1 (U ) \ {0 x }, then we will say that the Lagrange fundamental function L is regular and we obtain the real functionsL ab locally defined by Definition 37 If L is a Lagrange fundamental function, then we build the Legendre bundles morphism for any vector local (m+r)-charts (U, s U ) and (U, * s U ) of (E, π, M ) and ( * E, * π, M ) respectively.
Using the above definition, we deduce that if u = u a s a belongs to Γ(E, π, M ), then we obtain its Legendre transformation Definition 38 If L is a Lagrange fundamental function positively homogenous of degree two, namely F 1 . L is positively 2-homogenous on the fibres of vector bundle (E, π, M ) ; F 2 . For any vector local (m + r)-chart (U, s U ) of (E, π, M ) , the hessian is positively define for any u x ∈ π −1 (U ) \ {0 x }, then L will be called Finsler fundamental function.
Proposition 39 If L is a Finsler fundamental function on the vector bundle (E, π, M ), then Proof. From (24) we have ϕ L (u x ) = u a (x)L ab (u x )s b (x). But, the Finsler fundamental function L satisfies because L is positively 2-homogenous on the fibres of (E, π, M ). This complete the proof.
Definition 40 A Hamilton fundamental function on the dual vector bundle for any vector local (m + r)-chart (U, s U ) of (E, π, M ) and for any vector local Using the above definition, we deduce that if * u = u a s a belongs to Γ( * E, * π, M ), then we obtain its Legendre transformation belongs to Γ(E, π, M ).
Theorem 46 If L is a Lagrange fundamental function on the vector bundle (E, π, M ) and H is a Hamiltonian on the dual vector bundle ( * E, * π, M ), then: Proof. Using definition (37) and (43), we deduce that if and only if L ab (u x ) H bc (ϕ L (u x )) = δ c a (u x ) , for any u x ∈ π −1 (U ). Thus we have (i). Similar, we can prove (ii).
for any vector local (m+r)-chart (U, * s U ) of ( * E, * π, M ), where u a (x), a ∈ {1, · · · , r}, are the components of the solution of the system of differentiable equations , u x ∈ π −1 (U ) , will be called the Legendre transformation of the Lagrangian L.
It is remarkable that in the general case, if L is a Lagrange fundamental function on the vector bundle (E, π, M ) and H is its Legendre transformation, then H • ϕ L = L, but in particular, if L is a Finsler fundamental function on the vector bundle (E, π, M ) and H is its Legendre transformation, then H •ϕ L = L.
for any vector local (m + r)-chart (U, s U ) of (E, π, M ) , where u a (x) , a ∈ 1, r are the components of the solution of the system of differentiable equations

Duality between vertical and complete lifts
Let L be a Lagrangian on the vector bundle (E, π, M ) and let H be its Legendre transformation.