Generalized Lie Algebroids and Connections over Pair of Diffeomorphic Base Manifolds

Extending the definition of Lie algebroid from one base manifold to a pair of diffeomorphic base manifolds, we obtain the generalized Lie algebroid. When the diffeomorphisms used are identities, then we obtain the definition of Lie algebroid. We extend the concept of tangent bundle, and the Lie algebroid generalized tangent bundle is obtained. In the particular case of Lie algebroids, a similar Lie algebroid with the prolongation Lie algebroid is obtained. A new point of view over (linear) connections theory of Ehresmann type on a fiber bundle is presented. These connections are characterized by a horizontal distribution of the Lie algebroid generalized tangent bundle. Some basic properties of these generalized connections are investigated. Special attention to the class of linear connections is paid. The recently studied Lie algebroids connections can be recovered as special cases within this more general framework. In particular, all results are similar with the classical results. Formulas of Ricci and Bianchi type and linear connections of Levi-Civita type are presented. MSC 2010: 00A69, 58B34, 53B05


Introduction
In general, if C is a category, then we denote |C| the class of objects. For any A, B ∈ |C|, we denote C(A, B) the set of morphisms of A source and B target, and Iso C (A, B) the set of C-isomorphisms of A source and B target. Let Liealg, Mod, Man, B, and B v be the category of Lie algebras, modules, manifolds, fiber bundles, and vector bundles, respectively.
We know that if (E, π, M ) ∈ |B v |, Γ(E, π, M ) = {u ∈ Man(M, E) : u • π = Id M } and F (M ) = Man(M, R), then (Γ(E, π, M ), +, ·) is a F (M )-module. If (ϕ, ϕ 0 ) ∈ B v ((E, π, M ), (E , π , M )) such that ϕ 0 ∈ Iso Man (M, M ), then, using the operation, For the second time there appeared an idea to change in the previous diagrams the identities morphisms with arbitrary Man-isomorphisms h and η as in the following diagrams: is an operation with the following properties: GLA 1 the equality holds good for all u, v ∈ Γ(F, ν, N ) and f ∈ F (N ). We will say that the triple is a generalized Lie algebroid. The couple ([ , ] F,h , (ρ, η)) will be called generalized Lie algebroid structure.
So we extend the notion of Lie algebroid from one base manifold to a pair of diffeomorphic base manifolds, and we obtain the notion of generalized Lie algebroid. So we can discuss about the category GLA of generalized Lie algebroids. Examples of objects of this category are presented in Section 2. We remark that GLA is a subcategory of the category B v .
Using this new notion, we build the Lie algebroid generalized tangent bundle in Theorems 7 and 10. Particularly, if ((F, ν, N ), [ , ] F , (ρ, Id N )) is a Lie algebroid, (E, π, M ) = (F, ν, N ) and h = Id M , then we obtain a similar Lie algebroid with the prolongation Lie algebroid (see [7]). New and important results are presented in [8,10,11,17,19]. See also [14,15,16,18]. Using this general framework, in Section 4, we propose and develop a (linear) connections theory of Ehresmann type for fiber bundles in general and for vector bundles in particular. It covers all types of connections mentioned. In this general framework, we can define the covariant derivatives of sections of a fiber bundle (E, π, M ) with respect to sections of a generalized Lie algebroid In particular, if we use the generalized Lie algebroid structure: of covariant derivatives for the vector bundle (E, π, M ) such that for any X ∈ X (M ) and u, v ∈ Γ(E, π, M ), is very important, because the Yang-Mills theory is a variational theory that use (see [1]) the Yang-Mills functional: Cov 0 (E,π,M) where R D X is the curvature. Using our linear connections theory, we succeed to extend the set Cov 0 (E,π,M) of Yang-Mills theory, because using all generalized Lie algebroid structures for the tangent bundle (T M, τ M , M), we obtain all possible linear connections for the vector bundle (E, π, M ).
More importantly, it may bring within the reach of connection theory certain geometric structures that have not yet been considered from such a point of view. Finally, using our theory of linear connections, the formulas of Ricci and Bianchi type and linear connections of Levi-Civita type are presented.

Journal of Generalized Lie
In particular, using arbitrary isometries (symmetries, translations, rotations, etc.) for the Euclidean threedimensional space Σ, and arbitrary basis for the module of sections, we obtain a lot of generalized Lie algebroid structures for the tangent vector bundle (T Σ, τ Σ , Σ). Remark 5. If (E, π, M ) ∈ |B|, then we obtain the B v -morphism

The Lie algebroid generalized tangent bundle
We consider the following diagram: We take (x i , y a ) as canonical local coordinates on (E, π, M ), where i ∈ 1,m and a ∈ 1,r. Let be a change of coordinates on (E, π, M ). Then the coordinates y a change to y a according to the rule:

Journal of Generalized Lie Theory and Applications
In particular, if (E, π, M ) is vector bundle, then the coordinates y a change to y a according to the rule: Easily, we obtain the following Using the operation If z = z α t α ∈ Γ(F, ν, N ), then we obtain the section be the base sections for the Lie F (E)-algebra

For any sections
we obtain the section Since we have it implies Z α = 0, α ∈ 1,p and Y a = 0, a ∈ 1,r.
Therefore, the sections∂ 1 , . . . ,∂ p , ·∂ 1 , . . . , ·∂ r are linearly independent. We consider the vector subbundle π, E), +, ·), generated by the set of The base sections (∂ α , ·∂ a ) will be called the natural (ρ, η)-base. The matrix of coordinate transformation on ((ρ, η)T E, (ρ, η)τ E , E) at a change of fibred charts is In particular, if (E, π, M ) is a vector bundle, then the matrix of coordinate transformation on ((ρ, η)T E, (ρ, η)τ E , E) at a change of fibred charts is Easily, we obtain Using the operation Remark 8. In particular, if h = Id M , then the Lie algebroid is isomorphic with the usual Lie algebroid This is a reason for which the Lie algebroid will be called the Lie algebroid generalized tangent bundle.

The Lie algebroid generalized tangent bundle of dual vector bundle
Let (E, π, M ) ∈ |B v | be. We build the generalized tangent bundle of dual vector bundle ( * E, * π, M ) using the diagram: * We take (x i , p a ) as canonical local coordinates on ( * E, * π, M ), where i ∈ 1,m and a ∈ 1,r. Consider a change of coordinates on ( * E, * π, M ). Then the coordinates p a change to p a according to the rule: Easily, we obtain the following Using the operation

For any sections
we obtain the section Since we have We consider the vector subbundle Easily, we obtain 12 Journal of Generalized Lie Theory and Applications Using the operation The Lie algebroid generalized tangent bundle of the dual vector bundle ( * E, * π, M ) will be denoted:
We remark that the set { ·∂ a , a ∈ 1,r} is a base of the F (E)-module: Proposition 12. The short sequence of vector bundles is a split to the left in the previous exact sequence, will be called (ρ, η)connection for the fiber bundle (E, π, M ).
The (ρ, Id M )-connection will be called ρ-connection and will be denoted ρΓ and the (Id T M , Id M )-connection will be called connection and will be denoted Γ.
and will be called the horizontal vector subbundle.
Obviously, the components of are the real numbers it results that the components of are the real numbers Therefore, we have: After some calculations, we obtain: Remark 17. If we have a set of real local functions (ρ, η)Γ a γ that satisfies the relations of passing (4.1), then we have a (ρ, η)-connection (ρ, η)Γ for the fiber bundle (E, π, M ) . This (ρ, η)-connection will be called the (ρ, η)-connection associated to the connection Γ.
Proposition 25. The short sequence of vector bundles is a split to the left in the previous exact sequence, will be called (ρ, η)connection for the dual vector bundle ( * E, * π, M ). The differentiable real local functions (ρ, η)Γ bγ will be called the components of (ρ, η)-connection (ρ, η)Γ.
The Let {s a , a ∈ 1,r} be the dual base of the base {s a , a ∈ 1,r}.

Example 29.
If Γ is an Ehresmann connection for the vector bundle ( * E, * π, M ) on components Γ bk , then the differentiable real local functions are the components of a (ρ, η)-connection (ρ, η)Γ for the vector bundle ( * E, * π, M ), which will be called the (ρ, η)connection associated to the connection Γ.

Torsion and curvature. Formulas of Ricci and Bianchi type
We apply our theory for the diagram: Let ρΓ be a linear ρ-connection for the vector bundle (E, π, M ) by components ρΓ a bα . Using the components of linear ρ-connection ρΓ, then we obtain a linear ρ-connection ρΓ for the vector bundle (E, π, M ) given by the diagram:
In the particular case of Lie algebroids, h = Id M , we obtain the application: for any z, v ∈ Γ(F, ν, M ) and u ∈ Γ(E, π, M ), which will be called ρ-curvature associated to the linear ρ-connection ρΓ.
In the classical case, (ρ, h) = (Id T M , Id M ), we obtain the curvature R associated to the linear connection Γ.
Proposition 40. The (ρ, h)-curvature (ρ, h)R associated to the linear ρ-connection ρΓ, is R-linear in each argument and antisymmetric in the first two arguments. If In the particular case of Lie algebroids, h = Id M , we obtain ρR(t β , t α )s b put = ρR a bαβ s a , and

(5.2 )
Theorem 41. For any u a s a ∈ Γ(E, π, M ), we will use the notation and we verify the equality: After some calculations, we obtain: In the particular case of Lie algebroids, h = Id M , the relations (5.3) become: In the classical case, (ρ, h) = (Id T M , Id M ), the relations (5.3 ) become: After some calculations, we obtain: Lemma 42. If (E, π, M ) = (F, ν, N ), then, for any u a s a ∈ Γ(E, π, M ), we have that u a |c , a, c ∈ 1,n are the components of a tensor field of (1, 1) type.
Proof. Let U and U be two vector local (m + n) charts such that Summing the equalities (5.4) and (5.5), it results the conclusion of lemma.
Theorem 43. If (E, π, M ) = (F, ν, N ), then, for any u a s a ∈ Γ(E, π, M ), we will use the notation and we verify the formulas of Ricci type In the particular case of Lie algebroids, h = Id M , the relations (5.6) become: In the classical case, (ρ, h) = (Id T M , Id M ), the relations (5.6 ) become: Theorem 44. For any u a s a ∈ Γ( * E, * π, M ) we will use the notation: and we verify the equality: After some calculations, we obtain: where u a s a ∈ Γ(E, π, M ) such that u a u b = δ b a . In the particular case of Lie algebroids, h = Id M , the relations (5.7) become: In the classical case, (ρ, h) = (Id T M , Id M ), the relations (5.7 ) become: Proof. Since After some calculations, we obtain: Since Journal of Generalized Lie Theory and Applications and Lemma 45. If (E, π, M ) = (F, ν, N ), then, for any we have that u b|c , b, c ∈ 1,n are the components of a tensor field of (0, 2) type.
Proof. Let U and U be two vector local (m + n) charts such that U ∩ U = φ.
Summing the equalities (5.8) and (5.9), it results the conclusion of lemma.
Theorem 46. If (E, π, M ) = (F, ν, N ), then, for any we will use the notation and we verify the formulas of Ricci type In the particular case of Lie algebroids, h = Id M , the relations (5.10) become: (5.10 ) In the classical case, (ρ, h) = (Id T M , Id M ), the relations (5.10 ) become: (5.10 ) Theorem 47. For any tensor field: we verify the equality: In the particular case of Lie algebroids, h = Id M , the relations (5.11) become: In the classical case, (ρ, h) = (Id T M , Id M ), the relations (5.11 ) become: Theorem 48. If (E, π, M ) = (F, ν, N ), then we obtain the following formulas of Ricci type: In the particular case of Lie algebroids, h = Id M , the relations (5.12) become: (5.12 ) In the classical case, (ρ, h) = (Id T M , Id M ), the relations (5.12 ) become: We observe that if the structure functions of generalized Lie algebroid: the (ρ, h)-torsion associated to linear ρ-connection ρΓ, and the (ρ, h)-curvature associated to linear ρ-connection ρΓ are null, then we have the equality: respectively. This identities will be called the first and the second identity of Bianchi type, respectively. 1 2g ad (g de T e bc − g be T e dc + g ec T e bd ) (6.5) are the components of a linear ρ-connection compatible with the (pseudo)metrical structure g, where ρΓ a bc are the components of linear ρ-connection of Levi-Civita type (6.1). Therefore, the vector bundle (h * E, h * π, M ) becomes ρ-(pseudo)metrizable and the tensor field T is the (ρ, h)-torsion tensor field.
In the particular case of Lie algebroids, h = Id M , g ∈ T 0 2 (E, π, M ) is a (pseudo)metrical structure and T ∈ T 1 2 (E, π, M ) such that its components are skew symmetric in the lover indices, then the local real functions ρΓ a bc = ρΓ a bc + 1 2g ad (g de T e bc − g be T e dc + g ec T e bd ) (6.5') are the components of a linear ρ-connection compatible with the (pseudo)metrical structure g, where ρΓ a bc are the components of linear ρ-connection of Levi-Civita type (6.1 ).
In the particular case of Lie algebroids, h = Id M , g ∈ T 0 2 (E, π, M ) is a (pseudo)metrical structure and ρΓ is the linear ρ-connection (6.5') for the vector bundle (E, π, M ), then the local real functions:  are the components of a linear ρ-connection such that the vector bundle (h * E, h * π, M ) becomes ρ-(pseudo)metrizable.