Intersection between the geometry of generalized Lie algebroids and some aspects of interior and exterior differential systems
aa r X i v : . [ m a t h . DG ] N ov Intersection between the geometry of generalizedLie algebroids and some aspects of interior andexterior differential systems
Constantin M. ARCUS¸SECONDARY SCHOOL “CORNELIUS RADU”,RADINESTI VILLAGE, 217196, GORJ COUNTY, ROMANIAe-mail: c [email protected], c [email protected] 4, 2018
Abstract
An exterior differential calculus in the general framework of general-ized Lie algebroids is presented. A theorem of Maurer-Cartan type isobtained. All results with details proofs are presented and a new point ofview over exterior differential calculus for Lie algebroids is obtained. Us-ing the theory of linear connections of Ehresmann type presented in thefirst reference, the identities of Cartan and Bianchi type are presented.Supposing that any vector subbundle of the pull-back Lie algebroid ofa generalized Lie algebroid is interior differential system (IDS) for thatgeneralized Lie algebroid, then the involutivity of the
IDS in a theorem ofFrobenius type is characterized. Extending the classical notion of exteriordifferential system (EDS) to generalized Lie algebroids, then the involu-tivity of an
IDS in a theorem of Cartan type is characterized.
Keywords: vector bundle, (generalized) Lie algebroid, interior differ-ential system, exterior differential calculus, exterior differential system,Cartan identities, Bianchi identities.
Contents Torsion and curvature forms. Identities of Cartan and Bianchitype 184 Interior and exterior differential systems 235 A new direction by research 286 Acknowledgment 29Acknowledgment 29References 29
In general, if C is a category, then we denote |C| the class of objects andfor 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 and B v be the category of Lie algebras, modules, manifoldsand 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 ) ∈ B v (( E, π, M ) , ( E ′ , π ′ , M ′ )) such that ϕ ∈ Iso
Man ( M, M ′ ) , then, using the operation F ( M ) × Γ ( E ′ , π ′ , M ′ ) ·−−−−→ Γ ( E ′ , π ′ , M ′ )( f, u ′ ) f ◦ ϕ − · u ′ it results that (Γ ( E ′ , π ′ , M ′ ) , + , · ) is a F ( M )-module and we obtain the Mod -morphism Γ (
E, π, M ) Γ( ϕ,ϕ ) −−−−−−−−−−→ Γ ( E ′ , π ′ , M ′ ) u Γ ( ϕ, ϕ ) u defined by Γ ( ϕ, ϕ ) u ( y ) = ϕ (cid:16) u ϕ − ( y ) (cid:17) = (cid:0) ϕ ◦ u ◦ ϕ − (cid:1) ( y ) , for any y ∈ M ′ . If M, N ∈ |
Man | , h ∈ Iso
Man ( M, N ), η ∈ Iso
Man ( N, M ) and (
F, ν, N ) ∈| B v | so that there exists( ρ, η ) ∈ B v (( F, ν, N ) , ( T M, τ M , M ))2nd an operation Γ ( F, ν, N ) × Γ (
F, ν, N ) [ , ] F,h −−−→
Γ (
F, ν, N )( u, v ) [ u, v ] F,h with the following properties:
GLA . the equality holds good [ u, f · v ] F,h = f [ u, v ] F,h + Γ (
T h ◦ ρ, h ◦ η ) ( u ) f · v, for all u, v ∈ Γ (
F, ν, N ) and f ∈ F ( N ) .GLA . the -tuple (cid:16) Γ (
F, ν, N ) , + , · , [ , ] F,h (cid:17) is a Lie F ( N ) -algebra, GLA . the Mod -morphism
Γ (
T h ◦ ρ, h ◦ η ) is a LieAlg -morphism of (cid:16)
Γ (
F, ν, N ) , + , · , [ , ] F,h (cid:17) source and (Γ (
T N, τ N , N ) , + , · , [ , ] T N ) target, then the triple(1 . (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) is an object of the category GLA of generalized Lie algebroids. (see [1])Using the exterior differential algebra of the generalized Lie algebroid (1 . . A new theorem of Maurer-Cartan type is presented in thisgeneral framework. As any Lie algebroid can be regarded as a generalized Liealgebroid, then a new point of view over exterior differential calculus for Liealgebroids is obtained. (see also: [4 , , , ρ, h )-torsion and ( ρ, h )-curvature presented in [1] , we obtain the ( ρ, h )-torsion and ( ρ, h )-curvature forms and identities of Cartanand Bianchi type in Section 3 . Using the
Cartan’s moving frame method , there exists the following
Theorem (E. Cartan) If N ∈ | Man n | is a Riemannian manifold and X α = X iα ∂∂x i , α ∈ , n is an ortonormal moving frame, then there exists a colection of -forms Ω αβ , α, β ∈ , n uniquely defined by the requirements Ω αβ = − Ω βα and d F Θ α = Ω αβ ∧ Θ β , α ∈ , n where (cid:8) Θ α , α ∈ , n (cid:9) is the coframe. (see [12] , p. 151)We know that an r -dimensional distribution on a manifold N is a mapping D defined on N, which assignes to each point x of N an r -dimensional linear3ubspace D x of T x N. A vector fields X belongs to D if we have X x ∈ D x foreach x ∈ N. When this happens we write X ∈ Γ ( D ) . The distribution D on a manifold N is said to be differentiable if for any x ∈ N there exists r differentiable linearly independent vector fields X , ..., X r ∈ Γ ( D ) in a neighborhood of x. The distribution D is said to be involutive if forall vector fields X, Y ∈ Γ ( D ) we have [ X, Y ] ∈ Γ ( D ) . In the classical theory we have the following
Theorem (Frobenius)
The distribution D is involutive if and only if for each x ∈ N there exists a neighborhood U and n − r linearly independent -forms Θ r +1 , ..., Θ n on U which vanish on D and satisfy the condition d F Θ α = Σ β ∈ r +1 ,p Ω αβ ∧ Θ β , α ∈ r + 1 , n. for suitable -forms Ω αβ , α, β ∈ r + 1 , n. (see [9] , p. 58)Extending the notion of distribution, in Section 4 , we obtain the definition ofan IDS of a generalized Lie algebroid and a characterization of the ivolutivity ofan
IDS in a result of Frobenius type is presented in
Theorem 4.1.
In particular, h = Id M = η, then we obtain the theorem of Frobenius type for Lie algebroids.(see: [2] , p. 248)In Section 4 of this paper we will show that there exists very close linksbetween EDS and the geometry of generalized Lie algebroids. In the classi-cal sense, an
EDS is a pair ( M, I ) consisting of a smooth manifold M and ahomogeneous, differentially closed ideal I in the algebra of smooth differentialforms on M . ( see: [3 , , , EDS to generalized Liealgebroids, the involutivity of an
IDS in a result of Cartan type is presented inthe
Theorem 4.3.
In particular, h = Id M = η, then we obtain the theorem ofCartan type for Lie algebroids. (see: [2] , p. 249)Finally, in Section 5 , we present a new direction by research in SymplecticGeometry. We propose an exterior differential calculus in the general framework ofgeneralized Lie algebroids. As any Lie algebroid can be regarded as a generalizedLie algebroid, in particular, we obtain a new point of view over the exteriordifferential calculus for Lie algebroids.Let (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) ∈ | GLA | be. Definition 2.1
For any q ∈ N we denote by (Σ q , ◦ ) the permutations groupof the set { , , ..., q } . Definition 2.2
We denoted by Λ q ( F, ν, N ) the set of q -linear applicationsΓ ( F, ν, N ) q ω −−−→ F ( N )( z , ..., z q ) ω ( z , ..., z q )4uch that ω (cid:0) z σ (1) , ..., z σ ( q ) (cid:1) = sgn ( σ ) · ω ( z , ..., z q )for any z , ..., z q ∈ Γ (
F, ν, N ) and for any σ ∈ Σ q .The elements of Λ q ( F, ν, N ) will be called differential forms of degree q or differential q -forms . Remark 2.1 If ω ∈ Λ q ( F, ν, N ), then ω ( z , ..., z, ..., z, ...z q ) = 0 . Therefore,if ω ∈ Λ q ( F, ν, N ), then ω ( z , ..., z i , ..., z j , ...z q ) = − ω ( z , ..., z j , ..., z i , ...z q ) . Theorem 2.1 If q ∈ N , then (Λ q ( F, ν, N ) , + , · ) is a F ( N ) -module. Definition 2.3 If ω ∈ Λ q ( F, ν, N ) and θ ∈ Λ r ( F, ν, N ), then the ( q + r )-form ω ∧ θ defined by ω ∧ θ ( z , ..., z q + r ) = P σ (1) <...<σ ( q ) σ ( q +1) <...<σ ( q + r ) sgn ( σ ) ω (cid:0) z σ (1) , ..., z σ ( q ) (cid:1) θ (cid:0) z σ ( q +1) , ..., z σ ( q + r ) (cid:1) = 1 q ! r ! P σ ∈ Σ q + r sgn ( σ ) ω (cid:0) z σ (1) , ..., z σ ( q ) (cid:1) θ (cid:0) z σ ( q +1) , ..., z σ ( q + r ) (cid:1) , for any z , ..., z q + r ∈ Γ (
F, ν, N ) , will be called the exterior product of the forms ω and θ. Using the previous definition, we obtain
Theorem 2.2
The following affirmations hold good: If ω ∈ Λ q ( F, ν, N ) and θ ∈ Λ r ( F, ν, N ) , then (2 . ω ∧ θ = ( − q · r θ ∧ ω. For any ω ∈ Λ q ( F, ν, N ) , θ ∈ Λ r ( F, ν, N ) and η ∈ Λ s ( F, ν, N ) we obtain (2 .
2) ( ω ∧ θ ) ∧ η = ω ∧ ( θ ∧ η ) . For any ω, θ ∈ Λ q ( F, ν, N ) and η ∈ Λ s ( F, ν, N ) we obtain (2 .
3) ( ω + θ ) ∧ η = ω ∧ η + θ ∧ η. For any ω ∈ Λ q ( F, ν, N ) and θ, η ∈ Λ s ( F, ν, N ) we obtain (2 . ω ∧ ( θ + η ) = ω ∧ θ + ω ∧ η. For any f ∈ F ( N ), ω ∈ Λ q ( F, ν, N ) and θ ∈ Λ s ( F, ν, N ) we obtain (2 .
5) ( f · ω ) ∧ θ = f · ( ω ∧ θ ) = ω ∧ ( f · θ ) . Theorem 2.3 If Λ (
F, ν, N ) = ⊕ q ≥ Λ q ( F, ν, N ) , then (Λ ( F, ν, N ) , + , · , ∧ ) is a F ( N ) -algebra. This algebra will be called theexterior differential algebra of the vector bundle ( F, ν, N ) . emark 2.2 If (cid:8) t α , α ∈ , p (cid:9) is the coframe associated to the frame (cid:8) t α , α ∈ , p (cid:9) of the vector bundle ( F, ν, N ) in the vector local ( n + p )-chart U , then(2 . t α ∧ ... ∧ t α q (cid:0) z α t α , ..., z αq t α (cid:1) = q ! det (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) z α ... z α q ... ... ...z α q ... z α q q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , for any q ∈ , p. Remark 2.3 If (cid:8) t α , α ∈ , p (cid:9) is the coframe associated to the frame (cid:8) t α , α ∈ , p (cid:9) of the vector bundle ( F, ν, N ) in the vector local ( n + p )-chart U , then, for any q ∈ , p we define C qp exterior differential forms of the type t α ∧ ... ∧ t α q such that 1 ≤ α < ... < α q ≤ p. The set { t α ∧ ... ∧ t α q , ≤ α < ... < α q ≤ p } is a base for the F ( N )-module(Λ q ( F, ν, N ) , + , · ) . Therefore, if ω ∈ Λ q ( F, ν, N ), then ω = ω α ...α q t α ∧ ... ∧ t α q . In particular, if ω is an exterior differential p -form ω, then we can written ω = a · t ∧ ... ∧ t p , where a ∈ F ( N ) . Definition 2.4 If ω = ω α ...α q t α ∧ ... ∧ t α q ∈ Λ q ( F, ν, N )such that ω α ...α q ∈ C r ( N ) , for any 1 ≤ α < ... < α q ≤ p , then we will say that the q -form ω is differentiableof C r -class. Definition 2.5
For any z ∈ Γ (
F, ν, N ), the F ( N )-multilinear applicationΛ ( F, ν, N ) L z −−−−−→ Λ (
F, ν, N ) , defined by(2 . L z ( f ) = Γ ( T h ◦ ρ, h ◦ η ) z ( f ) , ∀ f ∈ F ( N )and(2 . L z ω ( z , ..., z q ) = Γ ( T h ◦ ρ, h ◦ η ) z ( ω ( z , ..., z q )) − q P i =1 ω (cid:16)(cid:16) z , ..., [ z, z i ] F,h , ..., z q (cid:17)(cid:17) , ω ∈ Λ q ( F, ν, N ) and z , ..., z q ∈ Γ (
F, ν, N ) , will be called the covariantLie derivative with respect to the section z. In the particular case of Lie algebroids, ( η, h ) = ( Id M , Id M ) , then the co-variant Lie derivative with respect to the section z is defined by(2 . ′ ) L z ( f ) = Γ ( ρ, Id M ) z ( f ) , ∀ f ∈ F ( M )and(2 . ′ ) L z ω ( z , ..., z q ) = Γ ( ρ, Id M ) z ( ω ( z , ..., z q )) − q P i =1 ω (( z , ..., [ z, z i ] F , ..., z q )) , for any ω ∈ Λ q ( F, ν, M ) and z , ..., z q ∈ Γ (
F, ν, M ) . In addition, if ρ = Id T M , then the covariant Lie derivative with respect tothe vector field z is defined by(2 . ′′ ) L z ( f ) = z ( f ) , ∀ f ∈ F ( M )and(2 . ′′ ) L z ω ( z , ..., z q ) = z ( ω ( z , ..., z q )) − q P i =1 ω (( z , ..., [ z, z i ] T M , ..., z q )) , for any ω ∈ Λ q ( T M, ν, M ) and z , ..., z q ∈ Γ (
T M, ν, M ) . Theorem 2.4 If z ∈ Γ (
F, ν, N ) , ω ∈ Λ q ( F, ν, N ) and θ ∈ Λ r ( F, ν, N ) ,then (2 . L z ( ω ∧ θ ) = L z ω ∧ θ + ω ∧ L z θ. Proof.
Let z , ..., z q + r ∈ Γ (
F, ν, N ) be arbitrary. Since L z ( ω ∧ θ ) ( z , ..., z q + r ) = Γ ( T h ◦ ρ, h ◦ η ) z (( ω ∧ θ ) ( z , ..., z q + r )) − q + r P i =1 ( ω ∧ θ ) (cid:16)(cid:16) z , ..., [ z, z i ] F,h , ..., z q + r (cid:17)(cid:17) = Γ ( T h ◦ ρ, h ◦ η ) z P σ (1) <...<σ ( q ) σ ( q +1) <...<σ ( q + r ) sgn ( σ ) · ω (cid:0) z σ (1) , ..., z σ ( q ) (cid:1) · θ (cid:0) z σ ( q +1) , ..., z σ ( q + r ) (cid:1)(cid:1) − q + r P i =1 ( ω ∧ θ ) (cid:16)(cid:16) z , ..., [ z, z i ] F,h , ..., z q + r (cid:17)(cid:17) = P σ (1) <...<σ ( q ) σ ( q +1) <...<σ ( q + r ) sgn ( σ ) · Γ (
T h ◦ ρ, h ◦ η ) z (cid:0) ω (cid:0) z σ (1) , ..., z σ ( q ) (cid:1)(cid:1) · θ (cid:0) z σ ( q +1) , ..., z σ ( q + r ) (cid:1) + P σ (1) <...<σ ( q ) σ ( q +1) <...<σ ( q + r ) sgn ( σ ) · ω (cid:0) z σ (1) , ..., z σ ( q ) (cid:1) · Γ (
T h ◦ ρ, h ◦ η ) z (cid:0) θ (cid:0) z σ ( q +1) , ..., z σ ( q + r ) (cid:1)(cid:1) − P σ (1) <...<σ ( q ) σ ( q +1) <...<σ ( q + r ) sgn ( σ ) · q P i =1 ω (cid:16) z σ (1) , ..., (cid:2) z, z σ ( i ) (cid:3) F,h , ..., z σ ( q ) (cid:17) · θ (cid:0) z σ ( q +1) , ..., z σ ( q + r ) (cid:1) P σ (1) <...<σ ( q ) σ ( q +1) <...<σ ( q + r ) sgn ( σ ) q + r P i = q +1 ω (cid:0) z σ (1) , ..., z σ ( q ) (cid:1) · θ (cid:16) z σ ( q +1) , ..., (cid:2) z, z σ ( i ) (cid:3) F,h , ..., z σ ( q + r ) (cid:17) = P σ (1) <...<σ ( q ) σ ( q +1) <...<σ ( q + r ) sgn ( σ ) L z ω (cid:16) z σ (1) , ..., (cid:2) z, z σ ( i ) (cid:3) F,h , ..., z σ ( q ) (cid:17) · θ (cid:0) z σ ( q +1) , ..., z σ ( q + r ) (cid:1) + P σ (1) <...<σ ( q ) σ ( q +1) <...<σ ( q + r ) sgn ( σ ) q + r P i = q +1 ω (cid:0) z σ (1) , ..., z σ ( q ) (cid:1) · L z θ (cid:16) z σ ( q +1) , ..., (cid:2) z, z σ ( i ) (cid:3) F,h , ..., z σ ( q + r ) (cid:17) = ( L z ω ∧ θ + ω ∧ L z θ ) ( z , ..., z q + r )it results the conclusion of the theorem. q.e.d. Definition 2.6
For any z ∈ Γ (
F, ν, N ), the F ( N )-multilinear applicationΛ ( F, ν, N ) i z −−−→ Λ (
F, ν, N )Λ q ( F, ν, N ) ∋ ω i z ω ∈ Λ q − ( F, ν, N ) , where i z ω ( z , ..., z q ) = ω ( z, z , ..., z q ) , for any z , ..., z q ∈ Γ (
F, ν, N ), will be called the interior product associated tothe section z. For any f ∈ F ( N ), we define i z f = 0 . Remark 2.4 If z ∈ Γ (
F, ν, N ) , ω ∈ Λ p ( F, ν, N ) and U is an open subset of N such that z | U = 0 or ω | U = 0 , then ( i z ω ) | U = 0 . Theorem 2.5 If z ∈ Γ (
F, ν, N ) , then for any ω ∈ Λ q ( F, ν, N ) and θ ∈ Λ r ( F, ν, N ) we obtain (2 . i z ( ω ∧ θ ) = i z ω ∧ θ + ( − q ω ∧ i z θ. Proof.
Let z , ..., z q + r ∈ Γ (
F, ν, N ) be arbitrary. We observe that i z ( ω ∧ θ ) ( z , ..., z q + r ) = ( ω ∧ θ ) ( z , z , ..., z q + r )= P σ (1) <...<σ ( q ) σ ( q +1) <...<σ ( q + r ) sgn ( σ ) · ω (cid:0) z σ (1) , ..., z σ ( q ) (cid:1) · θ (cid:0) z σ ( q +1) , ..., z σ ( q + r ) (cid:1) = P σ (1) <σ (2) <...<σ ( q ) σ ( q +1) <...<σ ( q + r ) sgn ( σ ) · ω (cid:0) z , z σ (2) , ..., z σ ( q ) (cid:1) · θ (cid:0) z σ ( q +1) , ..., z σ ( q + r ) (cid:1) + P σ (1) <...<σ ( q )1= σ ( q +1) <σ ( q +2) <...<σ ( q + r ) sgn ( σ ) · ω (cid:0) z σ (1) , ..., z σ ( q ) (cid:1) · θ (cid:0) z , z σ ( q +2) , ..., z σ ( q + r ) (cid:1) = P σ (2) <...<σ ( q ) σ ( q +1) <...<σ ( q + r ) sgn ( σ ) · i z ω (cid:0) z σ (2) , ..., z σ ( q ) (cid:1) · θ (cid:0) z σ ( q +1) , ..., z σ ( q + r ) (cid:1) + P σ (1) <...<σ ( q ) σ ( q +2) <...<σ ( q + r ) sgn ( σ ) · ω (cid:0) z σ (1) , ..., z σ ( q ) (cid:1) · i z θ (cid:0) z σ ( q +2) , ..., z σ ( q + r ) (cid:1) .
8n the second sum, we have the permutation σ = (cid:18) ... q q + 1 q + 2 ... q + rσ (1) ... σ ( q ) 1 σ ( q + 2) ... σ ( q + r ) (cid:19) . We observe that σ = τ ◦ τ ′ , where τ = (cid:18) ... q + 1 q + 2 ... q + r σ (1) ... σ ( q ) σ ( q + 2) ... σ ( q + r ) (cid:19) and τ ′ = (cid:18) ... q q + 1 q + 2 ... q + r ... q + 1 1 q + 2 ... q + r (cid:19) . Since τ (2) < ... < τ ( q + 1) and τ ′ has q inversions, it results that sgn ( σ ) = ( − q · sgn ( τ ) . Therefore, i z ( ω ∧ θ ) ( z , ..., z q + r ) = ( i z ω ∧ θ ) ( z , ..., z q + r )+ ( − q P τ (2) <...<τ ( q ) τ ( q +2) <...<τ ( q + r ) sgn ( τ ) · ω (cid:0) z τ (2) , ..., z τ ( q ) (cid:1) · i z θ (cid:0) z τ ( q +2) , ..., z τ ( q + r ) (cid:1) = ( i z ω ∧ θ ) ( z , ..., z q + r ) + ( − q ( ω ∧ i z θ ) ( z , ..., z q + r ) . q.e.d. Theorem 2.6
For any z, v ∈ Γ (
F, ν, N ) we obtain (2 . L v ◦ i z − i z ◦ L v = i [ z,v ] F,h . Proof.
Let ω ∈ Λ q ( F, ν, N ) be arbitrary. Since i z ( L v ω ) ( z , ...z q ) = L v ω ( z, z , ...z q )= Γ ( T h ◦ ρ, h ◦ η ) v ( ω ( z, z , ..., z q )) − ω (cid:16) [ v, z ] F,h , z , ..., z q (cid:17) − q P i =2 ω (cid:16)(cid:16) z, z , ..., [ v, z i ] F,h , ..., z q (cid:17)(cid:17) = Γ ( T h ◦ ρ, h ◦ η ) v ( i z ω ( z , ..., z q )) − q P i =2 i z ω (cid:16) z , ..., [ v, z i ] F,h , ..., z q (cid:17) − i [ v,z ] F,h ( z , ..., z q ) = (cid:16) L v ( i z ω ) − i [ v,z ] F,h (cid:17) ( z , ..., z q ) , for any z , ..., z q ∈ Γ (
F, ν, N ) it result the conclusion of the theorem. q.e.d.
Definition 2.7 If f ∈ F ( N ) and z ∈ Γ (
F, ν, N ) , then the exterior differ-ential operator is defined by(2 . d F f ( z ) = Γ ( T h ◦ ρ, h ◦ η ) ( z ) f. In the particular case of Lie algebroids, ( η, h ) = ( Id M , Id M ) , then the exte-rior differential operator is defined by(2 . ′ ) d F f ( z ) = Γ ( ρ, Id M ) ( z ) f.
9n addition, if ρ = Id T M , then the exterior differential operator is definedby(2 . ′′ ) d T M f ( z ) = z ( f ) . Theorem 2.7
The F ( N ) -multilinear application Λ q ( F, ν, N ) d F −−−→ Λ q +1 ( F, ν, N ) ω dω defined by (2 . d F ω ( z , z , ..., z q ) = q P i =0 ( − i Γ (
T h ◦ ρ, h ◦ η ) z i ( ω ( z , z , ..., ˆ z i , ..., z q ))+ P i F, ν, N ) , is unique with the following property: (2 . L z = d F ◦ i z + i z ◦ d F , ∀ z ∈ Γ ( F, ν, N ) . This F ( N )-multilinear application will be called the exterior differentia-tion operator for the exterior differential algebra of the generalized Lie algebroid (( F, ν, N ) , [ , ] F,h , ( ρ, η )) . In the particular case of Lie algebroids, ( η, h ) = ( Id M , Id M ) , then the exte-rior differentiation operator for the exterior differential algebra of the Lie alge-broid (( F, ν, M ) , [ , ] F , ( ρ, Id M )) is defined by(2 . ′ ) d F ω ( z , z , ..., z q ) = q P i =0 ( − i Γ ( ρ, Id M ) z i ( ω ( z , z , ..., ˆ z i , ..., z q ))+ P i 13) Since (cid:0) i z ◦ d F (cid:1) ω ( z , ..., z q ) = dω ( z , z , ..., z q )= q P i =0 ( − i Γ ( T h ◦ ρ, h ◦ η ) z i ( ω ( z , z , ..., ˆ z i , ..., z q ))+ P ≤ i T h ◦ ρ, h ◦ η ) z i ( ω ( z , z , ..., ˆ z i , ..., z q ))+ q P i =1 ( − i ω (cid:16) [ z , z i ] F,h , z , ..., ˆ z i , ..., z q (cid:17) + P ≤ i T h ◦ ρ, h ◦ η ) z i ( i z ω (( z , ..., ˆ z i , ..., z q ))) − P ≤ i F, ν, N ) it results that the property (2 . 13) is satisfied.In the following, we verify the uniqueness of the operator d F . Let d ′ F be an another exterior differentiation operator satisfying the property(2 . . Let S = (cid:8) q ∈ N : d F ω = d ′ F ω, ∀ ω ∈ Λ q ( F, ν, N ) (cid:9) be.Let z ∈ Γ ( F, ν, N ) be arbitrary.We observe that (2 . 13) is equivalent with(1) i z ◦ (cid:0) d F − d ′ F (cid:1) + (cid:0) d F − d ′ F (cid:1) ◦ i z = 0 . Since i z f = 0 , for any f ∈ F ( N ) , it results that (cid:0)(cid:0) d F − d ′ F (cid:1) f (cid:1) ( z ) = 0 , ∀ f ∈ F ( N ) . Therefore, we obtain that(2) 0 ∈ S. In the following, we prove that(3) q ∈ S = ⇒ q + 1 ∈ S Let ω ∈ Λ p +1 ( F, ν, N ) be arbitrary . Since i z ω ∈ Λ q ( F, ν, N ), using theequality (1), it results that i z ◦ (cid:0) d F − d ′ F (cid:1) ω = 0 . 11e obtain that, (cid:0)(cid:0) d F − d ′ F (cid:1) ω (cid:1) ( z , z , ..., z q ) = 0 , for any z , ..., z q ∈ Γ ( F, ν, N ) . Therefore d F ω = d ′ F ω, namely q + 1 ∈ S. Using the Peano’s Axiom and the affirmations (2) and (3) it results that S = N . Therefore, the uniqueness is verified. q.e.d. Note that if ω = ω α ...α q t α ∧ ... ∧ t α q ∈ Λ q ( F, ν, N ), then d F ω (cid:0) t α , t α , ..., t α q (cid:1) = q P i =0 ( − i θ ˜ kα i ∂ω α ,..., c α i ...α q ∂ κ ˜ k + P i The exterior differentiation operator d F given by the previoustheorem has the following properties: For any ω ∈ Λ q ( F, ν, N ) and θ ∈ Λ r ( F, ν, N ) we obtain (2 . d F ( ω ∧ θ ) = d F ω ∧ θ + ( − q ω ∧ d F θ. For any z ∈ Γ ( F, ν, N ) we obtain(2 . L z ◦ d F = d F ◦ L z . d F ◦ d F = 0 . Proof. 1. Let S = (cid:8) q ∈ N : d F ( ω ∧ θ ) = d F ω ∧ θ + ( − q ω ∧ d F θ, ∀ ω ∈ Λ q ( F, ν, N ) (cid:9) be. Since d F ( f ∧ θ ) ( z, v ) = d F ( f · θ ) ( z, v )= Γ ( T h ◦ ρ, h ◦ η ) z ( f ω ( v )) − Γ ( T h ◦ ρ, h ◦ η ) v ( f ω ( z )) − f ω (cid:16) [ z, v ] F,h (cid:17) = Γ ( T h ◦ ρ, h ◦ η ) z ( f ) · ω ( v ) + f · Γ ( T h ◦ ρ, h ◦ η ) z ( ω ( v )) − Γ ( T h ◦ ρ, h ◦ η ) v ( f ) · ω ( z ) − f · Γ ( T h ◦ ρ, h ◦ η ) v ( ω ( z )) − f ω (cid:16) [ z, v ] F,h (cid:17) = d F f ( z ) · ω ( v ) − d F f ( v ) · ω ( z ) + f · d F ω ( z, v )= (cid:0) d F f ∧ ω (cid:1) ( z, v ) + ( − f · d F ω ( z, v )= (cid:0) d F f ∧ ω (cid:1) ( z, v ) + ( − (cid:0) f ∧ d F ω (cid:1) ( z, v ) , ∀ z, v ∈ Γ ( F, ν, N ) , 12t results that(1 . 1) 0 ∈ S. In the following we prove that(1 . q ∈ S = ⇒ q + 1 ∈ S. Without restricting the generality, we consider that θ ∈ Λ r ( F, ν, N ) . Since d F ( ω ∧ θ ) ( z , z , ..., z q + r ) = i z ◦ d F ( ω ∧ θ ) ( z , ..., z q + r )= L z ( ω ∧ θ ) ( z , ..., z q + r ) − d F ◦ i z ( ω ∧ θ ) ( z , ..., z q + r )= ( L z ω ∧ θ + ω ∧ L z θ ) ( z , ..., z q + r ) − (cid:2) d F ◦ ( i z ω ∧ θ + ( − q ω ∧ i z θ ) (cid:3) ( z , ..., z q + r )= (cid:0) L z ω ∧ θ + ω ∧ L z θ − (cid:0) d F ◦ i z ω (cid:1) ∧ θ (cid:1) ( z , ..., z q + r ) − (cid:16) ( − q − i z ω ∧ d F θ + ( − q d F ω ∧ i z θ (cid:17) ( z , ..., z q + r ) − ( − q ω ∧ d F ◦ i z θ ( z , ..., z q + r )= (cid:0)(cid:0) L z ω − d F ◦ i z ω (cid:1) ∧ θ (cid:1) ( z , ..., z q + r )+ ω ∧ (cid:0) L z θ − d F ◦ i z θ (cid:1) ( z , ..., z q + r )+ (cid:0) ( − q i z ω ∧ d F θ − ( − q d F ω ∧ i z θ (cid:1) ( z , ..., z q + r )= h(cid:0)(cid:0) i z ◦ d F (cid:1) ω (cid:1) ∧ θ + ( − q +1 d F ω ∧ i z θ i ( z , ..., z q + r )+ (cid:2) ω ∧ (cid:0)(cid:0) i z ◦ d F (cid:1) θ (cid:1) + ( − q i z ω ∧ d F θ (cid:3) ( z , ..., z q + r )= (cid:2) i z (cid:0) d F ω ∧ θ (cid:1) + ( − q i z (cid:0) ω ∧ d F θ (cid:1)(cid:3) ( z , ..., z q + r )= (cid:2) d F ω ∧ θ + ( − q ω ∧ d F θ (cid:3) ( z , ..., z q + r ) , for any z , z , ..., z q + r ∈ Γ ( F, ν, N ), it results (1 . . Using the Peano’s Axiom and the affirmations (1 . 1) and (1 . 2) it results that S = N . Therefore, it results the conclusion of affirmation 1.2. Let z ∈ Γ ( F, ν, N ) be arbitrary.Let S = (cid:8) q ∈ N : (cid:0) L z ◦ d F (cid:1) ω = (cid:0) d F ◦ L z (cid:1) ω, ∀ ω ∈ Λ q ( F, ν, N ) (cid:9) be.Let f ∈ F ( N ) be arbitrary. Since (cid:0) d F ◦ L z (cid:1) f ( v ) = i v ◦ (cid:0) d F ◦ L z (cid:1) f = (cid:0) i v ◦ d F (cid:1) ◦ L z f = ( L v ◦ L z ) f − (cid:0)(cid:0) d F ◦ i v (cid:1) ◦ L z (cid:1) f = ( L v ◦ L z ) f − L [ z,v ] F,h f + d F ◦ i [ z,v ] F,h f − d F ◦ L z ( i v f )= ( L v ◦ L z ) f − L [ z,v ] F,h f + d F ◦ i [ z,v ] F,h f − 0= ( L v ◦ L z ) f − L [ z,v ] F,h f + d F ◦ i [ z,v ] F,h f − L z ◦ d F ( i v f )= ( L z ◦ i v ) (cid:0) d F f (cid:1) − L [ z,v ] F,h f + d F ◦ i [ z,v ] F,h f = ( i v ◦ L z ) (cid:0) d F f (cid:1) + L [ z,v ] F,h f − L [ z,v ] F,h f = i v ◦ (cid:0) L z ◦ d F (cid:1) f = (cid:0) L z ◦ d F (cid:1) f ( v ) , ∀ v ∈ Γ ( F, ν, N ) , it results that(2 . 1) 0 ∈ S. 13n the following we prove that(2 . q ∈ S = ⇒ q + 1 ∈ S. Let ω ∈ Λ q ( F, ν, N ) be arbitrary. Since (cid:0) d F ◦ L z (cid:1) ω ( z , z , ..., z q ) = i z ◦ (cid:0) d F ◦ L z (cid:1) ω ( z , ..., z q )= (cid:0) i z ◦ d F (cid:1) ◦ L z ω ( z , ..., z q )= (cid:2) ( L z ◦ L z ) ω − (cid:0)(cid:0) d F ◦ i z (cid:1) ◦ L z (cid:1) ω (cid:3) ( z , ..., z q )= h ( L z ◦ L z ) ω − L [ z,z ] F,h ω i ( z , ..., z q )+ h d F ◦ i [ z,z ] F,h ω − d F ◦ L z ( i z ω ) i ( z , ..., z q ) ip. = h ( L z ◦ L z ) ω − L [ z,z ] F,h ω i ( z , ..., z q )+ h d F ◦ i [ z,z ] F,h ω − L z ◦ d F ( i z ω ) i ( z , ..., z q )= h ( L z ◦ i z ) (cid:0) d F ω (cid:1) − L [ z,z ] F,h ω + d F ◦ i [ z,z ] F,h ω i ( z , ..., z q )= h ( i z ◦ L z ) (cid:0) d F ω (cid:1) + L [ z,z ] F,h ω − L [ z,z ] F,h ω i ( z , ..., z q )= i z ◦ (cid:0) L z ◦ d F (cid:1) ω ( z , ..., z q )= (cid:0) L z ◦ d F (cid:1) ω ( z , z , ..., z q ) , ∀ z , z , ..., z q ∈ Γ ( F, ν, N ) , it results (2 . . Using the Peano’s Axiom and the affirmations (2 . 1) and (2 . 2) it results that S = N . Therefore, it results the conclusion of affirmation 2.3. It is remarked that i z ◦ (cid:0) d F ◦ d F (cid:1) = (cid:0) i z ◦ d F (cid:1) ◦ d F = L z ◦ d F − (cid:0) d F ◦ i z (cid:1) ◦ d F = L z ◦ d F − d F ◦ L z + d F ◦ (cid:0) d F ◦ i z (cid:1) = (cid:0) d F ◦ d F (cid:1) ◦ i z , for any z ∈ Γ ( F, ν, N ) . Let ω ∈ Λ q ( F, ν, N ) be arbitrary. Since (cid:0) d F ◦ d F (cid:1) ω ( z , ..., z q +2 ) = i z q +2 ◦ ... ◦ i z ◦ (cid:0) d F ◦ d F (cid:1) ω = ... = i z q +2 ◦ (cid:0) d F ◦ d F (cid:1) ◦ i z q +1 ( ω ( z , ..., z q ))= i z q +2 ◦ (cid:0) d F ◦ d F (cid:1) (0) = 0 , ∀ z , ..., z q +2 ∈ Γ ( F, ν, N ) , it results the conclusion of affirmation 3. q.e.d. Theorem 2.9 If d F is the exterior differentiation operator for the exte-rior differential F ( N ) -algebra (Λ( F, ν, N ) , + , · , ∧ ), then we obtain the structureequations of Maurer-Cartan type ( C ) d F t α = − L αβγ t β ∧ t γ , α ∈ , p nd ( C ) d F κ ˜ ı = θ ˜ ıα t α , ˜ ı ∈ , n, where (cid:8) t α , α ∈ , p (cid:9) is the coframe of the vector bundle ( F, ν, N ) . This equations will be called the structure equations of Maurer-Cartan typeassociated to the generalized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) . In the particular case of Lie algebroids, ( η, h ) = ( Id M , Id M ) , then the struc-ture equations of Maurer-Cartan type become( C ′ ) d F t α = − L αβγ t β ∧ t γ , α ∈ , p and ( C ′ ) d F x i = ρ iα t α , i ∈ , m. In the particular case of standard Lie algebroid, ρ = Id T M , then the structureequations of Maurer-Cartan type become( C ′′ ) d T M dx i = 0 , i ∈ , m and ( C ′′ ) d T M x i = dx i , i ∈ , m. Proof. Let α ∈ , p be arbitrary. Since d F t α ( t β , t γ ) = − L αβγ , ∀ β, γ ∈ , p it results that(1) d F t α = − P β<γ L αβγ t β ∧ t γ . Since L αβγ = − L αγβ and t β ∧ t γ = − t γ ∧ t β , for nay β, γ ∈ , p, it results that(2) P β<γ L αβγ t β ∧ t γ = 12 L αβγ t β ∧ t γ Using the equalities (1) and (2) it results the structure equation ( C ) . Let ˜ ı ∈ , n be arbitrarily. Since d F κ ˜ ı ( t α ) = θ ˜ ıα , ∀ α ∈ , p it results the structure equation ( C ) . q.e.d. Let (cid:16) ( F ′ , ν ′ , N ′ ) , [ , ] F ′ ,h ′ , ( ρ ′ , η ′ ) (cid:17) be an another generalized Lie algebroid.15n the category GLA , we defined (see [1]) the set of morphisms of (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) source and (cid:16) ( F ′ , ν ′ , N ′ ) , [ , ] F ′ ,h ′ , ( ρ ′ , η ′ ) (cid:17) target as being the set { ( ϕ, ϕ ) ∈ B v (( F, ν, N ) , ( F ′ , ν ′ , N ′ )) } such that ϕ ∈ Iso Man ( N, N ′ ) and the Mod -morphism Γ ( ϕ, ϕ ) is a LieAlg -morphism of (cid:16) Γ ( F, ν, N ) , + , · , [ , ] F,h (cid:17) source and (cid:16) Γ ( F ′ , ν ′ , N ′ ) , + , · , [ , ] F ′ ,h ′ (cid:17) target. Definition 2.8 For any GLA -morphism ( ϕ, ϕ ) of (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) source and (cid:16) ( F ′ , ν ′ , N ′ ) , [ , ] F ′ ,h ′ , ( ρ ′ , η ′ ) (cid:17) target we define the applicationΛ q ( F ′ , ν ′ , N ′ ) ( ϕ,ϕ ) ∗ −−−−−−→ Λ q ( F, ν, N ) ω ′ ( ϕ, ϕ ) ∗ ω ′ , where (cid:0) ( ϕ, ϕ ) ∗ ω ′ (cid:1) ( z , ..., z q ) = ω ′ (Γ ( ϕ, ϕ ) ( z ) , ..., Γ ( ϕ, ϕ ) ( z q )) , for any z , ..., z q ∈ Γ ( F, ν, N ) . Remark 2.5 It is remarked that the B v -morphism ( T h ◦ ρ, h ◦ η ) is a GLA -morphism of (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) source and (cid:16) ( T N, τ N , N ) , [ , ] T N,Id N , ( Id T N , Id N ) (cid:17) target.Moreover, for any ˜ ı ∈ , n , we obtain( T h ◦ ρ, h ◦ η ) ∗ (cid:0) d κ ˜ ı (cid:1) = d F κ ˜ ı , d is the exterior differentiation operator associated to the exterior differ-ential Lie F ( N )-algebra (Λ ( T N, τ N , N ) , + , · , ∧ ) . Theorem 2.11 If ( ϕ, ϕ ) is a morphism of (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) source and (cid:16) ( F ′ , ν ′ , N ′ ) , [ , ] F ′ ,h ′ , ( ρ ′ , η ′ ) (cid:17) target, then the following affirmations are satisfied: For any ω ′ ∈ Λ q ( F ′ , ν ′ , N ′ ) and θ ′ ∈ Λ r ( F ′ , ν ′ , N ′ ) we obtain (2 . 18) ( ϕ, ϕ ) ∗ (cid:0) ω ′ ∧ θ ′ (cid:1) = ( ϕ, ϕ ) ∗ ω ′ ∧ ( ϕ, ϕ ) ∗ θ ′ . For any z ∈ Γ ( F, ν, N ) and ω ′ ∈ Λ q ( F ′ , ν ′ , N ′ ) we obtain (2 . i z (cid:0) ( ϕ, ϕ ) ∗ ω ′ (cid:1) = ( ϕ, ϕ ) ∗ (cid:0) i Γ( ϕ,ϕ ) z ω ′ (cid:1) . If N = N ′ and ( T h ◦ ρ, h ◦ η ) = ( T h ′ ◦ ρ ′ , h ′ ◦ η ′ ) ◦ ( ϕ, ϕ ) , then we obtain (2 . 20) ( ϕ, ϕ ) ∗ ◦ d F ′ = d F ◦ ( ϕ, ϕ ) ∗ . Proof. 1. Let ω ′ ∈ Λ q ( F ′ , ν ′ , N ′ ) and θ ′ ∈ Λ r ( F ′ , ν ′ , N ′ ) be arbitrary. Since( ϕ, ϕ ) ∗ (cid:0) ω ′ ∧ θ ′ (cid:1) ( z , ..., z q + r ) = (cid:0) ω ′ ∧ θ ′ (cid:1) (Γ ( ϕ, ϕ ) z , ..., Γ ( ϕ, ϕ ) z q + r )= 1( q + r )! P σ ∈ Σ q + r sgn ( σ ) · ω ′ (Γ ( ϕ, ϕ ) z , ..., Γ ( ϕ, ϕ ) z q ) · θ ′ (Γ ( ϕ, ϕ ) z q +1 , ..., Γ ( ϕ, ϕ ) z q + r )= 1( q + r )! P σ ∈ Σ q + r sgn ( σ ) · ( ϕ, ϕ ) ∗ ω ′ ( z , ..., z q ) ( ϕ, ϕ ) ∗ θ ′ ( z q +1 , ..., z q + r )= (cid:0) ( ϕ, ϕ ) ∗ ω ′ ∧ ( ϕ, ϕ ) ∗ θ ′ (cid:1) ( z , ..., z q + r ) , for any z , ..., z q + r ∈ Γ ( F, ν, N ), it results the conclusion of affirmation 1.2. Let z ∈ Γ ( F, ν, N ) and ω ′ ∈ Λ q ( F ′ , ν ′ , N ′ ) be arbitrary. Since i z (cid:0) ( ϕ, ϕ ) ∗ ω ′ (cid:1) ( z , ..., z q ) = ω ′ (Γ ( ϕ, ϕ ) z, Γ ( ϕ, ϕ ) z , ..., Γ ( ϕ, ϕ ) z q )= i Γ( ϕ,ϕ ) z ω ′ (Γ ( ϕ, ϕ ) z , ..., Γ ( ϕ, ϕ ) z q )= ( ϕ, ϕ ) ∗ (cid:0) i Γ( ϕ,ϕ ) z ω ′ (cid:1) ( z , ..., z q ) , for any z , ..., z q ∈ Γ ( F, ν, N ), it results the conclusion of affirmation 2 . 17. Let ω ′ ∈ Λ q ( F ′ , ν ′ , N ′ ) and z , ..., z q ∈ Γ ( F, ν, N ) be arbitrary. Since (cid:16) ( ϕ, ϕ ) ∗ d F ′ ω ′ (cid:17) ( z , ..., z q ) = (cid:16) d F ′ ω ′ (cid:17) (Γ ( ϕ, ϕ ) z , ..., Γ ( ϕ, ϕ ) z q )= q P i =0 ( − i Γ ( T h ′ ◦ ρ ′ , h ′ ◦ η ′ ) (Γ ( ϕ, ϕ ) z i ) · ω ′ (cid:16)(cid:16) Γ ( ϕ, ϕ ) z , Γ ( ϕ, ϕ ) z , ..., \ Γ ( ϕ, ϕ ) z i , ..., Γ ( ϕ, ϕ ) z q (cid:17)(cid:17) + P ≤ i T h ◦ ρ, h ◦ η ) ( z i ) · (cid:0) ( ϕ, ϕ ) ∗ ω ′ (cid:1) ( z , ..., b z i , ..., z q )+ P ≤ i T h ◦ ρ, h ◦ η ) ( z i ) · ω ′ (cid:16) Γ ( ϕ, ϕ ) z , ..., \ Γ ( ϕ, ϕ ) z i , ..., Γ ( ϕ, ϕ ) z q (cid:17) + P ≤ i For any q ∈ , n we define Z q ( F, ν, N ) = (cid:8) ω ∈ Λ q ( F, ν, N ) : d F ω = 0 (cid:9) , the set of closed differential exterior q -forms and B q ( F, ν, N ) = (cid:8) ω ∈ Λ q ( F, ν, N ) : ∃ η ∈ Λ q − ( F, ν, N ) | d F η = ω (cid:9) , the set of exact differential exterior q -forms . Using the theory of linear connections of Eresmann type presented in [1] forthe diagram:(3 . E (cid:16) F, [ , ] F,h , ( ρ, Id N ) (cid:17) π ↓ ↓ νM h −−−−−−−→ N , E, π, M ) ∈ | B v | and (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, Id N ) (cid:17) ∈ | GLA | , we obtain alinear ρ -connection ρ Γ for the vector bundle ( E, π, M ) by components ρ Γ abα . Using the components of this linear ρ -connection, we obtain a linear ρ -connection ρ ˙Γ for the vector bundle ( E, π, M ) given by the diagram:(3 . E (cid:18) h ∗ F, [ , ] h ∗ F , (cid:18) h ∗ F ρ , Id M (cid:19)(cid:19) π ↓ ↓ h ∗ νM Id M −−−−−−−−→ M If ( E, π, M ) = ( F, ν, N ) , then, using the components of the same linear ρ -connection ρ Γ , we can consider a linear ρ -connection ρ ¨Γ for the vector bundle( h ∗ E, h ∗ π, M ) given by the diagram:(3 . h ∗ E (cid:18) h ∗ E, [ , ] h ∗ E , (cid:18) h ∗ E ρ , Id M (cid:19)(cid:19) h ∗ π ↓ ↓ h ∗ πM Id M −−−−−−−−→ M Definitiona 3.1 For any a, b ∈ , n we define the connection form Ω ab = ρ Γ abc S c . In the particular case of Lie algebroid, h = Id M , we obtain the connec-tion form ω ab = ρ Γ abc s c . In addition, if h = Id M , then we obtain the connectionform ω ij = Γ ijk dx k . Definition 3.2 If ( E, π, M ) = ( F, ν, N ), then the application(3 . 4) Γ ( h ∗ E, h ∗ π, M ) ρ,h ) T −−−−−→ Γ ( h ∗ E, h ∗ π, M )( U, V ) −→ ρ T ( U, V )defined by:(5 . 5) ( ρ, h ) T ( U, V ) = ρ ¨ D U V − ρ ¨ D V U − [ U, V ] h ∗ E , for any U, V ∈ Γ ( h ∗ E, h ∗ π, M ) , is the ( ρ, h ) - torsion associated to linear ρ -connection ρ Γ . If ( ρ, h ) T ( S a , S b ) put = ( ρ, h ) T cab S c , then the vector valued 2-form(3 . 6) ( ρ, h ) T = (( ρ, h ) T cab S c ) S a ∧ S b will be called the vector valued form of ( ρ, h ) -torsion ( ρ, h ) T . In the particular case of Lie algebroids, h = Id M , then the vector valued2-form becomes:(3 . ′ ρ T = ( ρ T cab s c ) s a ∧ s b . In the classical case, ρ = Id T M , then the vector valued 2-form (3 . ′ ) becomes:(3 . ′′ T = (cid:16) T ijk ∂∂x i (cid:17) dx j ∧ dx k . efinition 3.3 For each c ∈ , n we obtain the scalar -form of ( ρ, h ) -torsion ( ρ, h ) T (3 . 7) ( ρ, h ) T c = ( ρ, h ) T cab S a ∧ S b . In the particular case of Lie algebroids, h = Id M , then the scalar 2-form(3 . 7) becomes:(3 . ′′ ρ T c = ρ T cab s a ∧ s b . In the classical case, ρ = Id T M , then the scalar 2-form (3 . ′ ) becomes:(3 . ′′ T i = T ijk dx j ∧ dx k . Definition 3.4 The application(3 . 8) (Γ ( h ∗ F, h ∗ ν, M ) × Γ( E, π, M ) ( ρ,h ) R −−−−−→ Γ( E, π, M )(( Z, V ) , u ) −→ ρ R ( Z, V ) u defined by(3 . 9) ( ρ, h ) R ( Z, V ) u = ρ ˙ D Z (cid:16) ρ ˙ D V u (cid:17) − ρ ˙ D V (cid:16) ρ ˙ D Z u (cid:17) − ρ ˙ D [ Z,V ] h ∗ F u, for any Z, V ∈ Γ ( h ∗ F, h ∗ ν, M ) , u ∈ Γ ( E, π, M ) , is called ( ρ, h )-curvatureassociated to linear ρ -connection ρ Γ . If ( ρ, h ) R ( T β , T α ) s b put = ( ρ, h ) R ab αβ s a , then the vector mixed form(3 . 10) ( ρ, h ) R = (cid:16)(cid:16) ( ρ, h ) R ab αβ s a (cid:17) T α ∧ T β (cid:17) s b will be called the vector valued form of ( ρ, h ) -curvature ( ρ, h ) R . In the particular case of Lie algebroids, h = Id M , then the vector mixedform (3 . 10) becomes:(3 . ′ ρ R = (cid:16)(cid:16) ρ R ab αβ s a (cid:17) t α ∧ t β (cid:17) s b In the classical case, h = Id M , then the vector mixed form (3 . ′ becomes:(3 . ′′ R = (cid:0) ( R ab hk s a ) dx h ∧ dx k (cid:1) s b . Definition 3.5 For each a, b ∈ , n we obtain the scalar -form of ( ρ, h ) -curvature ( ρ, h ) R (3 . 11) ( ρ, h ) R ab = ( ρ, h ) R ab αβ T α ∧ T β . In the particular case of Lie algebroids, h = Id M , the scalar -form (3 . 11) becomes(3 . ′ ρ R ab = ρ R ab αβ t α ∧ t β . 20n the classical case, h = Id M , the scalar -form (3 . ′ becomes:(3 . ′′ R ab = R ab hk dx h ∧ dx k . Theorem 3.1 The identities ( C ) ( ρ, h ) T a = d h ∗ F S a + Ω ab ∧ S b , and ( C ) ( ρ, h ) R ab = d h ∗ F Ω ab + Ω ac ∧ Ω cb hold good. These will be called the first respectively the second identity of Cartantype.In the particular case of Lie algebroids, h = Id M , then the identities ( C ) and ( C ) become ( C ′ ) ρ T a = d F s a + ω ab ∧ s b , and ( C ′ ) ρ R ab = d F ω ab + ω ac ∧ ω cb respectively.In the classical case, ρ = Id T M , then the identities ( C ′ ) and ( C ′ ) become: ( C ′′ ) T i = ddx i + ω ij ∧ dx j = ω ij ∧ dx j and ( C ′′ ) R ij = dω ij + ω ih ∧ ω hj , respectively.Proof. To prove the first identity we consider that ( E, π, M ) = ( F, ν, M ) . Since d h ∗ F S a ( U, V ) S a = ((Γ( h ∗ F ρ , Id M ) U ) S a ( V ) − (Γ( h ∗ F ρ , Id M ) V )( S a ( U )) − S a ([ U, V ] h ∗ F )) S a = (Γ( h ∗ F ρ , Id M ) U )( V a ) − (Γ( h ∗ F ρ , Id M ) V )( U a ) − S a ([ U, V ] h ∗ F ) S a = ρ ¨ D U V − V b ρ ¨ D U S b − ρ ¨ D V U − U b ρ ¨ D V S b − [ U, V ] h ∗ F = ( ρ, h ) T ( U, V ) − ( ρ Γ abc V b U c − ρ Γ abc U b V c ) S a = (( ρ, h ) T a ( U, V ) − Ω ab ∧ S b ( U, V )) S a , it results the first identity.To prove the second identity, we consider that ( E, π, M ) = ( F, ν, M ) . Since( ρ, h ) R ab ( Z, W ) s a = ( ρ, h ) R (( W, Z ) , s b )= ρ ˙ D Z (cid:16) ρ ˙ D W s b (cid:17) − ρ ˙ D W (cid:16) ρ ˙ D Z s b (cid:17) − ρ ˙ D [ Z,W ] h ∗ F s b = ρ ˙ D Z (Ω ab ( W ) s a ) − ρ ˙ D W (Ω ab ( Z ) s a ) − Ω ab ([ Z, W ] h ∗ F ) s a + (Ω ac ( Z ) Ω cb ( W ) − Ω ac ( W ) Ω cb ( Z )) s a = (cid:0) d h ∗ F Ω ab ( Z, W ) + Ω ac ∧ Ω cb ( Z, W ) (cid:1) s a 21t results the second identity. q.e.d. Theorem 3.2 The identities ( B ) d h ∗ F ( ρ, h ) T a = ( ρ, h ) R ab ∧ S b − Ω ac ∧ ( ρ, h ) T c and ( B ) d h ∗ F ( ρ, h ) R ab = ( ρ, h ) R ac ∧ Ω cb − Ω ac ∧ ( ρ, h ) R cb , hold good. We will called these the first respectively the second identity ofBianchi type.If the ( ρ, h ) -torsion is null, then the first identity of Bianchi type becomes: ( ˜ B ) ( ρ, h ) R ab ∧ s b = 0 . In the particularcase of Lie algebroids, h = Id M , then the identities ( B ) and ( B ) become ( B ′ ) d F ρ T a = ρ R ab ∧ s b − ω ac ∧ ρ T c and ( B ′ ) d F ρ R ab = ρ R ac ∧ ω cb − ω ac ∧ ρ R cb , respectively.In the classical case, ρ = Id T M , then the identities ( B ′ ) and ( B ′ ) become: ( B ′′ ) d T i = R ij ∧ dx j − ω ik ∧ T k and ( B ′′ ) d R ij = R ih ∧ ω hj − ω ih ∧ R hj , respectively.Proof. We consider ( E, π, M ) = ( F, ν, M ) . Using the first identity of Cartantype and the equality d h ∗ F ◦ d h ∗ F = 0 , we obtain: d h ∗ F ( ρ, h ) T a = d h ∗ F Ω ab ∧ S b − Ω ac ∧ d h ∗ F S c . Using the second identity of Cartan type and the previous identity, we obtain: d h ∗ F ( ρ, h ) T a = (( ρ, h ) R ab − Ω ac ∧ Ω cb ) ∧ S b − Ω ac ∧ (cid:0) ( ρ, h ) T c − Ω cb ∧ S b (cid:1) . After some calculations, we obtain the first identity of Bianchi type.Using the second identity of Cartan type and the equality d h ∗ F ◦ d h ∗ F = 0 , we obtain: d h ∗ F Ω ac ∧ Ω cb − Ω ac ∧ d h ∗ F Ω cb = d h ∗ F ( ρ, h ) R ab . Using the second of Cartan type and the previous identity, we obtain: d h ∗ F ( ρ, h ) R ab = (( ρ, h ) R ac − Ω ae ∧ Ω ec ) ∧ Ω cb − Ω ac ∧ (( ρ, h ) R cb − Ω ce ∧ Ω eb ) . After some calculations, we obtain the second identity of Bianchi type. q.e.d. Interior and exterior differential systems Let (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) be an object of the category GLA .Let AF F be a vector fibred ( n + p )-atlas for the vector bundle ( F, ν, N ) andlet AF T M be a vector fibred ( m + m )-atlas for the vector bundle ( T M, τ M , M ).Let ( h ∗ F, h ∗ ν, M ) be the pull-back vector bundle through h. If ( U, ξ U ) ∈ AF T M and ( V, s V ) ∈ AF F such that U ∩ h − ( V ) = φ , then wedefine the application h ∗ ν − ( U ∩ h − ( V ))) ¯ s U ∩ h − V ) −−−−−−−→ (cid:0) U ∩ h − ( V ) (cid:1) × R p ( κ , z ( h ( κ ))) (cid:16) κ , t − V,h ( κ ) z ( h ( κ )) (cid:17) . Proposition 4.1 The set AF F put = S ( U,ξ U ) ∈AF TM , ( V,s V ) ∈AF F U ∩ h − ( V ) = φ (cid:8)(cid:0) U ∩ h − ( V ) , ¯ s U ∩ h − ( V ) (cid:1)(cid:9) is a vector fibred m + p -atlas for the vector bundle ( h ∗ F, h ∗ ν, M ) . If z = z α t α ∈ Γ ( F, ν, N ) , then we obtain the section Z = ( z α ◦ h ) T α ∈ Γ ( h ∗ F, h ∗ ν, M ) such that Z ( x ) = z ( h ( x )) , for any x ∈ U ∩ h − ( V ) . Theorem 4.1 Let (cid:16) h ∗ F ρ , Id M (cid:17) be the B v -morphism of ( h ∗ F, h ∗ ν, M ) sourceand ( T M, τ M , M ) target, where (4 . h ∗ F h ∗ F ρ −−→ T MZ α T α ( x ) (cid:0) Z α · ρ iα ◦ h (cid:1) ∂∂x i ( x ) Using the operation Γ ( h ∗ F, h ∗ ν, M ) × Γ ( h ∗ F, h ∗ ν, M ) [ , ] h ∗ F −−−−−−−−→ Γ ( h ∗ F, h ∗ ν, M ) defined by (4 . 2) [ T α , T β ] h ∗ F = (cid:16) L γαβ ◦ h (cid:17) T γ , [ T α , f T β ] h ∗ F = f (cid:16) L γαβ ◦ h (cid:17) T γ + (cid:0) ρ iα ◦ h (cid:1) ∂f∂x i T β , [ f T α , T β ] h ∗ F = − [ T β , f T α ] h ∗ F , for any f ∈ F ( M ) , it results that (cid:18) ( h ∗ F, h ∗ ν, M ) , [ , ] h ∗ F , (cid:18) h ∗ F ρ , Id M (cid:19)(cid:19) 23s a Lie algebroid which is called the pull-back Lie algebroid of the generalizedLie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) . Definition 4.1 Any vector subbundle ( E, π, M ) of the pull-back vectorbundle ( h ∗ F, h ∗ ν, M ) will be called interior differential system (IDS) of thegeneralized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) . In particular, if h = Id N = η , then we obtain the definition of IDS of a Liealgebroid. (see [2]) Remark 4.1 If ( E, π, M ) is an IDS of the generalized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) , then we obtain a vector subbundle (cid:0) E , π , M (cid:1) of the vector bundle (cid:18) ∗ h ∗ F , ∗ h ∗ ν, M (cid:19) such thatΓ (cid:0) E , π , M (cid:1) put = (cid:26) Ω ∈ Γ (cid:18) ∗ h ∗ F , ∗ h ∗ ν, M (cid:19) : Ω ( S ) = 0 , ∀ S ∈ Γ ( E, π, M ) (cid:27) . The vector subbundle (cid:0) E , π , M (cid:1) will be called the annihilator vector sub-bundle of the IDS ( E, π, M ) . Proposition 4.2 If ( E, π, M ) is an IDS of the generalized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) such that Γ ( E, π, M ) = h S , ..., S r i , then it exists Θ r +1 , ..., Θ p ∈ Γ (cid:18) ∗ h ∗ F , ∗ h ∗ ν, M (cid:19) linearly independent such that Γ (cid:0) E , π , M (cid:1) = (cid:10) Θ r +1 , ..., Θ p (cid:11) . Definition 4.2 The IDS ( E, π, M ) of the generalized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) will be called involutive if [ S, T ] h ∗ F ∈ Γ ( E, π, M ) , for any S, T ∈ Γ ( E, π, M ) . Proposition 4.3 If ( E, π, M ) is an IDS of the generalized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) and { S , ..., S r } is a base for the F ( M ) -submodule (Γ ( E, π, M ) , + , · ) then ( E, π, M ) is involutive if and only if [ S a , S b ] h ∗ F ∈ Γ ( E, π, M ) , for any a, b ∈ , r. Theorem 4.2 ( of Frobenius type) Let ( E, π, M ) be an IDS of the general-ized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) . If (cid:8) Θ r +1 , ..., Θ p (cid:9) is a base for the ( M ) -submodule (cid:0) Γ (cid:0) E , π , M (cid:1) , + , · (cid:1) , then the IDS ( E, π, M ) is involutive ifand only if it exists Ω αβ ∈ Λ ( h ∗ F, h ∗ ν, M ) , α, β ∈ r + 1 , p such that d h ∗ F Θ α = Σ β ∈ r +1 ,p Ω αβ ∧ Θ β , α ∈ r + 1 , p. Proof: Let { S , ..., S r } be a base for the F ( M )-submodule (Γ ( E, π, M ) , + , · )Let { S r +1 , ..., S p } ∈ Γ ( h ∗ F, h ∗ ν, M ) such that { S , ..., S r , S r +1 , ..., S p } is a base for the F ( M )-module(Γ ( h ∗ F, h ∗ ν, M ) , + , · ) . Let Θ , ..., Θ r ∈ Γ (cid:18) ∗ h ∗ F , ∗ h ∗ ν, M (cid:19) such that (cid:8) Θ , ..., Θ r , Θ r +1 , ..., Θ p (cid:9) is a base for the F ( M )-module (cid:18) Γ (cid:18) ∗ h ∗ F , ∗ h ∗ ν, M (cid:19) , + , · (cid:19) . For any a, b ∈ , r and α, β ∈ r + 1 , p , we have the equalities:Θ a ( S b ) = δ ab Θ a ( S β ) = 0Θ α ( S b ) = 0Θ α ( S β ) = δ αβ We remark that the set of the 2-forms (cid:8) Θ a ∧ Θ b , Θ a ∧ Θ β , Θ α ∧ Θ β , a, b ∈ , r ∧ α, β ∈ r + 1 , p (cid:9) is a base for the F ( M )-module (cid:0) Λ ( h ∗ F, h ∗ ν, M ) , + , · (cid:1) . Therefore, we have(1) d h ∗ F Θ α = Σ b E, π, M ) is an involutive IDS of the generalized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) . As [ S b , S c ] h ∗ F ∈ Γ ( E, π, M ) , ∀ b, c ∈ , r it results that Θ α ([ S b , S c ] h ∗ F ) = 0 , ∀ (cid:0) b, c ∈ , r ∧ α ∈ r + 1 , p (cid:1) . Therefore, A αbc = 0 , ∀ (cid:0) b, c ∈ , r ∧ α ∈ r + 1 , p (cid:1) and we obtain d h ∗ F Θ α = Σ b,γ B αbγ Θ b ∧ Θ γ + C αβγ Θ β ∧ Θ γ = (cid:16) B αbγ Θ b + C αβγ Θ β (cid:17) ∧ Θ γ . As Ω αγ put = B αbγ Θ b + 12 C αβγ Θ β ∈ Λ ( h ∗ F, h ∗ ν, M ) , ∀ α, β ∈ r + 1 , p it results the first implication.Conversely, we admit that it existsΩ αβ ∈ Λ ( h ∗ F, h ∗ ν, M ) , α, β ∈ r + 1 , p such that(4) d h ∗ F Θ α = Σ β ∈ r +1 ,p Ω αβ ∧ Θ β , ∀ α ∈ r + 1 , p. Using the affirmations (1) , (2) and (4) we obtain that A αbc = 0 , ∀ (cid:0) b, c ∈ , r ∧ α ∈ r + 1 , p (cid:1) . Using the affirmation (3), we obtainΘ α ([ S b , S c ] h ∗ F ) = 0 , ∀ (cid:0) b, c ∈ , r ∧ α ∈ r + 1 , p (cid:1) . Therefore, [ S b , S c ] h ∗ F ∈ Γ ( E, π, M ) , ∀ b, c ∈ , r. Proposition 4.2 , we obtain the second implication. q.e.d. Let (cid:18) ( h ∗ F, h ∗ ν, M ) , [ , ] h ∗ F , (cid:18) h ∗ F ρ , Id M (cid:19)(cid:19) be the pull-back Lie algebroid ofthe generalized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) . Definition 4.3 Any ideal ( I , + , · ) of the exterior differential algebra ofthe pull-back Lie algebroid (cid:18) ( h ∗ F, h ∗ ν, M ) , [ , ] h ∗ F , (cid:18) h ∗ F ρ , Id M (cid:19)(cid:19) closed underdifferentiation operator d h ∗ F , namely d h ∗ F I ⊆ I , will be called differential idealof the generalized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) . In particular, if h = Id N = η , then we obtain the definition of the differentialideal of a Lie algebroid.(see[2]) Definition 4.5 Let ( I , + , · ) be a differential ideal of the generalized Liealgebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) .If it exists an IDS ( E, π, M ) such that for all k ∈ N ∗ and ω ∈ I∩ Λ k ( h ∗ F, h ∗ ν, M )we have ω ( u , ..., u k ) = 0 , for any u , ..., u k ∈ Γ ( E, π, M ) , then we will say that( I , + , · ) is an exterior differential system (EDS) of the generalized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) . In particular, if h = Id N = η , then we obtain the definition of the EDS of aLie algebroid.(see[2]) Theorem 4.3 ( of Cartan type) The IDS ( E, π, M ) of the generalized Liealgebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) is involutive, if and only if the ideal gen-erated by the F ( M ) -submodule (cid:0) Γ (cid:0) E , π , M (cid:1) , + , · (cid:1) is an EDS of the samegeneralized Lie algebroid.Proof. Let ( E, π, M ) be an involutive IDS of the generalized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) . Let (cid:8) Θ r +1 , ..., Θ p (cid:9) be a base for the F ( M )-submodule (cid:0) Γ (cid:0) E , π , M (cid:1) , + , · (cid:1) . We know that I (cid:0) Γ (cid:0) E , π , M (cid:1)(cid:1) = ∪ q ∈ N { Ω α ∧ Θ α , { Ω r +1 , ..., Ω p } ⊂ Λ q ( h ∗ F, h ∗ ν, M ) } . Let q ∈ N and { Ω r +1 , ..., Ω p } ⊂ Λ q ( h ∗ F, h ∗ ν, M ) be arbitrary.Using the Theorems 3.8 and 3.10 we obtain d h ∗ F (Ω α ∧ Θ α ) = d h ∗ F Ω α ∧ Θ α + ( − q +1 Ω β ∧ d h ∗ F Θ β = (cid:16) d h ∗ F Ω α + ( − q +1 Ω β ∧ Ω βα (cid:17) ∧ Θ α . As d h ∗ F Ω α + ( − q +1 Ω β ∧ Ω βα ∈ Λ q +2 ( h ∗ F, h ∗ ν, M )27t results that d h ∗ F (cid:0) Ω β ∧ Θ β (cid:1) ∈ I (cid:0) Γ (cid:0) E , π , M (cid:1)(cid:1) Therefore, d h ∗ F I (cid:0) Γ (cid:0) E , π , M (cid:1)(cid:1) ⊆ I (cid:0) Γ (cid:0) E , π , M (cid:1)(cid:1) . Conversely, let ( E, π, M ) be an IDS of the generalized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) such that the F ( M )-submodule (cid:0) I (cid:0) Γ (cid:0) E , π , M (cid:1)(cid:1) , + , · (cid:1) is an EDS of the gen-eralized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) . Let (cid:8) Θ r +1 , ..., Θ p (cid:9) be a base for the F ( M )-submodule (cid:0) Γ (cid:0) E , π , M (cid:1) , + , · (cid:1) . As d h ∗ F I (cid:0) Γ (cid:0) E , π , M (cid:1)(cid:1) ⊆ I (cid:0) Γ (cid:0) E , π , M (cid:1)(cid:1) it results that it existsΩ αβ ∈ Λ ( h ∗ F, h ∗ ν, M ) , α, β ∈ r + 1 , p such that d h ∗ F Θ α = Σ β ∈ r +1 ,p Ω αβ ∧ Θ β ∈ I (cid:0) Γ (cid:0) E , π , M (cid:1)(cid:1) . Using the Theorem 4.2 , it results that ( E, π, M ) is an involutive IDS. q.e.d. We know that the set of morphisms of (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) source and (cid:16) ( F ′ , ν ′ , N ′ ) , [ , ] F ′ ,h ′ , ( ρ ′ , η ′ ) (cid:17) target is the set { ( ϕ, ϕ ) ∈ B v (( F, ν, N ) , ( F ′ , ν ′ , N ′ )) } such that ϕ ∈ Iso Man ( N, N ′ ) and the Mod -morphism Γ ( ϕ, ϕ ) is a LieAlg -morphism of (cid:16) Γ ( F, ν, N ) , + , · , [ , ] F,h (cid:17) source and (cid:16) Γ ( F ′ , ν ′ , N ′ ) , + , · , [ , ] F ′ ,h ′ (cid:17) the simplectic space as being a pair (cid:16)(cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) , ω (cid:17) consisting of a generalized Lie algebroid (cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) and a nonde-generate close 2-form ω ∈ Λ ( F, ν, N ) . If (cid:16)(cid:16) ( F ′ , ν ′ , N ′ ) , [ , ] F ′ ,h ′ , ( ρ ′ , η ′ ) (cid:17) , ω ′ (cid:17) is an another simplectic space, then we can define the set of morphisms of (cid:16)(cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) , ω (cid:17) source and (cid:16)(cid:16) ( F ′ , ν ′ , N ′ ) , [ , ] F ′ ,h ′ , ( ρ ′ , η ′ ) (cid:17) , ω ′ (cid:17) target as being the set n ( ϕ, ϕ ) ∈ GLA (cid:16)(cid:16) ( F, ν, N ) , [ , ] F,h , ( ρ, η ) (cid:17) , (cid:16) ( F ′ , ν ′ , N ′ ) , [ , ] F ′ ,h ′ , ( ρ ′ , η ′ ) (cid:17)(cid:17)o such that ( ϕ, ϕ ) ∗ ( ω ′ ) = ω. So, we can discuss about the category of simplectic spaces as being a subcat-egory of the category of generalized Lie algebroids. The study of the geometryof objects of this category is a new direction by research.Very interesting will be a result of Darboux type in this general frameworkand the connections with the Poisson bracket. I would like to thank R˘adine¸sti-Gorj Cultural Scientifique Society for financialsupport. 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