2.1 The prolongation of the Lie algebroid over the vector bun- dle projection

Liviu Popescu

Abstract.In this paper the problem of compatibility between a nonlinear connection and other geometric structures on Lie algebroids is studied.

The notion of dynamical covariant derivative is introduced and a metric nonlinear connection is found in the more general case of Lie algebroids.

We prove that the canonical nonlinear connection induced by a regular Lagrangian on a Lie algebroid is the unique connection which is metric and compatible with the symplectic structure.

M.S.C. 2010: 53C05, 17B66, 70S05.

Key words: Lie algebroids, dynamical covariant derivative, metric nonlinear connec- tion.

1 Introduction

The notions of nonlinear connection and metric structure are important tools in the differential geometry of the tangent bundle of a manifold. The metric compatibility of a nonlinear connection generalize the compatibility between a Riemannian metric and the linear connection and it is known as one of the Helmholtz conditions for the inverse problem of Lagrangian Mechanics (see for instance [2, 3, 5, 8, 11, 13]).

In this paper we generalize the metric compatibility of a nonlinear connection at the level of Lie algebroids. The notion of Lie algebroid and its prolongation over the vector bundle projection generalize the concept of tangent bundle. Mackenzie [10]

has been achieved a unitary study of Lie algebroids and together with Higgins [6]

have introduced the notion of prolongation of a Lie algebroid over a smooth map.

Weinstein [22] shows that is possible to give a common description of the most inter- esting classical mechanical systems. He developed a generalized theory of Lagrangian Mechanics and obtained the equations of motions, using the Poisson structure [21] on the dual of a Lie algebroid and the Legendre transformation associated with a regular Lagrangian. In the last years the problems raised by Weinstein and related topics have been investigated by many authors (see for instance [1, 7, 9, 12, 15, 16, 17, 19]).

In the present paper we study the problem of compatibility between a nonlinear connection and some other geometric structures on Lie algebroid and its prolongation over the vector bundle projection. The paper is organized as follows. The second

Balkan Journal of Geometry and Its Applications, Vol.16, No.1, 2011, pp. 111-121.∗

°c Balkan Society of Geometers, Geometry Balkan Press 2011.

section contains the preliminary results on Lie algebroids. In the section three, the compatibility between a nonlinear connection and a pseudo-Riemannian metric is studied. The notion of dynamical covariant derivative on the prolongation of Lie al- gebroid over the vector bundle projection is introduced and its action on the Berwald basis is given. We find the expression of the Jacobi endomorphism on Lie algebroids and the relation with the curvature of the nonlinear connection. We prove that the canonical nonlinear connection induced by a regular Lagrangian is the unique con- nection which is metric and compatible with a symplectic structure. Also, using the notions of v-covariant derivative, dynamical covariant derivative and Jacobi endomor- phism, we obtain the Helmholtz conditions in the framework of a Lie algebroid.

2 Preliminaries on Lie algebroids

Let M be a real, C^∞-differentiable, n-dimensional manifold and (T M, πM, M) its tangent bundle. A Lie algebroidover a manifold M is a triple (E,[·,·]_E, σ), where (E, π, M) is a vector bundle of rankmoverM,which satisfies the following conditions:

a) theC^∞(M)-module of sections Γ(E) is equipped with a Lie algebra structure [·,·]E. b) σ : E → T M is a bundle map (called the anchor) which induces a Lie algebra homomorphism (also denotedσ) from the Lie algebra of sections (Γ(E),[·,·]E) to the Lie algebra of vector fields (χ(M),[·,·]) satisfying the Leibniz rule

[s1, f s2]E =f[s1, s2]E+ (σ(s1)f)s2, ∀s1, s2∈Γ(E), f∈C^∞(M).

From the above definition it results:

1^◦ [·,·]E is a R-bilinear operation,

2^◦ [·,·]E is skew-symmetric, i.e. [s1, s2]E=−[s2, s1]E, ∀s1, s2∈Γ(E), 3^◦ [·,·]E verifies the Jacobi identity

[s1,[s2, s3]E]E+ [s2,[s3, s1]E]E+ [s3,[s1, s2]E]E= 0,

andσbeing a Lie algebra homomorphism, means thatσ[s1, s2]E= [σ(s1), σ(s2)].For ω∈V_k

(E^∗) theexterior derivatived^Eω∈V_k+1

(E^∗) is given by the formula d^Eω(s₁, ..., s_k+1) =

k+1X

i=1

(−1)ⁱ⁺¹σ(s_i)ω(s₁, ...,s^ˆ_i, ..., s_k+1) +

+ X

1≤i<j≤k+1

(−1)^i+jω([si,sj]E, s1, ...,s^ˆi, ...,s^ˆj, ...sk+1).

where si ∈ Γ(E), i = 1, k+ 1, and it results that (d^E)² = 0. For ξ ∈ Γ(E) the Lie derivative with respect toξ is given by Lξ = iξ◦d^E+d^E◦iξ, where iξ is the contraction withξ.

If we take the local coordinates (xⁱ) on an openU ⊂ M, a local basis {sα} of the sections of the bundleπ⁻¹(U) →U generates local coordinates (xⁱ, y^α) on E. The local functionsσⁱ_α(x),L^γ_αβ(x) onM given by

σ(sα) =σ_αⁱ ∂

∂xⁱ, [sα, sβ]E=L^γ_αβsγ, i= 1, n, α, β, γ= 1, m,

are called thestructure functionsof the Lie algebroid and satisfy thestructure equa- tionson Lie algebroid

σ^j_α∂σ_βⁱ

∂x^j −σ_β^j∂σ_αⁱ

∂x^j =σ_γⁱL^γ_αβ, X

(α,β,γ)

Ã

σⁱ_α∂L^δ_βγ

∂xⁱ +L^δ_αηL^η_βγ

!

= 0,

2.1 The prolongation of the Lie algebroid over the vector bun- dle projection

Let (E, π, M) be a vector bundle. For the projection π :E →M we can construct the prolongation ofE(see [6, 9, 12, 15]). The associated vector bundle is (TE, π2, E) where

TE = ∪

w∈E

TwE, TwE={(ux, vw)∈Ex×TwE|σ(ux) =Twπ(vw), π(w) =x∈M}, and the projection π2(ux, vw) =πE(vw) =w, where πE : T E → E is the tangent projection. We also have the canonical projectionπ1:TE→Egiven byπ1(u, v) =u.

The projection onto the second factorσ¹:TE→T E,σ¹(u, v) =vwill be the anchor of a new Lie algebroid over the manifoldE. An element ofTEis said to be vertical if it is in the kernel of the projectionπ1. We will denote (VTE, π2|VTE, E) the vertical bundle of (TE, π2, E) and σ¹|VTE : VTE → V T E is an isomorphism. The local basis of Γ(TE) is given by{Xα,Vα}, where [12]

Xα(u) = µ

sα(π(u)), σⁱ_α ∂

∂xⁱ

¯¯

¯¯

u

¶

, Vα(u) = µ

0, ∂

∂y^α

¯¯

¯¯

u

¶ ,

and (∂/∂xⁱ, ∂/∂y^α) is the local basis onT E.The structure functions ofTEare given by the following formulas

σ¹(Xα) =σⁱ_α ∂

∂xⁱ, σ¹(Vα) = ∂

∂y^α,

[Xα,Xβ]E =L^γ_αβXγ, [Xα,Vβ]E= 0, [Vα,Vβ]E= 0.

The local expression of the differential of a function L on E is d^EL =σⁱ_α∂x^∂LiX^α+

∂L

∂y^αV^α,where{X^α,V^α}denotes the corresponding dual basis of{Xα,Vα}andd^Exⁱ= σ_αⁱX^α, d^Ey^α=V^α. The differential of sections of (TE)^∗ is determined by

d^EX^α=−1

2L^α_βγX^β∧ X^γ, d^EV^α= 0.

The other canonical geometric objects (see [9]) areEuler sectionC=y^αVα and the vertical endomorphism or tangent structure J = X^α⊗ Vα. A section S of TE is calledsemispray(or second order differential equation -SODE) ifJ(S) =C.In local coordinates a semispray has the expressionS(x, y) =y^αXα+S^α(x, y)Vα. If we have the relation [C,S]E =S, thenSis called spray and the functionsS^αare homogeneous functions of degree 2 iny^α.

A nonlinear connection onTE is an almost product structureN onπ2:TE→E (i.e. a bundle morphismN :TE→ TE, such thatN²=id) smooth onTE\{0}such

thatVTE = ker(id+N). IfN is a connection on TE thenHTE = ker(id−N) is the horizontal subbundle associated toN and TE=VTE⊕HTE. The connection N onTE induces two projectors h,v :TE→ TE such that h(ρ) =ρ^hand v(ρ) =ρ^v for everyρ∈Γ(TE), where h = ¹₂(id+N) and v = ¹₂(id−N).The sections

δα= (Xα)^h=Xα−N_α^βVβ,

generate a local basis of HTE. The frame {δ_α,V_α} is a local basis of TE called Berwald basis. The dual basis is {X^α, δV^α} where δV^α = V^α+N_β^αX^β. The Lie brackets of the Berwald basis{δα,Vα}are [15]

[δα, δβ]E=L^γ_αβδγ+R^γ_αβVγ, [δα,Vβ]E= ∂N_α^γ

∂y^β Vγ, [Vα,Vβ]E= 0, where

(2.1) R^γ_αβ=δ_β(N_α^γ)−δ_α(N_β^γ) +L^ε_αβN_ε^γ. The curvature of the connectionN is given by Ω =−Nhwhere

Nh(z, w) = [hz,hw]E−h[hz, w]E−h[z,hw]E+ h²[z, w]E, is the Nijenhuis tensor of h. In local coordinates we have

Ω =−1

2R^γ_αβX^α∧ X^β⊗ Vγ,

where R^γ_αβ are given by (2.1) and represent the local coordinates expression of the curvature.

3 Dynamical covariant derivative and metric non- linear connection

Definition 3.1. A map∇:T(TE\{0})→T(TE\{0}) is said to be a tensor deriva- tion onTE\{0} if the following conditions are satisfied:

i)∇ isR-linear

ii)∇is type preserving, i.e. ∇(T^r_s(TE\{0})⊂T^r_s(TE\{0}), for each (r, s)∈N×N.

iii)∇ obeys the Leibnitz rule∇(P⊗S) =∇P⊗S+P⊗ ∇S, for any tensorsP, S on TE\{0}.

iv)∇commutes with any contractions, whereT^•_•(TE\{0}) is the space of tensors on TE\{0}.

For a semisprayS we consider theR-linear map∇0: Γ(TE\{0})→Γ(TE\{0}) given by

∇0ρ= h[S,hρ]E+ v[S,vρ]E, ∀ρ∈Γ(TE\{0}).

It results that

∇0(f ρ) =S(f)ρ+f∇0ρ, ∀f ∈C^∞(E), ρ∈Γ(TE\{0}).

Any tensor derivation onTE\{0}is completely determined by its actions on smooth functions and sections onTE\{0}(see [20] generalized Willmore’s theorem, p. 1217).

Therefore there exists a unique tensor derivation∇onTE\{0} such that

∇ |C^∞(E)=S, ∇ |Γ(TE\{0})=∇0.

We will call the tensor derivation∇, thedynamical covariant derivative induced by the semisprayS and a nonlinear connectionN.

Proposition 3.1. The following formulas hold

[S,Vβ]E=−δβ− µ

N_β^α+∂S^α

∂y^β

¶ Vα

[S, δ_β]_E=¡

N_β^α−L^α_βεy^ε¢

δ_α+R^γ_βV_γ where

R^γ_β=−σ_βⁱ∂S^γ

∂xⁱ − S(N_β^γ) +N_β^αN_α^γ+N_β^α∂S^γ

∂y^α +N_ε^γL^ε_αβy^α.

The action of the dynamical covariant derivative on the Berwald basis is given by

∇Vβ= v[S,Vβ]E=− µ

N_β^α+∂S^α

∂y^β

¶ Vα

∇δβ = h[S, δβ]E =¡

N_β^α−L^α_βεy^ε¢ δα.

It is not difficult to extend the action of∇to the algebra of tensors by requiring for

∇ to preserve the tensor product. For a pseudo-Riemannian metric g onVTE (i.e.

a (2,0)-type symmetric tensorg=gαβ(x, y)V^α⊗ V^β of rankmonVTE) we have (∇g)(ρ1, ρ2) =S(g(ρ1, ρ2))−g(∇ρ1, ρ2)−g(ρ1,∇ρ2),

and in local coordinates we get

(3.1) g_αβ/:= (∇g)(V_α,V_β) =S(g_αβ) +g_γβ µ

N_α^γ+∂S^γ

∂y^α

¶ +g_γα

µ

N_β^γ+∂S^γ

∂y^β

¶ . Definition 3.2. The nonlinear connectionN is called metric or compatible with the metric tensorg if5g= 0, that is

S(g(ρ1, ρ2)) =g(∇ρ1, ρ2) +g(ρ1,∇ρ2).

If S be a semispray, N a nonlinear connection and ∇ the dynamical covariant derivative induced by (S, N), then we set:

Proposition 3.2. The nonlinear connectionNe with the coefficients given by Ne_β^α=N_β^α−1

2g^αγgγβ/, is a metric nonlinear connection.

Proof. Since N_β^α are the coefficients of a nonlinear connection and g^αγgγβ/, are the components of a tensor of type (1,1) it results thatNe_β^α are also the coefficients of a nonlinear connection. We consider the dynamical covariant derivative induced by (S,Ne) and we have

(∇g)(Vα,Vβ) = S(gαβ) +gγβ

µ

Ne_α^γ+∂S^γ

∂y^α

¶ +gγα

µ

Ne_β^γ+∂S^γ

∂y^β

¶

= S(gαβ) +gγβ

µ

N_α^γ+∂S^γ

∂y^α

¶ +gγα

µ

N_β^γ+∂S^γ

∂y^β

¶

−gγβ1

2g^γεgεα/

− gγα1

2g^γεgεβ/=gαβ/−1

2gαβ/−1

2gαβ/= 0

that is the connectionNe is metric. ¤

3.1 The case of SODE connection

A semispray (SODE) given byS=y^αXα+S^αVαdetermines an associated nonlinear connection

N=−LSJ, with local coefficients

(3.2) N_α^β= 1

2 µ

−∂S^β

∂y^α +y^εL^β_αε

¶ .

Proposition 3.3. The following equations hold (3.3) [S,Vβ]E =−δβ+¡

N_β^α−L^α_βεy^ε¢ Vα,

(3.4) [S, δβ]E=¡

N_β^α−L^α_βεy^ε¢

δα+R^α_βVα, where

(3.5) R^α_β=−σ_βⁱ∂S^α

∂xⁱ − S(N_β^α)−N_γ^αN_β^γ+ (L^γ_εβN_γ^α+L^α_γεN_β^γ)y^ε.

The dynamical covariant derivative induced by S and associated nonlinear con- nection is characterized by

∇Vβ= v[S,Vβ]E=¡

N_β^α−L^α_βεy^ε¢

Vα=−1 2

µ∂S^α

∂y^β +L^α_βεy^ε

¶ Vα,

∇δβ = h[S, δβ]E =¡

N_β^α−L^α_βεy^ε¢ δα. (3.6) g_αβ/:= (∇g)(Vα,Vβ) =S(gαβ)−gγβN_α^γ−gγαN_β^γ+³

gγβL^γ_αε+gγαL^γ_βε´ y^ε,

which is equivalent to

(3.7) (∇g)(Vα,Vβ) =S(gαβ) +1 2

∂S^γ

∂y^αgγβ+1 2

∂S^γ

∂y^βgγα+1 2

³

gγβL^γ_αε+gγαL^γ_βε

´ y^ε.

Definition 3.3. The Jacobi endomorphism is given by Φ = v[S,hρ]E.

Locally, from (3.4) we obtain that Φ =R^α_βVα⊗ X^β,where R^α_β is given by (3.5) and represent the local coefficients of the Jacobi endomorphism.

Proposition 3.4. The following result holds

Φ =iSΩ + v[vS,hρ]E.

Proof. Indeed, Φ(ρ) = v[S,hρ]E= v[hS,hρ]E+ v[vS,hρ]E and Ω(S, ρ) = v[hS,hρ]E,

which yields Φ(ρ) = Ω(S, ρ) + v[vS,hρ]E. ¤

If S is a spray, then the coefficients S^α are 2-homogeneous with respect to the variablesy^β and it results

2S^α= ∂S^α

∂y^βy^β=−2N_β^αy^β+L^α_βγy^βy^γ =−2N_β^αy^β. S= hS =y^αδα, vS = 0, N_β^α= ∂N_ε^α

∂y^β y^ε+L^α_βεy^ε, which yields Φ =iSΩ,and locally we getR^α_β =R^α_εβy^ε.

3.1.1 Lagrangian case

Let us consider a regular LagrangianLonE, that is the matrix gαβ= ∂²L

∂y^α∂y^β,

has constant rankm. The symplectic structure induced by the regular Lagrangian is [12]

ω_L= ∂²L

∂y^α∂y^βV^β∧ X^α+1 2

µ ∂²L

∂xⁱ∂y^βσ_αⁱ − ∂²L

∂xⁱ∂y^ασⁱ_β− ∂L

∂y^γL^γ_αβ

¶

X^α∧ X^β.

Let us consider the energy function given by EL:=y^α∂L

∂y^α −L, then the symplectic equation

iSωL=−d^EEL, S ∈Γ(TE),

and the regularity condition of the Lagrangian determine the components of the semis- pray

(3.8) S^ε=g^εβ

µ σ_βⁱ ∂L

∂xⁱ −σⁱ_α ∂²L

∂xⁱ∂y^βy^α−L^θ_βαy^α∂L

∂y^θ

¶ , wheregαβg^βγ =δ^γ_α.

The connectionN with the coefficients given by (3.2), determined by the semispray

(3.8) will be called thecanonical nonlinear connectioninduced by a regular Lagrangian L. Its coefficients are given by

(3.9) N_β^α= 1

2g^αε

·

S(gεβ) +σ_βⁱ ∂²L

∂xⁱ∂y^ε −σ_εⁱ ∂²L

∂xⁱ∂y^β −L^γ_βε∂L

∂y^γ +³

gγεL^γ_βθ+gγβL^γ_εθ´ y^θ

¸

Theorem 3.5.The canonical nonlinear connectionNinduced by a regular Lagrangian Lis a metric nonlinear connection.

Proof. Introducing the expression of the semispray (3.8) into the equation (3.7) we obtain

(∇g)(Vα,Vβ) = y^εσⁱ_ε∂gαβ

∂xⁱ +g^εγ µ

σⁱ_γ∂L

∂xⁱ −σ_θⁱ ∂²L

∂xⁱ∂y^γy^θ−L^θ_γτy^τ ∂L

∂y^θ

¶∂gαβ

∂y^ε +1

2 µ

gγβ∂g^γε

∂y^α +gγα∂g^γε

∂y^β

¶ µ σⁱ_ε∂L

∂xⁱ −σⁱ_θ ∂²L

∂xⁱ∂y^εy^θ−L^θ_ετy^τ ∂L

∂y^θ

¶

+1 2

µ

σⁱ_β ∂²L

∂xⁱ∂y^α +σⁱ_α ∂²L

∂xⁱ∂y^β

¶

−σ_εⁱ∂gαβ

∂xⁱ y^ε

−1 2

µ

σⁱ_α ∂²L

∂xⁱ∂y^β +σⁱ_β ∂²L

∂xⁱ∂y^α

¶

−1 2

∂L

∂y^ε

¡L^ε_βα+L^ε_αβ¢

−1 2

³

gγβL^γ_αε+gγαL^γ_βε

´ y^ε+1

2

³

gγβL^γ_αε+gγαL^γ_βε

´ y^ε.

By direct computation, using the equalities gγβ∂g^γε

∂y^α =−g^γε∂gγβ

∂y^α =−g^γε∂gαβ

∂y^γ , L^θ_αβ=−L^θ_βα

it results (∇g)(Vα,Vβ) = 0,which ends the proof. ¤ Theorem 3.6. The canonical nonlinear connection induced by a regular Lagrangian is a unique connection which is metric and compatible with the symplectic structure ωL, that is

(3.10) ∇g= 0,

(3.11) ωL(hρ,hν) = 0, ∀ρ, ν∈Γ(TE\{0}) Proof. Using the equationV^α=δV^α−N_β^αX^β it results

ωL=gαβ(δV^β−N_γ^βX^γ)∧ X^α+1 2

µ ∂²L

∂xⁱ∂y^βσ_αⁱ − ∂²L

∂xⁱ∂y^ασⁱ_β− ∂L

∂y^εL^ε_αβ

¶

X^β∧ X^α

=gαβδV^β∧X^α+1 2

µ

gαγN_β^γ−gβγN_α^β+ ∂²L

∂xⁱ∂y^βσ_αⁱ − ∂²L

∂xⁱ∂y^ασ_βⁱ − ∂L

∂y^εL^ε_αβ

¶

X^β∧X^α

=gαβδV^β∧ X^α+1 2

µ

Nαβ−Nβα+ ∂²L

∂xⁱ∂y^βσ_αⁱ − ∂²L

∂xⁱ∂y^ασⁱ_β− ∂L

∂y^εL^ε_αβ

¶

X^β∧ X^α,

whereNαβ:=gαγN_β^γ.We have thatωL(hρ,hν) = 0 if and only if the second part of the above relation vanishes, that is

N[αβ]=1

2(Nαβ−Nβα) = 1 2

µ ∂²L

∂xⁱ∂y^ασ_βⁱ − ∂²L

∂xⁱ∂y^βσⁱ_α+ ∂L

∂y^εL^ε_αβ

¶ .

It result that the skew symmetric part ofNαβis uniquely determined by the condition (3.11). The symmetric part ofNαβis completely determined by the metric condition (3.10). Indeed

S(gαβ) = gγβN_α^γ+gγαN_β^γ−³

gγβL^γ_αε+gγαL^γ_βε´ y^ε

= Nβα+Nαβ−³

gγβL^γ_αε+gγαL^γ_βε´ y^ε

= 2N_(αβ)−³

gγβL^γ_αε+gγαL^γ_βε´ y^ε. that is

2N(αβ)=S(gαβ) +

³

gγβL^γ_αε+gγαL^γ_βε

´ y^ε.

The equations (3.10) and (3.11) uniquely determine the coefficients of the nonlinear connection

N_β^γ = g^γαNαβ=g^γα(N(αβ)+N[αβ])

= 1

2g^γα

·

S(gαβ) + ∂²L

∂xⁱ∂y^ασ_βⁱ − ∂²L

∂xⁱ∂y^βσⁱ_α− ∂L

∂y^εL^ε_βα+

³

gγβL^γ_αε+gγαL^γ_βε

´ y^ε

¸

Conversely, introducing (3.8) into (3.2) we have (3.9) which ends the proof. ¤ From [18, 4] we have:

Definition 3.4. A linear connection on Lie algebroid is a mapD: Γ(E)×Γ(E)→ Γ(E) which satisfies the rules

i)Dρ+ωη=Dρη+Dωη, ii)D_ρ(η+ω) =D_ρη+D_ρω, iii)Df ρη=fD_ρη,

iv)Dρ(f η) = (σ(ρ)f)η+fD_ρη,

for any functionf ∈C^∞(M) and sectionsρ, η, ω∈Γ(E).

Forρ, η∈Γ(E), the sectionDρη ∈Γ(E) is called thecovariant derivative of the sectionηwith respect to the sectionρ.LetN be a nonlinear connection, then a linear connectionDon Lie algebroid (E,[·,·]E, σ) is calledN−linear connection if [14]

i)Dpreserves by parallelism the horizontal distributionHTE.

ii) The tangent structureJ is absolute parallel with respect toD, that isDJ = 0.

Consequently, the following properties hold:

(Dρη^h)^v= 0, (Dρη^v)^h= 0, Dρh = 0, Dρv = 0, Dρ(Jη^h) =J(Dρη^h), Dρ(Jη^v) =J(Dρη^v).

If we denoteD_ρ^hη=D_ρhη,D_ρ^vη=Dρ^vη then the following decomposition is obtained Dρ=D^h_ρ+D^v_ρ, ρ∈Γ(E)

We remark thatD^handD^v are not covariant derivative, because D_ρ^hf =σ(ρ^h)f 6=σ(ρ)f, D^v_ρf =σ(ρ^v)f 6=σ(ρ)f

but, it still preserves many properties of D. Indeed,D^h and D^v satisfy the Leibniz rule, and D^h and D^v will be called the h−covariant derivation and v− covariant derivation, respectively.

Remark 3.7. The invariant form of Helmholtz conditions on Lie algebroids is given by:

D_ρ^vg(ν, θ) =D_θ^vg(ν, ρ),

∇g= 0,

g(Φρ, ν) =g(Φν, ρ), forν, ρ, θ∈Γ(E), which in local coordinates yield

∂gαβ

∂y^ε =∂gαε

∂y^β, S(gαβ)−gγβN_α^γ−gγαN_β^γ =y^ε

³

gγβL^γ_εα+gγαL^γ_εβ

´ , gαγ

µ σⁱ_β∂S^γ

∂xⁱ +SN_β^γ+N_β^εN_ε^γ−(L^δ_εβN_δ^γ+L^γ_δεN_β^δ)y^ε

¶

= gβγ

µ σ_αⁱ∂S^γ

∂xⁱ +SN_α^γ+N_α^εN_ε^γ−(L^δ_εαN_δ^γ+L^γ_δεN_α^δ)y^ε

¶ .

In the case of standard Lie algebroid (T M,[·,·], id) we obtain the classical Helmholtz conditions [11].

Acknowledgments. This work was supported by the strategic grant POS- DRU/89/1.5 /S/61928, Project ID 61928 (2009) co-financed by the European Social Fund within the Sectorial Operational Programme Human Resources Development 2007-2013.

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Author’s address:

Liviu Popescu

University of Craiova, Faculty of Economics, Dept. of Mathematics and Statistics,

13 Al. I. Cuza st., RO-200585 Craiova, Romania.

e-mail: [email protected] , [email protected]