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
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, π
M, 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∈Vk(E∗) theexterior derivativedEω∈Vk+1
(E∗) is given by the formula
dEω(s1, ..., sk+1) = k+1
X
i=1
(−1)i+1σ(si)ω(s1, ..., ˆ
si, ..., sk+1) +
+ X
1≤i<j≤k+1
(−1)i+jω([si,sj]E, s1, ..., ˆ
si, ..., ˆ
sj, ...sk+1).
where si ∈ Γ(E), i = 1, k+ 1, and it results that (dE)2 = 0. For ξ ∈ Γ(E) the
Lie derivative with respect toξ is given by Lξ = iξ◦dE+dE◦iξ, where iξ is the
contraction withξ.
If we take the local coordinates (xi) on an openU ⊂ M, a local basis {s
α} of the
sections of the bundleπ−1(U) →U generates local coordinates (xi, yα) on E. The
local functionsσi
α(x),Lγαβ(x) onM given by
σ(sα) =σαi
∂
∂xi, [sα, sβ]E=L γ
are called thestructure functionsof the Lie algebroid and satisfy thestructure equa-tionson Lie algebroid
σjα
∂σβi
∂xj −σ j β
∂σαi
∂xj =σ i γL
γ αβ,
X
(α,β,γ)
Ã
σiα
∂Lδβγ
∂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)), σiα
∂ ∂xi
¯ ¯ ¯ ¯u
¶
, Vα(u) =
µ
0, ∂ ∂yα
¯ ¯ ¯ ¯u
¶
,
and (∂/∂xi, ∂/∂yα) is the local basis onT E.The structure functions ofTEare given
by the following formulas
σ1(X
α) =σiα
∂ ∂xi, σ
1(V α) =
∂ ∂yα,
[Xα,Xβ]E =LγαβXγ, [Xα,Vβ]E= 0, [Vα,Vβ]E= 0.
The local expression of the differential of a function L on E is dEL =σi α∂x∂LiX
α+ ∂L
∂yαVα,where{Xα,Vα}denotes the corresponding dual basis of{Xα,Vα}anddExi=
σi
αXα, dEyα=Vα. The differential of sections of (TE)∗ is determined by
dEXα=−1 2L
α
βγXβ∧ Xγ, dEVα= 0.
The other canonical geometric objects (see [9]) areEuler sectionC=yαV
α and the
vertical endomorphism or tangent structure J = Xα⊗ V
α. 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) =yαX
α+Sα(x, y)Vα. If we have
the relation [C,S]E =S, thenSis called spray and the functionsSαare homogeneous
functions of degree 2 inyα.
A nonlinear connection onTE is an almost product structureN onπ2:TE→E
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) =ρhand v(ρ) =ρv
for everyρ∈Γ(TE), where h = 12(id+N) and v = 12(id−N).The sections
δα= (Xα)h=Xα−NαβVβ,
generate a local basis of HTE. The frame {δα,Vα} is a local basis of TE called
Berwald basis. The dual basis is {Xα, δVα} where δVα = Vα+Nα
βXβ. The Lie
brackets of the Berwald basis{δα,Vα}are [15]
[δα, δβ]E=Lγαβδγ+RγαβVγ, [δα,Vβ]E=
∂Nγ α
∂yβ Vγ, [Vα,Vβ]E= 0,
where
(2.1) Rγαβ=δβ(Nαγ)−δα(Nβγ) +L ε αβNεγ.
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∧ Xβ⊗ V γ,
where Rγαβ 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. ∇(Tr
s(TE\{0})⊂Trs(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
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 derivativeinduced by the semisprayS and a nonlinear connectionN.
Proposition 3.1. The following formulas hold
[S,Vβ]E=−δβ−
µ
Nβα+
∂Sα
∂yβ
¶
Vα
[S, δβ]E=
¡
Nβα−Lαβεyε
¢
δα+RγβVγ
where
Rγβ=−σβi
∂Sγ
∂xi − S(N γ β) +N
α
βNαγ+Nβα
∂Sγ
∂yα +N γ εLεαβyα.
The action of the dynamical covariant derivative on the Berwald basis is given by
∇Vβ= v[S,Vβ]E=−
µ
Nβα+
∂Sα
∂yβ
¶
Vα
∇δβ = h[S, δβ]E =¡Nβα−Lαβεyε
¢
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 metricg onVTE (i.e. a (2,0)-type symmetric tensorg=gαβ(x, y)Vα⊗ Vβ 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) gαβ/:= (∇g)(Vα,Vβ) =S(gαβ) +gγβ
µ
Nαγ+
∂Sγ
∂yα
¶
+gγα
µ
Nβγ+∂S
γ
∂yβ
¶
.
Definition 3.2. The nonlinear connectionN is called metric or compatible with the metric tensorg if▽g= 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
e
Nβα=Nβα−
1 2g
αγg γβ/,
Proof. Since Nα
β are the coefficients of a nonlinear connection and gαγgγβ/, are the
components of a tensor of type (1,1) it results thatNeβα are also the coefficients of
a nonlinear connection. We consider the dynamical covariant derivative induced by (S,Ne) and we have
that is the connectionNe is metric. ¤
3.1
The case of SODE connection
A semispray (SODE) given byS=yαX
α+SαVαdetermines an associated nonlinear
connection
Proposition 3.3. The following equations hold
(3.3) [S,Vβ]E =−δβ+
The dynamical covariant derivative induced by S and associated nonlinear con-nection is characterized by
∇Vβ= v[S,Vβ]E=¡Nβα−Lαβεyε
which is equivalent to
Definition 3.3. The Jacobi endomorphism is given by Φ = v[S,hρ]E.
Locally, from (3.4) we obtain that Φ =Rα
βVα⊗ Xβ,where Rαβ 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 Sα are 2-homogeneous with respect to the
variablesyβ and it results
2Sα= ∂Sα
∂yβy
β=−2Nα
βyβ+Lαβγyβyγ =−2Nβαyβ.
S= hS =yαδα, vS = 0, Nβα=
∂Nεα
∂yβ y ε+Lα
βεyε,
which yields Φ =iSΩ,and locally we getRαβ =Rαεβyε.
3.1.1 Lagrangian case
Let us consider a regular LagrangianLonE, that is the matrix
gαβ=
∂2L
∂yα∂yβ,
has constant rankm. The symplectic structure induced by the regular Lagrangian is [12]
ωL=
∂2L
∂yα∂yβV
β∧ Xα+1
2
µ
∂2L
∂xi∂yβσ i α−
∂2L
∂xi∂yασ i β−
∂L ∂yγL
γ αβ
¶
Xα∧ Xβ.
Let us consider the energy function given by
EL:=yα
∂L ∂yα −L,
then the symplectic equation
iSωL=−dEEL, S ∈Γ(TE),
and the regularity condition of the Lagrangian determine the components of the semis-pray
(3.8) Sε=gεβ
µ
σβi
∂L ∂xi −σ
i α
∂2L
∂xi∂yβy α−Lθ
βαyα
∂L ∂yθ
¶
,
wheregαβgβγ =δγα.
(3.8) will be called thecanonical nonlinear connectioninduced by a regular Lagrangian
L. Its coefficients are given by (3.9)
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
By direct computation, using the equalities
gγβ
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
whereNαβ:=gαγNβγ.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∂yασ i β−
∂2L
∂xi∂yβσ i α+
∂L ∂yεL
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) = gγβNαγ+gγαNβγ−
³
gγβLγαε+gγαLγβε
´
yε
= Nβα+Nαβ−
³
gγβLγαε+gγαLγβε
´
yε
= 2N(αβ)−
³
gγβLγαε+gγαLγβε
´
yε.
that is
2N(αβ)=S(gαβ) +
³
gγβLγαε+gγαLγβε
´
yε.
The equations (3.10) and (3.11) uniquely determine the coefficients of the nonlinear connection
Nβγ = gγαNαβ=gγα(N(αβ)+N[αβ])
= 1
2g
γα
·
S(gαβ) +
∂2L
∂xi∂yασ i β−
∂2L
∂xi∂yβσ i α−
∂L ∂yεL
ε βα+
³
gγβLγαε+gγαLγβε
´
yε
¸
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)Dρ(η+ω) =Dρη+Dρω,
iii)Df ρη=fDρη,
iv)Dρ(f η) = (σ(ρ)f)η+fDρη,
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:
(Dρηh)v= 0, (Dρηv)h= 0, Dρh = 0, Dρv = 0,
Dρ(Jηh) =J(Dρηh), Dρ(Jηv) =J(Dρηv).
If we denoteDh
ρη=Dρhη,Dv
We remark thatDhandDv are not covariant derivative, because
Dρhf =σ(ρh)f 6=σ(ρ)f, Dvρf =σ(ρv)f 6=σ(ρ)f
but, it still preserves many properties of D. Indeed,Dh and Dv satisfy the Leibniz
rule, and Dh and Dv 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:
Dv
ρg(ν, θ) =Dθvg(ν, ρ),
∇g= 0,
g(Φρ, ν) =g(Φν, ρ),
forν, ρ, θ∈Γ(E), which in local coordinates yield
∂gαβ
∂yε =
∂gαε
∂yβ,
S(gαβ)−gγβNαγ−gγαNβγ =y ε³g
γβLγεα+gγαLγεβ
´
,
gαγ
µ
σiβ
∂Sγ
∂xi +SN γ β +N
ε
βNεγ−(LδεβN γ δ +L
γ δεN
δ β)yε
¶
=
gβγ
µ
σαi
∂Sγ
∂xi +SN γ
α+NαεNεγ−(LδεαN γ δ +L
γ δεN
δ α)yε
¶
.
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.
References
[1] M. Anastasiei, Metrizable linear connections in a Lie algebroid, J. Adv. Math. Stud. 3, 1 (2010) 9-18.
[2] I. Buc˘ataru, Metric nonlinear connection, Differential Geom. Appl. 25 (2007) 335-343.
[3] I. Buc˘ataru, M. F. Dahl, Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations. J. Geom. Mech. 1, 2 (2009), 159– 180.
[4] J. Cortez, E. Martinez,Mechanical control systems on Lie algebroids, IMA Math. Control Inform., 21 (2004), 457-492.
[6] P. J. Higgins, K. Mackenzie,Algebraic constructions in the category of Lie alge-broids, J. Algebra 129 (1990) 194-230.
[7] D. Hrimiuc, L. Popescu,Nonlinear connections on dual Lie algebroids, Balkan J. Geom. Appl. 11, 1 (2006) 73-80.
[8] O. Krupkova, R. Malikova, Helmholtz conditions and their generalizations,
Balkan J. Geom. Appl. 15, 1 (2010) 80-89.
[9] M. de Leon, J. C. Marrero, E. Martinez,Lagrangian submanifolds and dynamics on Lie algebroidsJ. Phys. A: Math. Gen. 38 (2005) 241-308.
[10] K. Mackenzie,Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society, Lecture Note Series 124, Cambridge, 1987.
[11] E. Mart´ınez, J.F. Carinena, W. Sarlet, Derivations of differential forms along the tangent bundle projection II, Differential Geom. Appl. 3, 1 (1993), 1–29. [12] E. Mart´ınez,Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67,
(2001) 295-320.
[13] T. Mestdag, M. Crampin, Nonholonomic systems as restricted Euler-Lagrange systems, Balkan J. Geom. Appl., 15, 2 (2010) 70-81.
[14] R. Miron, M. Anastasiei,The geometry of Lagrange spaces. Theory and applica-tions, Kluwer Acad. Publish. 59, 1994.
[15] L. Popescu,Geometrical structures on Lie algebroids, Publ. Math. Debrecen 72, 1-2 (2008), 95-109.
[16] L. Popescu,Lie algebroids framework for distributional systems, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 55, 2 (2009) 257-274.
[17] L. Popescu,A note on Poisson-Lie algebroids (I), Balkan J. Geom. Appl., 14, 2 (2009) 79-89.
[18] P. Popescu,Almost Lie structures, derivations andR-curvature on relative tan-gent spaces. Rev. Roum. Math. Pures Appl. 37, 9 (1992) 779-789.
[19] M. Popescu, P. Popescu,Geometric objects defined by almost Lie structures, Lie algebroids and related topics, Banach Centre Publ., 54 (2001) 217-233.
[20] J. Szilasi,A setting for spray and Finsler geometry, Handbook of Finsler Geom-etry, Kluwer Academic Publishers, (2003), 1183-1437.
[21] I. Vaisman,Lectures on the geometry of Poisson manifolds, Birkh¨auser Verlag, 1994.
[22] A. Weinstein,Lagrangian mechanics and groupoids, Fields Inst. Com. 7 (1996) 206-231.
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.
