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Metrizable linear connections in a lie algebroid.

1. INTRODUCTION

Le algebroids as particular anchored vector bundles [21] have now an important place in differential geometry and algebraic geometry. Initially defined as infinitesimal part of Lie grupoids, their algebra [12] and geometry is independently and largely developed [5, 8, 11, 20]. Besides they have proved to be useful in Mechanics [2, 4, 7, 16, 24], in the theory of nonholonomic systems [3, 9, 18] in control theory [6], in field theory [16], in quantum and classical gravity [22, 23]. The cohomology of Lie algebroids started in [17]. There the adapted exterior differential d was for the first time introduced. See also [13]. Holonomy and characteristic classes have been studied in [10]. For more references see the monograph [14]

Let A be a Lie algebroid and D an ^-connection (defined in the Section 3) in a vector bundle (F, g, M). We say that D is metrizable if there exists a Riemannian metric h in (F, g, M) such that Dh = 0. The tangent bundle TM is a trivial Lie algebroid and a TM-connection in (F, g, M) is nothing but an usual linear connection in this vector bundle. In [1] we provided conditions for metrizability of a TMconnection in any vector bundle as well as in vector bundles endowed with Finsler functions. In this paper we extend some results from [1] to A-connection in (F, g, M). The notations from [19] are used.

2. LIE ALGEBROIDS

Let [xi] = (E7, g, M) be a vector bundle of rank m. Here E and M are smooth i.e. [[C.sup.[infinity] manifolds with dimM = n, dimE = n + m and p: E [right arrow] M is a smooth submersion. The fibres [E.sub.x] = [p-1](x), [member of] M are linear spaces of dimension m which are isomorphic with the type fibre [R.sup.m].

Let F(M) be the ring of smooth real functions on M. We denote by [GAMMA](E) and X(M) the 3r(M)-module of sections of [xi] and of the tangent bundle (TM,t,M), respectively. If (U, (([x.sup.i])) i, j, k 1,2, n, is a local chart on M, then [partial derivative]/[partial derivative][X.sup.i] provide a local basis for X(U). Let [s.sub.a]: U [right arrow] [P.sup.-1](U), a, b, c,... = 1, 2, ..., m be a local basis for T ([p.sup.-1](U)). Any section s over U has the form s = [y.sup.a][s.sub.a](x), [member of] G U and we will take (V,?/a) as local coordinates on [p.sup.-1](U). A change of these coordinates ([x.sup.i][y.sup.a]) [right arrow] ([x.sup.[??], [y.sub.[??]) has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let [xi.sup.*] = ([E.sup.*],[p.sup.*],M) be the dual of vector bundle [xi] and [[theta].sup.a]: U [right arrow] [P.sup.*-]{U), x [right arrow] [[theta].sup.a](x) [member of] [E.sup.*.sub.x] a local basis for [GAMMA] [p.sup.*-1](U)) such that [theta].sup.a]([s.sub.b]) = [[delta].sup.a].sub.b]

Next, we may consider the tensor bundle of type (r, 5), [T.sup.t].sub.s](E) over M and its sections. For g [member of] [GAMMA] ([E.sup.*] <g> [E.sup.*]) we have g = [g.sub.ab](x)[[theta].sup.a] As ([E.sup.*] [cross product] [theta].sup.b] As ([E.sup.*] |[cross product] [E.sup.*]) [congruent to][L.sub.2] (E, R)| we may regard g as a smooth mapping x [right arrow] g(x): [E.sub.x] x [E.sub.x] [M with g(x) a bilinear mapping given by g(x)(sa,Sb) = gab(x), x [member of] M.

If the mapping g(x) is symmetric i.e. [g.sub.ab] = [g.sub.ba] and positive definite i.e. gab(x)[[xi].sup.a] [[xi].sup.b] > 0 for every ([[xi].sup.a]) [not equal to] 0, one says that g defines a Riemannian metric in the vector bundle [xi].

Let us assume that

(i) [GAMMA](E) is endowed with a Lie algebra structure [,] over R,

(ii) There exists a bundle map [rho]: E [right arrow] TM, called anchor map. It induces a Lie algebra homomorphism (denoted also by [rho]) from [GAMMA](E) to X(M), (iii) For any sections [s.sub.1],[s.sub.2] [member of] [member of] [GAMMA](E) and for any f [member of] F(M) the following identity holds

[s.sub.1] [fs.sub.2] = f[s.sub.1] [s.sub.2] + [rho]{[s.sub.1])[fs.sub.2]

Definition 2.1. The triplet A = ([xi],[,],[rho]) with the properties (i), (ii) and (iii) is called a Lie algebroid.

Examples:

1. The tangent bundle (TM,t,M) with the usual Lie bracket and [rho] equal to the identity map form a Lie algebroid.

2. Any integrable subbundle of TM with the Lie bracket defined by restriction and p the inclusion map is a Lie algebroid.

3. Let (F, g, M) be any vector bundle. On F we have the vertical distribution u [right arrow] [V.sub.u] F = [Kerq*,u]u [member of] F, where [q.sub.*] denotes the differential of q.

This distribution is integrable. If we regard it as a subbundle of TF, accordingly to Example 2 a Lie algebroid is obtained. Locally, we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

The structure function p\ and Lcah of the Lie algebroid A have to satisfy the following identities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.3)

3. CONNECTIONS IN LIE ALGEBROIDS

Let A = ([xi], [,],[rho]) be a Lie algebroid with [xi] = (F,p, M) and let (F, q , M) be any vector bundle.

Definition 3.1. An A-connection in the bundle (F, (p, M) is a mapping D: [GAMMA](E) x [GAMMA](F) [right arrow] [GAMMA], s[sigma] [right arrow] [D.sub.s][sigma]

with the properties:

1) [D.sub.s]+[s.sub.2][sigma]= [D.sub.s1][sigma]+[D.sub.s2][sigma],

2) [D.sub.fs][sigma] = f[D.sub.s][sigma],

3) [D.sub.s]([sigma.sub.1] + [[sigma].sub.2]) = [D.sub.s][sigma].sub.1] + [[D.sub.s][sigma].sub.2]

4) [D.sub.s]([[f.sub.[sigma]= [rho](s) f [sigma] + f [D.sub.s][sigma]

for s,[s.sub.1],[s.sub.2] [member of] [GAMMA](E), [sigma], [[sigma].sub.1], [[sigma].sub.2] [member of] [GAMMA] (F)| f [member of] F (M)

Notice that a TM-connection in the vector bundle (F, q, M) is nothing but a linear connection in this vector bundle.

Definition 3.2. An ^-connection in the bundle [xi] = (F,p, M) is called a linear connection in the Lie algebroid A.

The notion of tangent lift of a curve on M is generalized as follows.

Definition 3.3. Let A = ([xi] [,],[rho]) be a Lie algebroid with [xi] = (F,_p, M). A curve a: [0,1] [right arrow] E is called admissible or an A-path if [rho]([alpha](t)) = d/dt p([alpha](t)), t [member of] [0,1]. The curve [gamma](2) = p([alpha](t)) will be called the base path of [alpha]. The A-path [alpha] is called vertical if [rho]([alpha](t)) = 0. In this case 7 reduces to a point and the curve a is contained in the fibre in that point.

Locally, if [alpha](t) = [x.sup.i](t), [y.sup.a] (t)) then [gamma](t) = ([x.sup.i](t)) and [alpha] is an A-path if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.1)

and it is a vertical A-path if and only if

[[rho.sup.i].sub.a] (x(t))[Y.sup.a](t)= 0, t [member of] 0,1. (3.2)

Let ([sigma][alpha]) = [alpha], [beta], [gamma].. = K: = rank of (F, g, M), a local basis in [GAMMA] (F). Then a local section [sigma] has the form [sigma] = [[z.sup.[alpha] [sigma][alpha] and ([z.sup.][alpha]] are the coordinates in the fibres of (F,q,M).

For s = [y.sup.a][s.sub.a] and [sigma] = [[z.sup.[alpha] [[sigma].sub.alpha], by the Definition 3.1 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and if we put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For a linear connection D in the Lie algebroid A = ([xi], [,] [rho]) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let D be an A-connection in the vector bundle (F, q, M) and a: [0,1] [right arrow] E an ^4-path.

A smooth mapping [sigma]: [0,1] [right arrow] F is called an [alpha]-section if q([sigma](t)) = p([alpha](t)), t G [0,1].

Locally, if [alpha]{t) = ([x.sup.i](t), [y.sup.a](t)) then [sigma](t) = ([x.sup.i] (t), [z.sup.[alpha]] (t)).

Let [GAMMA](F)[alpha] be the linear space of [alpha]-section in the vector bundle (F,q,M). We define an operator [[D.sup.[alpha]: T(F)[alpha] [right arrow] [GAMMA](F)[alpha], [sigma](t) ([[D.sup.[sigma])(t) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.4)

whenever [sigma](t) = [[z.sup.[alpha](t)[[sigma].sub.[alpha].

The operator Da has the following properties:

(i) [[D.sup.[alpha] (C.sub.1][sigma].sub.1] + [c.sub.2][[sigma].sub.2]) = [c.sub.1] [[D.sup.[alpha] [[sigma].sub.1] + [c.sub.2] [[D.sup.[alpha] [[sigma].sub.2], [c.sub.1]

(ii) [D.sup.[alpha](fs) = df/dt [sigma] + f [[D.sup.[sigma] [sigma] or [sigma] [member of] [GAMMA] (F)[alpha] and f [0,1] [right arrow] R a smooth function,

(iii) If [??] is a local section that extends [sigma] [member of] [GAMMA] (F)[alpha] and [rho]([alpha](t)) [not equal to] 0 ([rho]([alpha](t)) = 0) then ([D].sup.[alpha][sigma])(t) = [[D.sup.[alpha]([??]) (resp. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The first two properties are immediate by (3.4). To prove (iii) one uses (3.4), (3.2) and (3.3)

By contradiction one proves that [[D.sup.[alpha] is the unique operator with the properties (i) , (ii) and (iii). Indeed, if [D.sup.[alpha] [GAMMA](F)[alpha] [right arrow] [GAMMA](F)[alpha] is another operator satisfying (i), (ii) and (iii), it easily follows that it is has the form that appears in the second hand of (3.4).

Definition 3.4. An a-section a in the vector bundle (F, g, M) is said to be parallel if [D.sup.[alpha][sigma] = 0.

Locally, the a-section [sigma](t) = [z.sup.alpha](t)[sigma][alpha] with [alpha](t) = [y.sup.a](t)[s.sub.a] is parallel if and only if the functions [z.sup.alpha](t) are solutions of the following linear system of ordinary differential equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.5)

This system has an unique solution t [right arrow] [sigma](t) with the initial condition [sigma](0) = [sigma].sub.0]. This fact allow us to define the parallel displacement along [alpha], denoted by

[P.sup.t].sub.[alpha]: F[gamma](t), [gamma](t)= p([alpha](t)=q ([sigma])(t)), [P.sup.t][alpha]([sigma].sub.0]= [sigma](t).

The maps [P.sup.t].sub.[alpha] are linear isomorphisms.

In particular, we may take a a loop based at x [member of] M i.e. [gamma](0) = [gamma](1) = x and we get the linear isomorphism [[P.sub.[alpha]: [F.sub.x] [right arrow] [F.sub.x]. Its inverse is [[P.sub.[alpha.sup-1] where [alpha.sup-1] is the reverse loop of [alpha] and if we consider the composite [alpha .sub.1] [alpha.sub.2] that is [alpha.sub.2] followed by [[alpha].sub.1] of two loops based on x it comes out that [P.sub.alpha1][[alpha].sub.2] = [[P.sub.[alpha].sub.2] 0 [[P.sub.[alpha].sub.1]- On this way one obtains a subgroup of the linear isomorphisms of [F.sub.x] called the holonomy group of D, denoted by [PHI](x)).

We fix t and consider ([P.sup.t].sub.[alpha]).sup.-1]: [F.sub.[gamma](t) [right arrow](0)- Locally, if ([P.sup.T][t.sub.alpha]).sup.-1]([sigma](tau])) = [??], then [??]t) = [??] [??] = [P.sup.t].sub.[alpha].sup.-1] and ([??]) are solutions of (3.5).

By Taylor's formula hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We divide this by t, take t [right arrow] 0 and obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Suppose now that D is a linear connection in the Lie algebroid A = (xi],[,],[rho])[xi] (E,p,M). An a-path is called geodesic if [D.sup.[alpha][alpha] = 0. Locally, if [alpha](t) = (x.sup.i](t), [y.sup.a](t)), then [D.sup.alpha] [alpha] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] sa and a is a geodesic if and only if the

functions |x.sup.i](t),[y.sup.a](t) are solution of the following system of ordinary differential equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It is clear that one has the existence and uniqueness of geodesic with a given base point x [member of] M and a given [y.sub.o] [member of] [E.sub.x0]. If for a pair ([x.sub.0][y.sub.0]) we have [p.sup.i].sub.[alpha]([x.sub.0])[y.sup.a].sub.0] = 0, the corresponding geodesic is contained in the fibre [E.sub.xo] i.e. it is a vertical A-path.

4. RIEMANNIAN METRICS IN LIE ALGEBROIDS

Let A = ([xi], [,],[rho]) be a Lie algebroid with [xi] = (F,p, M) and a vector bundle (F, g, M) endowed with an A-connection D whose local coefficients are ([[GAMMA.sup.[alpha][[beta].sub.a]).

A Riemannian metric in (F, g, M) is a mapping g that assigns to any x [member of] M a scalar product gx in Ex such that for any local section [sigma.sub.1][sigma.sub.2] [member of] [GAMMA](F), the function [g.sub.x] ([sigma].sub.1] [sigma].sub.2]) is smooth. Locally, we set [g.sub.x] ([[sigma].sub.[alpha], [sigma][beta] = [g.sub.[alpha] [beta] and so

gx([[sigma].sub.1][[sigma].sub.2] = [[g.sub.[alpha][beta] (x) z[alpha].sub.1][[z.sup.[alpha].sub.1] [z[beta].sub.2] if [sigma][beta].sub.2] [sigma].sub.1]= [z.sup.[alpha].sub.1] [[sigma.sub.][alpha] [sigma.sub.2] = z[beta].sub.2] [[sigma].sub.[beta]

The operator of covariant derivative D can be extended to the tensor algebra of (F, q, M) taking [D.sub.[sigma]f]= [rho]([sigma])f, assuming that it commutes with the contractions and behaves like a derivation with respect to tensor product. It comes out that if a; is a section in the dual (F*,g*,M) then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and if a is a section in [L.sup.2](F, M), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.1)

Definition 4.1. The Riemannian metric g is called compatible with the A-connection D if [D.sub.s]g = 0 for every s [member of] [GAMMA](E).

By (4.1) the condition of compatibility between g and D is equivalent to [rho](s)g([sigma.sub.1][sigma.sub.2]) = g([sigma].sub.1] [D.sub.s][sigma.sub.2]) [member of] [GAMMA] (E) [sigma.sub.1] [sigma.sub.2] [member of] [GAMMA] (F) (4.2) Locally, (4.2) is written as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The operator [[D.sup.[alpha] can be extended to a-section in the tensor bundles constructed with (F, g, M) and one deduces that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(F, q, M) coincides with (F,p, M) we have

Theorem 4.1. There exists an unique linear connection V in the Lie algebroid A such that

(i) [V.sub.s]g = 0,

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It is given by the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.3)

and its local coefficients are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.4)

Proof. In the condition (i) written for [s.sub.1], [s.sub.2], [s.sub.3] [member of] [GAMMA](E) we cyclically permute [s.sub.1], [s.sub.2], [s.sub.3] and so we obtain two new identities. We add these and from the result we subtract the first. Using (ii) some terms cancel each other and we get (4.3). Writing (4.3) in a local basis of sections we find (4.4). The uniqueness follows by contradiction.

If we put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

we get a section in the bundle L(E, E; E) that may be called the torsion of [nabla].

The curvature of [nabla] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The connection [nabla] given by the Theorem 4.1 is called the Levi-Civita connection of A.

We stress that the Theorem 4.1 says that given g there exists and is unique [nabla] such that [[nabla].sub.g] = 0 and [T.sub.[nabla]] = 0. Now we give a different proof of this theorem.

Given g we may associate to it the energy function E: E [right arrow] M, E(s) = g(s,s), s [member of] E. Locally, E(x,y) = [g.sub.ab]{x)[y.sup.a][y.sup.b], s = [y.sup.a][s.sub.a].

The energy function E is a regular Lagrangian on E i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In [2], we associated to any regular Lagrangian L on a Lie algebroid a semispray on E, that is a vector field

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.5)

where [g.sub.ab] = 1/2 [[delta].sup.2]L/[delta][y.sup.a][delta][y.sup.b] and ([g.sub.ab]) is the inverse of the matrix ([g.sub.ab])

Taking L = [epsilon] in (4.5), a direct calculation in which [L.sup.a.sub.cd][y.sup.c][y.sup.d] = 0 is used, shows that the semispray associated to E has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (46)

with [[GAMMA].sup.a.sub.cd] given by (4.4). These coefficients determines [nabla]. They are symmetric in bottom indices, hence [T.sub.[nabla]] = 0. The uniqueness of [nabla] follows by contradiction.

Note that (4.6) gives a 2-homogeneous semispray, that is a spray.

By (4.6) it follows

Theorem 4.2. The integral curves of S are just the geodesies of the Levi-Civita connection [nabla] in the Lie algebroid A.

For a different derivation of S from g we refer to [24].

Now we come back to the general framework and prove

Lemma 4.1. Let be the vector bundle (F, g, M) endowed with an A-connection D and a Riemannian metric g. Then for any A-path [alpha]: t [right arrow] [alpha](t), t [member of] [0,1] on E with base curve [gamma] = q o [alpha] = p o [alpha] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [[sigma].sub.1], [[sigma].sub.2] [member of] [F.sub.[gamma]](0) and [P.sup.t.sub.[alpha]] : [F.sub.[gamma](t)] [right arrow] [F.sub.[gamma](0)] is the parallel displacement defined by D along [alpha].

Proof Let [[??].sub.1] and [[??].sub.2] be the parallel [alpha]--sections in (F, g, M) such that [[??].sub.1](0) = [[sigma].sub.1], [[??].sub.2](0) = [[sigma].sub.2]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. By Taylor formula in the local basis ([sigma][beta]) we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and a similar formula for [P.sup.t.sub.[alpha]][[sigma].sup.2]. Recall that [alpha](t) = [y.sup.a](t)[s.sub.a]. Then by using again the Taylor formula and omitting the terms which contain [t.sup.2] and [t.sup.3], we may write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the terms in the parenthesis are computed for some values [tau] in (0,t).

Now dividing by t, taking t [right arrow] 0 and looking at (4.4), we obtain (4.10). ?

Definition 4.2. An A-conection D in (F, g, M) is said to be metrizable if there exists a Riemannian metric in (F, g, M) such that Dg = 0.

Based on Lemma 4.1 we obtain:

Theorem 4.3. Any A-conection D is metrizable with respect to g if and only if all its parallel displacements are isometries with respect to g.

In particular, the holonomy group [PHI](x) is made up of isometries of ([F.sub.x], g(x)).

Using a local chart around x we may put in an 1 : 1 correspondence with a subgroup of [??](x) of GL(k,R), k = rank F. A change of local chart moves that subgroup in a conjugate of it. We identify with this class of conjugate subgroups in GL(k,R). With this identification, by Theorem 4.6 we get

Theorem 4.4. A necessary condition for an A-connection to be metrizable is that the holonomy group [PHI](x) be a subgroup of the orthogonal group 0(k) for every x [member of] M.

Indeed, if [PHI](x) is made up of isometries of ([F.sub.x],g(x)), the elements of [??](x) are isometries of ([R.sup.k], <, >), with the inner product <, > induced by g(x).

Invited paper to celebrate Professor Constantin Udriste, on the occasion of his seventies

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MIHAI ANASTASIEI

"Al. I. Cuza" University of Iasi

Faculty of Mathematics,

700506; Iasi, Romania

E-mail address: anastas@uaic.ro
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Author:Anastasiei, Mihai
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Date:Jan 1, 2010
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