DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 161–169

DiffM DOES NOT ACT TRANSITIVELY ON THE SOLUTIONS OF AN INVARIANT VARIATIONAL PROBLEM ON F M

J. MUÑOZ MASQUÉ, M.E. ROSADO MARÍA

Abstract. Let π : F M → M be the bundle of linear frames of a C^{∞} m−manifold M, and let L : J^{1}(F M) → ℝ be an invariant under diffeomorphisms function. We set E(L) = {j_{x}^{2}s ∈ J^{2}(F M); E(L)(j_{x}^{2}s) = 0}, where E(L) is the Euler-Lagrange operator associated to the Lagrangian density Lθ^{1} ∧...∧θ^{m}, θ = (θ^{1}, ..., θ^{m}) being the canonical form on F M. Let Diff_{x}M be the group of germs of diffeomorphisms ϕ : (M, x) → (M, x) and let S_{x} be the set of germs of sections of π, defined around x ∈ M, which are extremals of the above density. The following result is proved. If (i) M and L are real analytic, (ii) E(L) → M is a fibered subbundle of J^{2}(F M), and (iii) E(L) is formally integrable, then: 1) Under the natural action of the diffeomorphisms on linear frames, Diff_{x}M leaves invariant S_{x}. 2) The action of Diff_{x}M on S_{x} is non-transitive.

1. Introduction

In General Relativity and gauge theories one usually works with Lagrangian densities invariant under a large group of transformations G: the group of all diffeomorphisms of the base manifold in the case of General Relativity and the gauge group (that is, the group of vertical automorphisms of a principal bundle) in the case of gauge theories. If S is the set of extremals of such an invariant Lagrangian density, the group G acts in a natural way on S, and one is interested in the structure of the quotient space S/G; that is, the space of solutions of the variational problem modulo the gauge group of the (geometric) structure that we consider. Note that it is essential for the Lagrangian density to be G-invariant in order for the group G to act on S, the space of solutions.

In the present work we analyse Lagrangian densities defined on the first jet prolongation of the bundle of linear frames of an m-dimensional manifold M, which are invariant under the group of all diffeomorphisms of M (cf. [GM]). We consider the bundle of linear frames π : F M → M of a C^{∞} manifold, and an invariant under diffeomorphisms function L : J^{1}(F M) → ℝ, associated uniquely to a Lagrangian density Ω_{m}; that is, Ω_{m} = Lθ^{1}∧...∧θ^{m}, where θ = (θ^{1}, ..., θ^{m}) is the canonical form on F M. For each Lagrangian function L ∈ C^{∞}(J^{1}(F M)) there exist (globally defined) unique functions E_{ij}(L) ∈ C^{∞}(J^{2}(F M)), such that the Euler-Lagrange equations of the Lagrangian density Ω_{m} can be globally written as: E_{ij}(L)(j_{x}^{2}s) = 0, 1 ≤ i, j ≤ m. In theorem 3.3 we prove that these functions E_{ij}(L) are natural respect to the group of diffeomorphisms on M. Therefore, assuming that (1) M and L are real analytic, (2) in order to be able to construct a fiber subbundle of the 2−jet bundle E(L) → M, called the Euler-Lagrange equation bundle, we assume that the number of functions E_{ij}(L) that are functionally independent is locally constant, and assuming that (3) the Euler-Lagrange equation bundle E(L) is formally integrable, taking into account that dim E_{x}(L) > dim G_{x}^{3}(M) and that there exists an analytic solution of any formally integrable partial differential equation (cf. [G]), we prove that the group Diff _{x}M acts on S_{x} in a natural and non-transitive way.

1991 Mathematics Subject Classification. 58E30, 57R25, 58D19, 58G35.

Key words and phrases. Invariant Lagrangians, Euler–Lagrange morphism.

This paper is in final form and no version of it will be submitted for publication elsewhere.

The importance of this result is that it ensures that S_{x}/Diff_{x}M is not trivial, so that this space is a good candidate to carry on it a multisymplectic structure. Indeed, in every case we have dealt with (see §5), the quotient space S_{x}/Diff_{x}M is not only non-trivial but also a (germ of) infinite dimensional functional space which we describe in some examples.

2. Invariant Lagrangians on FM

Let M be an m−dimensional connected C^{∞} manifold and let π : F M → M be the bundle of linear frames. F M is a principal fiber bundle over M whose structure group is the full linear group GL(m; ℝ). This bundle is endowed with a ℝ^{m}−valued 1−form θ = (θ^{1}, ..., θ^{m}), called the canonical form on F M (cf. [KN]). Any coordinate system (x_{i}), 1 ≤ i ≤ m, on an open subset U of M induces a coordinate system (x_{i}, z_{ij}), 1 ≤ i, j ≤ m, on π^{−1}(U) by the formula u = ((∂/∂x_{1})_{x}, ..., (∂/∂x_{m})_{x}) · (z_{ij}(u)), u ∈ π^{−1}(x). The canonical form is given in terms of an induced system by θ^{i} = ∑_{j} z^{ij}dx_{j}, 1 ≤ i ≤ m, where (z^{ij}) = (z_{ij})^{−1} stands for the inverse matrix. Any diffeomorphism ϕ of M induces an automorphism ϕ̃ of F M mapping the frame u = (X_{1}, ..., X_{m}) at x into the frame ϕ̃(u) = (ϕ_{∗}X_{1}, ..., ϕ_{∗}X_{m}) at ϕ(x), such that ϕ · π = π · ϕ̃. Given a fibered manifold p : P → M, let us denote by p^{1} : J^{1}(P) → M the 1−jet bundle of local sections of P. If we consider the bundle of linear frames of a m−dimensional manifold M, then each automorphism φ : F M → F M induces a diffeomorphism J^{1}(φ) : J^{1}(F M) → J^{1}(F M) by setting J^{1}(φ)(j_{x}^{1}s) = j_{φ_{M} (x)}^{1}(φ · s · φ_{M}^{−1}), where φ_{M} : M → M is the induced map on the base manifold.

Definition 2.1. A differentiable function f : J^{1}(F M) → ℝ is said to be invariant under diffeomorphisms if f · J^{1}(ϕ̃) = f, ∀ϕ diffeomorphism of M.

Definition 2.2. Given a fibered manifold p : P → M, with m =dimM, m + n =dimP, a Lagrangian density over P is an m−form Ω_{m} on J^{1}(P), that is horizontal on M with respect to the projection p^{1} : J^{1}(P) → M.

DiffM DOES NOT ACT TRANSITIVELY 163

Remark 2.3. If the manifold M is orientable and ω_{m} is a volume form on M, then every Lagrangian density Ω_{m} can be uniquely written as Ω_{m} = Lω_{m}, for some function L ∈ C^{∞}(J^{1}P), which is called the Lagrangian function associated to the volume form ω_{m}. The Lagrangian function L depends on the volume form ω_{m}. In the particular case of the bundle of linear frames, a Lagrangian density admits a canonical decomposition, such that there is a natural and 1:1 correspondence between Lagrangian densities and Lagrangian functions, namely

Proposition 2.4. Let π : F M → M be the principal fiber bundle of linear frames of an m−dimensional manifold M, and let (θ^{1}, ..., θ^{m}) be the components of the canonical form θ on F M. We have:

(i) The m−form θ^{1} ∧ ... ∧ θ^{m} is π−horizontal on F M and it does not vanish at every x ∈ M.

(ii) For each Lagrangian density Ω_{m} on J^{1}(F M) there is a unique function L ∈ C^{∞}(J^{1}(F M)), such that: Ω_{m} = Lθ^{1} ∧ ... ∧ θ^{m}. The function L will be called the Lagrangian function associated to the density Ω_{m}.

Definition 2.5. A Lagrangian density Ω_{m} on J^{1}(F M) is said to be invariant under diffeomorphisms if for every diffeomorphism ϕ of M we have: (J^{1}(ϕ̃))^{∗}Ω_{m} = Ω_{m}.

Proposition 2.6. Let L ∈ C^{∞}(J^{1}(F M)) be the Lagrangian associated to a density Ω_{m} on J^{1}(F M). Then Ω_{m} is invariant under diffeomorphisms if and only if L is a function invariant under diffeomorphisms.

3. Euler-Lagrange equations for an invariant Lagrangian

Let the m−form Ω_{m} = Ldx_{1}∧...∧dx_{m} be a Lagrangian density on J^{1}(F M). As is well-known ([KMS], XI, §49 ) there exists a morphism of fibered manifolds over M, called the Euler-Lagrange morphism associated to Ω_{m}, E(Ω_{m}) : J^{2}(F M) → V^{∗}(F M)⊗∧^{m}T^{∗}M, whose local expression is given by: E(Ω_{m}) = ∑_{i,j} E_{ij}(L)dz_{ij} ⊗ dx_{1} ∧ ... ∧ dx_{m}, where,

E_{ij}(L) = ∂L/∂z_{ij} − ∑_{k} d/dx_{k}(∂L/∂z_{k}^{ij}),

d/dx_{k} = ∂/∂x_{k} + z_{k}^{ij}∂/∂z_{ij} + z_{(kl)}^{ij}∂/∂z_{l}^{ij} is the formal derivative with respect to

x_{k}, and (x_{k}, z_{ij}, z_{k}^{ij}, z_{(kl)}^{ij}) is the coordinate system induced by (x_{k}, z_{ij}) on J^{2}(F M).

Proposition 3.1. Let E_{ij} ∈ 𝔤𝔩(m; ℝ) be the matrix such that the entry at the i-th row and the j-th column is 1 and the others entries are all zero, and let ε_{ij} ∈ V^{∗}(F M), 1 ≤ i, j ≤ m, be the dual basis of E_{ij}^{∗}, where E_{ij}^{∗} is the fundamental vector field associated to E_{ij} ( [KN], I,§4); that is, ε_{ij}(E_{kl}^{∗}) = δ_{ik}δ_{jl}. For each Lagrangian function L ∈ C^{∞}(J^{1}(F M)) associated to a Lagrangian density Ω_{m}, (i.e. Ω_{m} = Lθ^{1} ∧ ... ∧ θ^{m}), there exist (globally defined) unique functions E_{ij}(L) ∈ C^{∞}(J^{2}(F M)), such that E(Ω_{m})(j_{x}^{2}s) = ∑_{i,j} E_{ij}(L)(j_{x}^{2}s)(ε_{ij})_{s(x)} ⊗ (θ^{1} ∧ ... ∧ θ^{m})_{s(x)}. Accordingly, the Euler-Lagrange equations of a Lagrangian density Ω_{m} = Lθ^{1} ∧ ... ∧ θ^{m} on J^{1}(F M) can be globally written down as: E_{ij}(L)(j_{x}^{2}s) = 0, 1 ≤ i, j ≤ m.

Theorem 3.2. With the above notations, for any diffeomorphism ϕ of M, we have: E_{ij}(L · J^{1}(ϕ̃)) = E_{ij}(L) · J^{2}(ϕ̃), where J^{2}(ϕ̃) is the diffeomorphism induced by ϕ̃ on J^{2}(F M).

Proof. On a coordinate neighbourhood (J^{1}(F U); (x_{i}, z_{ij}, z_{k}^{ij})) of J^{1}(F M), the expression of the Euler-Lagrange equations for the Lagrangian L is (†) E_{ij}(L) = detz_{ab} ∑_{k} E_{kj}(L)z_{ki} where L ∈ C^{∞}(J^{1}(F M)) is the Lagrangian function associated to ω_{m} = dx_{1} ∧ ... ∧ dx_{m}, and so L = Ldet(z_{ij}). Let us calculate the Euler-Lagrange equations of (J^{1}(ϕ̃))^{∗}Ω_{m} = L̄det(Φ_{ab})dx_{1} ∧ ... ∧ dx_{m}, where Φ_{ab} = ∂ϕ_{a}/∂x_{b}, and L̄ = L · J^{1}(ϕ̃). Therefore,

(∗) E_{ij}(L̄det(Φ_{ab})) = ∂(L̄det(Φ_{ab}))/∂z_{ij} − ∑_{k} d/dx_{k}(∂(L̄det(Φ_{ab}))/∂z_{k}^{ij})

= ∂L̄/∂z_{ij}det(Φ_{ab}) − ∑_{k} d/dx_{k}(det(Φ_{ab})∂L̄/∂z_{k}^{ij})

= (∂L̄/∂z_{ij} − ∑_{k} d/dx_{k}(∂L̄/∂z_{k}^{ij}))det(Φ_{ab}) − ∑_{k} ∂L̄/∂z_{k}^{ij}∂(det(Φ_{ab}))/∂x_{k} = E_{ij}(L̄)det(Φ_{ab}) − ∑_{k} ∂L̄/∂z_{k}^{ij}∂(det(Φ_{ab}))/∂x_{k}, 1 ≤ i, j ≤ m.

So, in order to know E_{ij}(L̄ · det(Φ_{ab})), we need to calculate (1) ∂L̄/∂z_{ij} and (2) d/dx_{k}(∂L̄/∂z_{k}^{ij}). We have:

ϕ_{i} = x_{i} · J^{1}(ϕ̃) = x_{i} · ϕ,

ϕ_{ij} = z_{ij} · J^{1}(ϕ̃) = z_{ij} · ϕ̃ = ∑_{k}(∂ϕ_{i})/(∂x_{k})z_{kj},
ϕ_{k}^{ij} = z_{k}^{ij} · J^{1}(ϕ̃) = ∑_{l,k}((∂^{2}ϕ_{i})/(∂x_{l}∂x_{k})z_{kj} + Φ_{ik}z_{l}^{kj})Φ^{lk},

where Φ^{lk} = ∂(ϕ^{−1})_{l}/∂x_{k} ; then we have:

(1) ∂L̄/∂z_{ij} = ∑_{a}(∂L/∂z_{aj} · J^{1}(ϕ̃))Φ_{ai} + ∑_{a,c}(∂L/∂z_{c}^{aj} · J^{1}(ϕ̃))∂(Φ_{ai})/∂x_{l}Φ^{lc}

(2) d/dx_{k}(∂L̄/∂z_{k}^{ij}) = ∑_{a,c} d/dx_{k}[(∂L/∂z_{c}^{aj} · J^{1}(ϕ̃))Φ_{ai}Φ^{kc}]

= ∑_{a,c}[d/dx_{k}(∂L/∂z_{c}^{aj} · J^{1}(ϕ̃))Φ_{ai}Φ^{kc} + (∂L/∂z_{c}^{aj} · J^{1}(ϕ̃))∂(Φ_{ai}Φ^{kc})/∂x_{k}]

= ∑_{a,c} ∑_{l}(d/dx_{l}(∂L/∂z_{c}^{aj}) · J^{2}(ϕ̃))Φ_{ai}δ_{l}^{c} + ∑_{k}(∂L/∂z_{c}^{aj} · J^{1}(ϕ̃))∂(Φ_{ai}Φ^{kc})/∂x_{k}.

Hence, ∂L̄/∂z_{ij} − ∑_{k} d/dx_{k}(∂L̄/∂z_{k}^{ij}) = ∑_{a}[(∂L/∂z_{aj} · J^{1}(ϕ̃))

− ∑_{l}(d/dx_{l}(∂L/∂z_{l}^{aj}) · J^{2}(ϕ̃)) − ∑_{c}(∂L/∂z_{c}^{aj} · J^{1}(ϕ̃))∂Φ^{lc}/∂x_{l})]Φ_{ai}

= ∑_{a}[(E_{aj}(L) · J^{2}(ϕ̃)) − ∑_{cl}(∂L/∂z_{c}^{aj} · J^{1}(ϕ̃))∂Φ^{lc}/∂x_{l}]Φ_{ai}.

Therefore, replacing the above formula on (∗) we have:

E_{ij}(L̄det(Φ_{ab})) = E_{ij}(L̄)det(Φ_{ab}) − ∑_{l} ∂L̄/∂z_{l}^{ij} ∂(det(Φ_{ab}))/∂x_{l}

= ∑_{k}{(E_{kj}(L) · J^{2}(ϕ̃))det(Φ_{ab})

− ∑_{c}[(∂L/∂z_{c}^{kj} · J^{1}(ϕ̃)) ∑_{l} ∂(Φ^{lc}det(Φ_{ab}))/∂x_{l}]}Φ_{ki},

but ∑_{l} ∂(Φ^{lc}det(Φ_{ab}))/∂x_{l} = 0. In fact, we can define m vector fields X_{1}, ..., X_{m}, by imposing that: (∗∗) dϕ_{1} ∧ ... ∧ d̂ϕ_{c} ∧ ... ∧ dϕ_{m} = i_{X_{c}}(dx_{1} ∧ ... ∧ dx_{m}). Hence:

\begin{matrix}∑_{l} ∂(ϕ_{1}, ..., ϕ̂_{c}, ..., ϕ_{m})/∂(x_{1}, ..., x̂_{l}, ..., x_{m}) dx_{1} ∧ .. ∧ d̂x_{l} ∧ ... ∧ dx_{m} = \\ = ∑_{l}(−1)^{l−1}X_{c}(x_{l})dx_{1} ∧ ... ∧ d̂x_{l} ∧ ... ∧ dx_{m}, \end{matrix}

and accordingly, X_{c} = (−1)^{c−1} ∑_{l}((detΦ)Φ^{lc})∂/∂x_{l}. Moreover, divX_{c} = ∑_{l} ∂((detΦ)Φ^{lc})/∂x_{l} = 0, since the form on the left hand side of (∗∗) is closed. Consequently, we have:

E_{ij}((L · J^{1}(ϕ̃))det(Φ_{ab})) = ∑_{k}(E_{kj}(L) · J^{2}(ϕ̃))det(Φ_{ab})Φ_{ki},

and replacing the above formula on (†) the result follows,

E_{ij}(L · J^{1}(ϕ̃)) = E_{ij}((L · J^{1}(ϕ̃))det(Φ_{ab})det(z_{ij}))

= det(z_{ab}) ∑_{l} z_{li}E_{lj}((L · J^{1}(ϕ̃))det(Φ_{ab}))
= det((z · J^{1}(ϕ̃))_{ab}) ∑_{k}(E_{kj}(L) · J^{2}(ϕ̃))(z_{ki} · J^{1}(ϕ̃))
= E_{ij}(L) · J^{2}(ϕ̃).
□

4. The action of G^{3}(M) on the equation bundle

As is well-known ([KMS], XI, §49.3) a section s : U → F M of the principal bundle of linear frames F M of a manifold M is an extremal of the variational problem defined by a Lagrangian density Ω_{m} on J^{1}(F M) if and only if the 2−jet prolongation of s, satisfies the Euler-Lagrange equations of Ω_{m}; that is, (see prop. 3.2) if and only if E_{ij}(L)·j_{x}^{2}s = 0, 1 ≤ i, j ≤ m, ∀x ∈ U, where L ∈ C^{∞}(J^{1}(F M)) is the Lagrangian function associated to the Lagrangian density Ω_{m}.

Definition 4.1. Let us define for each x ∈ M, the set E_{x}(L) of 2−jet extension j_{x}^{2}s of sections s of F M on x, that satisfy the Euler-Lagrange equations of Ω_{m} on x; that is,

∀x ∈ M, E_{x}(L) = {j_{x}^{2}s ∈ J_{x}^{2}(F M); E_{ij}(L)(j_{x}^{2}s) = 0, ∀i, j}.

We set E(L) = ⋃_{x∈M} E_{x}(L). If E(L) is a fibered submanifold of π^{2} : J^{2}(F M) → M (see [G], def.7.1 ), then E(L) is a partial differential equation of order 2 on F M. In that case, the fiber bundle E(L) over M will be called the Euler-Lagrange equation bundle. Let G_{x}^{3}(M) be the Lie group (cf. [KMS], IV ,§12) of 3−jets, j_{x}^{3}ϕ, with source x ∈ M, of local diffeomorphisms ϕ of M, satisfying ϕ(x) = x, with the product given by j_{x}^{3}ϕ · j_{x}^{3}ϕ′ := j_{x}^{3}(ϕ · ϕ′), and differentiable structure given by the global chart:

G_{x}^{3}(M) ⟶ GL(m; ℝ) × (S^{2}(ℝ^{m})^{∗} ⊗ ℝ^{m}) × (S^{3}(ℝ^{m})^{∗} ⊗ ℝ^{m})
j_{x}^{3}ϕ ⟼ (Dϕ(x), D^{2}ϕ(x), D^{3}ϕ(x)),

where D, D^{2} and D^{3} stand for the first, second and third derivatives of ϕ with respect to a coordinate system defined around x ∈ M. Let J_{x}^{2}(F M) be the set of all 2−jet of sections s on the frame bundle F M, j_{x}^{2}s, with source x ∈ M. The Lie group G_{x}^{3}(M) acts (on the left) on the manifold J_{x}^{2}(F M) by setting: j_{x}^{3}ϕ · j_{x}^{2}s = J^{2}(ϕ̃)(j_{x}^{2}s) = j_{x}^{2}(ϕ̃ · s · ϕ^{−1}), where ϕ̃ is the diffeomorphism induced on F M by the diffeomorphism ϕ of M.

Proposition 4.2. If the function L ∈ C^{∞}(J^{1}(F M)) is invariant under diffeomorphisms then the Lie group G_{x}^{3}(M) acts on E_{x}(L), for all x ∈ M.

Notation 4.3. 1. Let Diff_{x}M be the group of germs of diffeomorphisms ϕ : (M, x) → (M, x).

2. Let S_{x} be the set of germs of sections of π, defined around x ∈ M, which are the extremals of the Lagrangian density Ω_{m}.

Theorem 4.4. Let M be an m−dimensional (m > 1) connected, analytic manifold and let us consider a variational problem given by a Lagrangian density Ω_{m} on J^{1}(F M). Let us assume the following three conditions:

1. The Lagrangian function L associated to Ω_{m} is analytic,

2. for every j_{x}^{2}s ∈ J^{2}(F M), the number of functions E_{ij}(L) : J_{x}^{2}(F M) → ℝ that are functionally independent is locally constant. Hence E(L) → M is a fiber subbundle of the 2−jet bundle of F M, and

3. the Euler-Lagrange equation bundle E(L) is formally integrable ( [G], Def. 7.2). Then the group Diff_{x}M acts on S_{x} in a natural and non-transitive way.

Proof. There exist two elements ξ, and ξ′ of the space E_{x}(L) that are not equivalent by the Lie group G_{x}^{3}(M). This is true because G_{x}^{3}(M) acts on E_{x}(L), and dim E_{x}(L) > dim G_{x}^{3}(M) for m > 1. In fact, we have: dim E_{x}(L) ≥ dim J_{x}^{2}(F M) − m^{2} = m^{2}(m + 1)(m + 2)/2 − m^{2} = m^{3}(m + 3)/2, the equality being true if all functions E_{ij}(L) : J_{x}^{2}(F M) → ℝ are functionally independent, and we also have for m > 1: dim G_{x}^{3}(M) = m^{2}(m^{2} + 6m + 11)/6, and so, dim E_{x}(L) > dim G_{x}^{3}(M). Therefore, (see [G], th.9.1) there exist two analytic solutions s and s′ of the Euler-Lagrange equations, over a neighbourhood U of x, such that: j_{x}^{2}s = ξ, j_{x}^{2}s′ = ξ′, ∀x ∈ U. Let us assume that there is ϕ ∈ Diff_{x}M such that ϕ̃ · s · ϕ^{−1} = s′; that is, the diffeomorphism ϕ of M transforms the solution s into the solution s′, then we will have: J^{2}(ϕ̃)(ξ) = ξ′ and we are led to a contradiction. □

5. The basis of invariant Lagrangians. Examples

In [GM] it is proved that a Lagrangian L : J^{1}(F M) → ℝ is invariant under diffeomorphisms if and only if L can be locally written as a differentiable function of a fixed system of invariant Lagrangians L_{jk}^{i} : J^{1}(F M) → ℝ, 1 ≤ i ≤ m, 1 ≤ j < k ≤ m; i.e., L = F · L_{jk}^{i}, where F ∈ C^{∞}(ℝ^{m^{2}(m−1)/2}). The Lagrangians L_{jk}^{i} have the following geometrical meaning: Given a point ξ ∈ J_{x}^{1}(F M), let s = (X_{1}, ..., X_{m}) be a local frame defined on a neighbourhood of x ∈ M, representing the above point; i.e., ξ = j_{x}^{1}s. Then, we define L_{jk}^{i} by the formula: [X_{j}, X_{k}]_{x} = ∑_{i} L_{jk}^{i}(ξ)(X_{i})_{x}. It is easily proved that the definition makes sense, because [X_{j}, X_{k}]_{x} only depends on j_{x}^{1}s. The local expression for L_{jk}^{i} is as follows: L_{jk}^{i} = ∑_{h,l}(z_{hj}z_{h}^{lk} − z_{hk}z_{h}^{lj})z^{il}. We shall study the variational problems defined by the invariant Lagrangians L_{jk}^{i} of the above basis for m = 2 or 3.

Proposition 5.1. The densities Ω_{m} = L_{jk}^{i}θ^{1} ∧ .. ∧ θ^{m} only define two essentially different kinds of variational problems according to i = j or i ≠ j, k.

Proof. Let ψ_{σ} : F M → F M be the fibre-preserving diffeomorphisms given by ψ_{σ}(u) = (X_{σ_{1}}, ..., X_{σ_{m}}), where σ = (σ_{1}, ..., σ_{m}) ∈ P erm{1, ..., m} and u = (X_{1}, ..., X_{m}). It is not difficult to prove that for every A ∈ GL(m; ℝ), u ∈ F M, we have: ψ_{σ}(u · A) = u · A^{σ}, where A^{σ} is the matrix A^{σ} = (A_{σ_{1}}, ..., A_{σ_{m}}), A_{1}, ..., A_{m} being the columns of A. Given two Lagrangians L_{jk}^{i} and L_{bc}^{a} as above, with 1 ≤ i ≤ m, 1 ≤ j < k ≤ m, i ≠ j, k, and 1 ≤ a ≤ m, 1 ≤ b < c ≤ m, a ≠ b, c, there is σ ∈ P erm{1, ..., m} such that σ_{i} = a, σ_{j} = b, and σ_{k} = c. Then, we have: (L_{jk}^{i} ·J^{1}(ψ_{σ}))(j_{x}^{1}s) = L_{jk}^{i}(j_{x}^{1}(ψ_{σ}(s(x)))) = ω_{σ_{i}}([X_{σ_{j}}, X_{σ_{k}}]) = ω_{a}([X_{b}, X_{c}]) = L_{bc}^{a}(j_{x}^{1}s). Hence, L_{jk}^{i} · J^{1}(ψ_{σ}) = L_{bc}^{a}. The diffeomorphisms ψ_{σ} preserve the fibres and leave invariant the canonical form on J^{1}(F M), so we know that the Lagrangians L_{jk}^{i}, with i ≠ j, k define the same variational problem. If we consider the Lagrangians L_{ij}^{i} and L_{ab}^{a}, it suffices to take a permutation σ such that σ_{i} = a, and σ_{j} = b, and the result follows. □

(I) dim M=2. According to the above proposition we only have to study the following Lagrangian density, Ω_{2} = L_{12}^{1}θ^{1} ∧ θ^{2}.

Theorem 5.2. With the above hypotheses and notations, for every local frame s = (X_{1}, X_{2}), with dual coframe (ω_{1}, ω_{2}), we have:

(j^{1}s)^{∗}Ω_{2} = ω_{1}([X_{1}, X_{2}])ω_{1} ∧ ω_{2} = −dω_{1},

or equivalently, Ω_{2} is the horizontal part of dθ^{1}. Consequently, Ω_{2} defines a trivial variational problem; i.e., every section is an extremal.

For a general germ of frame s = (X_{1}, X_{2}) around x ∈ M, the orbit of s under Diff_{x}M can be identified with the germs of non-singular vector fields at x ∈ M.

Proof. Let D be a compact submanifold of M, with ∂D ≠ ∅ and dimD=dimM = m, and let us assume that s : D → F M is a critical section for the Lagrangian density Ω_{2}, that is, d/dt ∫_{D}(j^{1}s_{t})^{∗}Ω_{2} = 0 for every one-parameter deformation s_{t} of the section s. By considering the one-parameter family of frames given by: s_{t}(x) = s(x) · A_{t}(x), s_{t} : D → F M, |t| < ϵ, s_{0} = s, where, A_{t}(x) = e^{tφ(x)E_{ij}}, 1 ≤ i, j ≤ 2, with φ ∈ C^{∞}(D), φ|_{∂D} = 0, for E_{11} and taking into account the Stokes theorem, we obtain:

0 = \frac{d}{dt} |_{t=0} ∫_{D} d(e^{tφ}ω_{1}) = ∫_{D} d(φω_{1}) = ∫_{∂D} φω_{1}.

Therefore, using the same argument for every E_{ij}, we have that every section is an extremal of the Lagrangian density Ω_{2}.

By considering a diffeomorphism ψ of M such that ψ · X_{1} = ∂/∂x_{1}, and ψ · X_{2} = a∂/∂x_{1} + b∂/∂x_{2} with a, b ∈ C^{∞}(U), U ⊂ D, we see that the space of solutions of the variational problem given by the Lagrangian density Ω_{2}, cocient by the group of diffeomorphisms of M can be identified with the space of vector fields at x ∈ M. □

(II) dim M=3. In this case, we have to study the following two Lagrangian densities, Ω_{3} = L_{12}^{1}θ^{1} ∧ θ^{2} ∧ θ^{3}, Ω′_{3} = L_{12}^{3}θ^{1} ∧ θ^{2} ∧ θ^{3}.

Theorem 5.3. With the above hypotheses and notations, we have

1) A local frame s = (X_{1}, X_{2}, X_{3}), with dual coframe (ω_{1}, ω_{2}, ω_{3}), is a critical section for the Lagrangian density Ω_{3} if and only if:

ω_{1} ∧ dω_{3} = 0, dω_{1} ∧ ω_{3} = 0, ω_{2} ∧ dω_{3} = 0,

dω_{1} ∧ ω_{1} = 0, ω_{3} ∧ dω_{3} = 0, dω_{1} ∧ ω_{2} = 0.

2) A local frame s = (X_{1}, X_{2}, X_{3}), with dual coframe (ω_{1}, ω_{2}, ω_{3}), is a critical section for the Lagrangian density Ω′_{3} if and only if:

dω_{3} ∧ ω_{3} = 0, ω_{1} ∧ dω_{3} + dω_{3} ∧ ω_{1} = 0, ω_{2} ∧ dω_{3} + dω_{3} ∧ ω_{2} = 0. Moreover, we have

1’) Given a local coordinate system (x_{1}, x_{2}, x_{3}) on a neighbourhood of x ∈ M, for every critical section s = (X_{1}, X_{2}, X_{3}) for the Lagrangian density Ω_{3}, there exists a local diffeomorphism ψ, with ψ_{M}(x) = x , such that: ψ·X_{1} = ∂/∂x_{1}, ψ· X_{2} = δ∂/∂x_{2}, δ ∈ C^{∞}(ℝ^{3}),

ψ · X_{3} = α∂/∂x_{1} + β∂/∂x_{2} + γ∂/∂x_{3}, α, γ ∈ C^{∞}(ℝ^{3}), β ∈ C^{∞}(ℝ). 2’) Given a local coordinate system (x_{1}, x_{2}, x_{3}) on a neighbourhood of x ∈ M,

for every critical section s = (X_{1}, X_{2}, X_{3}) for the Lagrangian density Ω′_{3}, there exists a local diffeomorphism ψ, keeping x fixed, such that: ψ · X_{1} = f_{11}∂/∂x_{1} + f_{21}∂/∂x_{2},

ψ · X_{2} = f_{12}∂/∂x_{1} + f_{22}∂/∂x_{2},

ψ · X_{3} = f_{13}∂/∂x_{1} + f_{23}∂/∂x_{2} + g∂/∂x_{3}, f_{ij} ∈ C^{∞}(ℝ^{3}), g ∈ C^{∞}(ℝ).

Proof. Let D be a compact submanifold of M, with ∂D ≠ ∅ and dimD=dimM = m, and let us assume that s : D → F M is a critical section for the Lagrangian density Ω_{3}, that is, d/dt ∫_{D}(j^{1}s_{t})^{∗}Ω_{3} = 0 for every one-parameter deformation s_{t} of the section s, where (j^{1}s)^{∗}Ω_{3} = ω_{1}([X_{1}, X_{2}])ω_{1} ∧ ω_{2} ∧ ω_{3} = dω_{1} ∧ ω_{3}, (up to sign). By considering the one-parameter family of frames given by: s_{t}(x) = s(x) · A_{t}(x), s_{t} : D → F M, |t| < ϵ, s_{0} = s, where, A_{t}(x) = e^{tφ(x)E_{ij}}, 1 ≤ i, j ≤ 3, with φ ∈ C^{∞}(D), φ|_{∂D} = 0, for E_{11} we obtain:

0 = \frac{d}{dt} |_{t=0} ∫_{D} d(e^{tφ}ω_{1}) ∧ ω_{3} = ∫_{D} d(φω_{1}) ∧ ω_{3} = ∫_{D} d(φω_{1} ∧ ω_{3}) + ∫_{D} φω_{1} ∧ dω_{3}

taking into account that φ is an arbitrary function and the Stokes theorem, we have, ∫_{D} d(φω_{1} ∧ ω_{3}) = ∫_{∂D} φω_{1} ∧ ω_{3} = 0. Consequently , ω_{1} ∧ dω_{3} = 0.

Using the same argument for every E_{ij}, we obtain all the Euler-Lagrange equations, and contracting them, we have: L_{ij}^{1} = 0, and L_{ij}^{3} = 0, i ≠ j, 1 ≤ i, j ≤ 3. Therefore the section s = (X_{1}, X_{2}, X_{3}) is a critical section of the action if it verifies: [X_{i}, X_{j}] = L_{ij}^{2}X_{2}, 1 ≤ i, j ≤ 3, i ≠ j. By considering a diffeomorphism ψ of M such that ψ · X_{1} = ∂/∂x_{1}, and imposing the above conditions we obtain that ψ · X_{2} = δ∂/∂x_{2}, where δ ∈ C^{∞}(ℝ^{3}) and ψ · X_{3} = α∂/∂x_{1} + β∂/∂x_{2} + γ∂/∂x_{3}, where α, γ ∈ C^{∞}(ℝ^{3}), and β ∈ C^{∞}(ℝ).

Using the same arguments as in the above case, we obtain for the Lagrangian density Ω′_{3} = L_{12}^{3}θ^{1} ∧ θ^{2} ∧ θ^{3} the following Euler-Lagrange equations: dω_{3} ∧ ω_{3} = 0, ω_{1} ∧ dω_{3} + dω_{3} ∧ ω_{1} = 0, and ω_{2} ∧ dω_{3} + dω_{3} ∧ ω_{2} = 0, or equivalently, L_{ij}^{3} = 0, 1 ≤ i, j ≤ 3, i ≠ j. Therefore, a section s = (X_{1}, X_{2}, X_{3}) is a critical section for the variational problem defined by Ω′_{3} if it verifies: (∗) [X_{i}, X_{j}] = L_{ij}^{k}X_{k}, k = 1, or 2, 1 ≤ i, j ≤ 3, i ≠ j. From the equation [X_{1}, X_{2}] = L_{12}^{1}X_{1} + L_{12}^{2}X_{2}, we know that (X_{1}, X_{2}) is involutive and as a calculation shows, there is a coordinate system such that:

X_{1} = f_{11}(x_{1}, x_{2}, x_{3})∂/∂x_{1} + f_{21}(x_{1}, x_{2}, x_{3})∂/∂x_{2},
X_{2} = f_{12}(x_{1}, x_{2}, x_{3})∂/∂x_{1} + f_{22}(x_{1}, x_{2}, x_{3})∂/∂x_{2}, det(f_{ij}) ≠ 0,
X_{3} = a(x_{1}, x_{2}, x_{3})X_{1} + b(x_{1}, x_{2}, x_{3})X_{2} + c(x_{1}, x_{2}, x_{3})∂/∂x_{3},

By imposing the others two conditions of (∗) to s we obtain that ∂c/∂x_{1} = 0 and ∂c/∂x_{2} = 0, therefore, X_{3} = α∂/∂x_{1}+β∂/∂x_{2}+γ∂/∂x_{3}, α, β ∈ C^{∞}(ℝ^{3}) and γ ∈ C^{∞}(ℝ). □

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