Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume VOL (YEAR), No. NO, 73-87.

Abstract

de León, Oubiña, Salgado: Characterization of Higher-order Tangent Bundles M. de León, J. A. Oubiña and M. Salgado

Manuel de León, Instituto de Matemáticas y Física Fundamental Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, Spain e-mail: CEEML02@CC.CSIC.ES

José A. Oubiña & Modesto Salgado, Departamento de Xeometría e Topoloxía, Facultade de Matemáticas, Universidade de Santiago de Compostela, Spain e-mail: oubina@zmat.usc.es, modesto@zmat.usc.es

T.angent bundles of higher order are the framework on which the higher order Lagrangian theories are developed. Moreover, almost tangent manifolds of higher order have been shown to be relevant in singular Lagrangian theories. The purpose of this paper is to prove that under some global hypotheses a higher order integrable almost tangent manifold is diffeomorphic to the tangent bundle of the same order of some manifold. An application to the reduction of singular Lagrangians is given. MSC 1991: 53C15, 58F05, 70H35.

1. Introduction

As we know, higher order tangent bundles are the natural framework on which the dynamics of Lagrangian systems of higher order is developed. In fact, if L : T^{k}Q ⟶ ℝ is a Lagrangian function of order k, where Q is the configuration manifold, then the dynamics is characterized by a suitable vector field X defined on the tangent bundle T^{2k−1}Q of order 2k −1. In fact, we can define from L the symplectic Poincaré–Cartan form ω_{L} (supposed L to be regular) and the energy E_{L} associated to L. Then there exists a unique vector field ξ_{L} on T^{2k−1}Q such that i_{ξ_{L}}ω_{L} = dE_{L}. The vector field ξ_{L} is a 2k–th order differential equation such that the projections of its integral curves onto Q are the solutions of the Euler–Lagrange equations for L. These results give a global symplectic formulation for higher order mechanics (see [11, 12]); for the corresponding local formulation see [20], for instance.

If L is singular then ω_{L} is not symplectic (we say that ω_{L} is presymplectic) and thus the equation i_{X}ω_{L} = dE_{L} has no solution in general, and even if a solution exists it will be neither

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} YEAR Heldermann Verlag, Berlin

74 Characterization of Higher-order Tangent Bundles^{1}

unique nor a 2k–th order differential equation. In [9], de León, Giraldo & Rodrigues have proved that, under some hypotheses on L, the space T^{2k−1}Q can be reduced to an integrable almost tangent manifold of the same order on which there exists a local regular Lagrangian gauge equivalent to the original one. Then the two Lagrangians contain the same dynamical information. To do this we project all the geometric ingredients of T^{2k−1}Q onto the quotient manifold T̂^{2k−1}Q = T^{2k−1}Q/ Ker ω_{L}. The projected canonical almost tangent structure of T^{2k−1}Q gives an integrable almost tangent structure on T̂^{2k−1}Q. Since de León & Rodrigues [14] have proved that the integrable almost tangent manifolds are sufficient to derive the Euler-Lagrange equations, then we obtain a local regular Lagrangian gauge equivalent to L (see [1, 2] for the order one).

From this discussion we see that the study of integrable almost tangent manifolds of higher order is relevant for Mechanics. Particularly, an interesting problem is to characterize when an integrable higher order almost tangent manifold is a higher order tangent bundle of some manifold. The local equivalence is given from the integrability (see [4, 8] for instance). The aim of this paper is to obtain a global result. In fact, we prove that if (M, J) is an integrable almost tangent manifold of order k satisfying some global hypothesis, then M is diffeomorphic to the tangent bundle T^{k}S of order k of some manifold S. Moreover, the diffeomorphism may be choosen in such a way that the almost tangent structure J on M is transported to the canonical almost tangent structure of T^{k}S. This result generalizes those of Crampin & Thompson [3] and Thompson & Schwardmann [17] for the order one.

For the order one, an early and different approach is due to Nagano [15] in the case of the characterization of cotangent bundles. In fact, Nagano has proved that if M is a differentiable manifold endowed with a vector field satisfying the same properties of those satisfied by the canonical vector field of a vector bundle, then there exists a unique vector bundle structure on M over the singular submanifold S of the vector field. If, moreover, M is an exact symplectic manifold then M is isomorphic to the cotangent bundle T^{∗}S, indeed as symplectic vector bundles. In [10] we have used the ideas of Nagano to give a characterization of tangent bundles. In fact, we prove that if M is an integrable almost tangent manifold endowed with a vector field C which satisfies the properties of the Liouville vector field of a tangent bundle, then there exists a unique vector bundle structure on M isomorphic to T S (S being the singular submanifold) and such that the Liouville vector field and the canonical almost tangent structure of T S are transported via the isomorphism to C and the almost tangent structure on M, respectively. This result was obtained independently by Filippo, Landi, Marmo & Vilasi [5], by using a different procedure. In [10] we have proved that the reduced manifold T̂Q = T Q/ ker ω_{L} is in fact a bona fide tangent bundle. The same is true in the higher order case. In fact, if L : T^{k}Q → ℝ is a singular Lagrangian of order k admitting a global dynamics, then the projected almost tangent manifold of order 2k − 1, T̂^{2k−1}Q = T^{2k−1}Q/ ker ω_{L}, is a tangent bundle of order 2k − 1 over the singular submanifold defined by the projected higher order Liouville vector field.

The paper is structured as follows. In §2 we recall the notations and main results which will be used in the sequel. In §3, we obtain the characterization of tangent bundles of order 2. This result is extended to the arbitrary higher order case in §4. In §5 we give an alternative characterization by using the ideas of Nagano and Filippo et al. By using the later characterization, in §6 we prove that T̂^{2k−1}Q is a true higher tangent bundle when L admits a global dynamics.

2. Background

Let Q be a differentiable manifold of dimension m. The tangent bundle of order k of Q is the (k + 1)m–dimensional manifold T^{k}Q whose points are k–jets at 0 ∈ ℝ of differentiable curves in Q. If k = 0 we identify T^{0}Q with Q and, if k = 1, T^{1}Q = T Q is the tangent bundle of Q. If r < k we denote by ρ_{r}^{k} : T^{k}Q → T^{r}Q the canonical projection given by

ρ_{r}^{k}(j_{0}^{k}σ) = j_{0}^{r}σ,

for all j_{0}^{k}σ ∈ T^{k}Q.

We shall now describe the local coordinates in T^{k}Q. Let (U, q^{i}) be a coordinate neighborhood of Q and σ : ℝ → Q a curve in Q such that σ(0) ∈ U. Then the k-jet j_{0}^{k}σ is represented in T^{k}U by (z_{0}^{i}, z_{1}^{i}, ⋯, z_{k}^{i}) where z_{r}^{i} = \frac{1}{r!}(d^{r}/dt^{r})_{|t=0}(q^{i} ∘ σ), 1 ≤ r ≤ k, 1 ≤ i ≤ m.

We denote by J_{1} the canonical tensor field of type (1, 1) on T^{k}Q locally given by

J_{1} = ∑_{r=0}^{k} \frac{∂}{∂z_{r+1}^{i}} ⊗ dz_{r}^{i} .  (1)

As J_{1}^{k+1} = 0 and rank J_{1} = km, the tensor field J_{1} defines a canonical almost tangent structure of order k on T^{k}Q (see [12, 4, 8, 21]). For each r, 2 ≤ r ≤ k, we set J_{r} = J_{1}^{r}, the composition of J_{1} r-times.

Also, let C_{1} be the canonical vector field on T^{k}Q which is locally given by

C_{1} = ∑_{r=1}^{k} r z_{r}^{i} \frac{∂}{∂z_{r}^{i}} ;  (2)

C_{1} is called the Liouville vector field of T^{k}Q and it is the infinitesimal generator of the positive homothetias of T^{k}Q. A direct computation shows that

L_{C_{1}} J_{1} = −J_{1} , J_{k}C_{1} = 0 .

We set C_{r} = J_{r−1}C_{1}, 2 ≤ r ≤ k.

As we know [6, 16] ρ_{k−1}^{k} : T^{k}Q → T^{k−1}Q is an affine bundle modelled on the vector bundle (ρ_{0}^{k−1})^{∗}(T Q) = T^{k−1}Q×_{Q}T Q, i.e., the induced vector bundle defined by the following commutative diagram:

[figure: T^{k−1}Q ×_{Q} T Q; T Q; T^{k−1}Q; Q; ρ_{0}^{1}; ρ_{0}^{k−1}]

We know that this vector bundle is canonically isomorphic to the vertical vector bundle Ker T ρ_{k−2}^{k−1} = V^{ρ_{k−2}^{k−1}}(T^{k−1}Q) (see [12]).

3. Characterization of tangent bundles of order 2

Let (M, J) be an almost tangent manifold of order 2 with almost tangent structure J, i.e., M is a 3n–dimensional manifold and J is a tensor field of type (1,1) and of constant rank 2n such that J^{3} = 0. Then we have

Im J^{2} ⊂ Im J , Ker J ⊂ Ker J^{2} ,
Im J^{2} = Ker J , Im J = Ker J^{2} .

We assume that J is integrable. Then the Nijenhuis tensor of J vanishes identically and hence Im J and Im J^{2} are integrable distributions. Consider the spaces of leaves M_{0} = M/Im J and M_{1} = M/Im J^{2} determined by the corresponding foliations, also denoted by Im J and Im J^{2}. We obtain the following commutative diagram:

[figure: M; M_{0}; M_{1}; π_{0}^{2}; π_{1}^{2}; π_{0}^{1}]

where π_{0}^{2} : M → M_{0} and π_{1}^{2} : M → M_{1} are the canonical projections. We denote by π_{0}^{1} : M_{1} → M_{0} the induced projection which maps the leaf of Im J^{2} passing through a point x ∈ M onto the leaf of Im J passing through x.

We shall consider an example of manifold M with an integrable almost tangent structure of order 2 such that the quotient space M_{0} = M/ Im J is a quotient manifold of M but not so is M_{1} = M/ Im J^{2}. Let M be the 3–torus 𝕋^{3} obtained as the quotient manifold of ℝ^{3} by the integral lattice ℤ^{3}. Let (x, y, z) be the natural coordinates of ℝ^{3}, which may be regarded as local coordinates on 𝕋^{3}. Let α ∈ ℝ, α ≠ 0. Then

J = \frac{∂}{∂x} ⊗ dx − \frac{1}{α} \frac{∂}{∂x} ⊗ dy + α\frac{∂}{∂y} ⊗ dx − \frac{∂}{∂y} ⊗ dy + \frac{∂}{∂x} ⊗ dz

determines an integrable almost tangent structure of order 2 on 𝕋^{3}. Moreover, J(∂/∂x) = X, J(∂/∂y) = (−1/α)X and J(∂/∂z) = ∂/∂x, where X is the vector field on 𝕋^{3} given by X = ∂/∂x + α ∂/∂y. Then Im J = ⟨∂/∂x, ∂/∂y⟩ = ⟨X, ∂/∂x⟩ and Im J^{2} = ⟨X⟩. If α is irrational, each leaf of Im J^{2} is dense in 𝕊^{1} × 𝕊^{1} × {y}, for some y ∈ 𝕊^{1}, and so the space 𝕋^{3}/ Im J^{2} is not a quotient manifold of 𝕋^{3}.

Now, we return to the general situation and suppose that Im J and Im J^{2} define fibrations, i.e., there exist differentiable structures on M_{0} and M_{1}, respectively such that π_{0}^{2} : M → M_{0} and π_{1}^{2} : M → M_{1} are submersions. In other words, we are assuming that M_{0} and M_{1} have the structure of quotient manifolds. Then M_{1} and M_{0} have dimension 2n and n, respectively. Further, π_{0}^{1} : M_{1} → M_{0} is a differentiable submersion and M_{0} is also a quotient manifold of M.

Proposition 3.1 J projects onto an integrable almost tangent structure J_{1} on M_{1}.

Proof. Let X_{1} ∈ T_{x_{1}}M_{1}, x_{1} ∈ M_{1}. Then we define

(J_{1})_{x_{1}}X_{1} = T π_{1}^{2}(JX),  (3)

where X ∈ T_{x}M is such that π_{1}^{2}(x) = x_{1}, T π_{1}^{2}X = X_{1}.

We shall prove that (3) does not depend on the choice of x and X. In fact, if Y ∈ T_{x}M projects also onto X_{1}, then we have T π_{1}^{2}(Y − X) = 0 which implies Y − X ∈ Im J^{2} and then JY = JX. On the other hand, (3) does not depend on the choice of the point x. In fact it is sufficient to check (see [1]) that Im (L_{X}J) ⊂ Im J^{2}, for all X ∈ Im J^{2}, which is a matter of routine.

Finally, we have

J_{1}^{2}(X_{1}) = J_{1}(T π_{1}^{2}(JX)) = (T π_{1}^{2}(J^{2}X)) = 0 ,

and hence J_{1}^{2} = 0. Thus, Im J_{1} ⊂ Ker J_{1}. On the other hand, let X_{1} ∈ Ker J_{1}. Then X_{1} = T π_{1}^{2}X, where JX ∈ Im J^{2}. Therefore there exists a tangent vector Y on M such that JX = J^{2}Y. Then X − JY ∈ Ker J = Im J^{2} = ker T π_{1}^{2}. Hence

X_{1} = T π_{1}^{2}(X) = T π_{1}^{2}(JY) = J_{1}(T π_{1}^{2}(JY)) ∈ Im J_{1}.

Since Ker J_{1} = Im J_{1} we deduce that J_{1} is an almost tangent structure on M_{1}. The integrability of J_{1} follows since the Nijenhuis tensor of J projects onto the Nijenhuis tensor of J_{1}. □

From Proposition 3.1 we deduce that (M_{1}, J_{1}) is an integrable almost tangent manifold. Moreover, it defines a fibration. In fact, since Ker(T π_{0}^{1}) = Im J_{1}, the space of leaves M_{1}/Im J_{1} may be canonically identified to M/Im J and then it possesses a differentiable structure of quotient manifold in such a way that the canonical projection π : M_{1} → M_{1}/Im J_{1} is identified to the projection π_{0}^{1}. Then we obtain the following commutative diagram:

[figure: M; M_{0} ≡ M_{1}/ Im J_{1}; M_{1}; π_{0}^{2}; π_{1}^{2}; π_{0}^{1} ≡ π]

Since (M_{1}, J_{1}) is integrable then there exists a linear symmetric connection ∇_{1} on M_{1} such that ∇_{1}J_{1} = 0. As we know (see [3]) ∇_{1} induces a flat connection on the fibres of π_{0}^{1}. According to Crampin & Thompson [3] we have the following Proposition 3.2 Suppose that each leaf of the foliation defined by Im J_{1} is simply connected and geodesically complete with respect to the flat connection induced on it by ∇_{1}. Then π_{0}^{1} : M_{1} → M_{0} is an affine bundle modelled on T M_{0}. Moreover, if M_{1} → M_{0} admits a global section then M_{1} is isomorphic as a vector bundle to T M_{0}, this isomorphism depending on the choice of the section.

For each π_{0}^{1}-vertical tangent vector X ∈ T_{x_{1}}M_{1} we define its vertical lift X^{v} to T_{x}M, π_{1}^{2}(x) = x_{1}, as follows:

X^{v} = JX̃ ,

where X̃ ∈ T_{x}M is such that T π_{1}^{2}X̃ = X. Then we have

T π_{1}^{2}X^{v} = T π_{1}^{2}J(X̃) = J_{1}(X) = 0 ,

since T π_{0}^{1}X = 0 implies that X ∈ Ker T π_{0}^{1} = Im J_{1} = Ker J_{1}. Thus, X^{v} is π_{1}^{2}-vertical. Moreover, every π_{1}^{2}–vertical vector is a vertical lift.

Remark 3.1 We can consider a differentiable subbundle S of T M complementary to Ker T π_{1}^{2} = Im J^{2} with respect to a Riemannian metric. Then, if X is a vector field on M_{1}, there exists a unique vector field X̃ on M π_{1}^{2}–related to X such that X̃_{x} ∈ S_{x} for all x ∈ M. Since S is a differentiable subbundle of T M, X̃ is a differentiable vector field on M. If the vector field X is π_{0}^{1}–vertical then X^{v} = JX̃ is a π_{1}^{2}–vertical differentiable vector field. Further, the vector field X̃ is π_{0}^{2}–vertical and then X̃ ∈ Im J.

Lemma 3.1 We have:

[X^{v}, Y^{v}] = 0 ,

for all π_{0}^{1}-vertical vector fields X and Y on M_{1}.

Proof. In fact, we have

[X^{v}, Y^{v}] = [JX̃, JỸ] = J[JX̃, Ỹ] + J[X̃, JỸ] = J[X^{v}, Ỹ] + J[X̃, Y^{v}] .

But T π_{1}^{2}[X^{v}, Ỹ] = [T π_{1}^{2}X^{v}, T π_{1}^{2}Ỹ] = [0, Y] = 0. Also, T π_{1}^{2}[X̃, Y^{v}] = 0. Hence [X^{v}, Ỹ] and [X̃, Y^{v}] belong to Ker T π_{1}^{2} = Im J^{2}, which implies the result. □

Since J is integrable, then there exists a symmetric linear connection ∇ on M such that ∇J = 0.

Proposition 3.3 The connection ∇ induces a flat connection on each fibre of the fibration π_{1}^{2} : M → M_{1}.

Proof. From ∇J = 0 and the fact that ∇ is symmetric, it follows

∇_{X^{v}} Y^{v} = ∇_{X^{v}} JỸ = J(∇_{X^{v}} Ỹ) = J(∇Ỹ X^{v}) + [X^{v}, Ỹ])
= J(∇Ỹ X^{v}) = J(∇Ỹ JX̃) = ∇Ỹ J^{2}X̃ = 0 ,

since X̃ ∈ Im J. Now, the vertical lifts X^{v} span the vertical bundle Ker T π_{1}^{2}. Then we have the result. □

Consider the vector bundle V (M_{1}) → M_{1} consisting of the tangent vectors to M_{1} which are vertical with respect to the fibration π_{0}^{1} : M_{1} → M_{0}.

Proposition 3.4 Suppose that each leaf of the foliation defined by Im J^{2} is simply connected and geodesically complete with respect to the flat connection induced on it by ∇. Then π_{1}^{2} : M → M_{1} is an affine bundle modelled on V (M_{1}).

Proof. We define a morphism

Φ : M ×_{M_{1}} V (M_{1}) → M,

as follows. Let x ∈ M, X ∈ V (M_{1}) such that X ∈ T_{x_{1}}M_{1}, x_{1} = π_{1}^{2}(x), T π_{0}^{1}(X) = 0. Consider the vertical lift X^{v} of X as a vector field on the fibre (π_{1}^{2})^{−1}(x_{1}).

By Proposition 3.3, we have ∇_{X^{v}} X^{v} = 0 and hence X^{v} is a geodesic vector field. From the assumptions of completeness of the fibres, we deduce that X^{v} is complete. Denote by ϕ_{X}(t, x) the integral curve of X^{v} on (π_{1}^{2})^{−1}(x_{1}) such that ϕ_{X}(0, x) = x. Now we define

Φ_{x_{1}}(x, X) = ϕ_{X}(1, x) ,

and we have a commutative diagram

[figure: Φ; M ×_{M_{1}} V (M_{1}); M; M_{1}]

Only rest to prove that Φ_{x_{1}} defines a transitive, free action of V (M_{1})_{x_{1}} on (π_{1}^{2})^{−1}(x_{1}). To do this we proceed as in Crampin & Thompson [3, pp. 69–70] (see also [13]). □

Theorem 3.1 Suppose that:

(1) each leaf of the foliation defined by Im J^{2} is simply connected and geodesically complete with respect to the flat connection induced on it by ∇;

(2) each leaf of the foliation defined by Im J_{1} is simply connected and geodesically complete with respect to the flat connection induced on it by ∇_{1}. Then we have:

(1) π_{0}^{1} : M_{1} → M_{0} is an affine bundle modelled on T M_{0};

(2) π_{1}^{2} : M → M_{1} is an affine bundle modelled on V (M_{1}).

Moreover, if π_{0}^{2} : M → M_{0} and π_{1}^{2} : M → M_{1} admit global sections then M_{1} is isomorphic as a vector bundle to T M_{0} and M is diffeomorphic to T^{2}M_{0}. Further, if we make the identification M_{1} ≡ T M_{0}, then the fibre bundles π_{0}^{2} : M → M_{0} and π_{1}^{2} : M → M_{1} may be identified to ρ_{0}^{2} : T^{2}M_{0} → M_{0} and ρ_{1}^{2} : T^{2}M_{0} → T M_{0}, respectively.

Proof. (1) and (2) are a direct consequence of Propositions 3.2 and 3.4. If the fibration π_{0}^{2} : M → M_{0} admits a global section s : M_{0} → M then π_{0}^{1} : M_{1} → M_{0} admits the global section π_{1}^{2} ∘ s. Therefore there exists a vector bundle isomorphism φ : M_{1} → T M_{0}. The restriction (T φ)_{|_{V (M_{1})}} : V (M_{1}) → V (T M_{0}) is also a vector bundle isomorphism. Moreover, if π_{1}^{2} admits a global section, there exist a diffeomorphism M → V (M_{1}) over M_{1}. Since ρ_{1}^{2} : T^{2}M_{0} → T M_{0} is a locally trivial bundle with fibre ℝ^{n}, there exists a global section of ρ_{1}^{2} and so we also have a diffeomorphism T^{2}M_{0} → V (T M_{0}) over T M_{0}. Summing up we have the following commutative diagram, where the top arrows define a diffeomorphism M → T^{2}M_{0}.

[figure: M; V (M_{1}); V (T M_{0}); T^{2}M_{0}; M_{1}; T M_{0}; M_{0}; (T φ)_{|V (M_{1})}; φ; π_{1}^{2}; ρ_{1}^{2}; π_{0}^{1}; ρ_{0}^{1}]

□

4. Characterization of tangent bundles of higher-order

In this section we shall generalize the results of the preceedent one for an arbitrary order k.

Let (M, J) be an integrable almost tangent manifold of order k with almost tangent structure J. Then M is a (k + 1)n-dimensional manifold, J^{k+1} = 0 and J has constant rank kn. Further we have

Im J^{k} ⊂ Im J^{k−1} ⊂ ⋯ ⊂ Im J^{2} ⊂ Im J ,

where Im J^{r} has dimension (k − r + 1)n. Also we have

Im J^{r} = Ker J^{k−r+1} .

We put M_{k} = M and denote by M_{r} = M/Im J^{r+1}, 0 ≤ r ≤ k − 1, the corresponding spaces of leaves and by π_{s}^{r} : M_{r} → M_{s} the canonical projections, 0 ≤ s < r ≤ k.

We assume that all the spaces of leaves M_{k−1}, M_{k−2}, ⋯, M_{1}, M_{0} have a structure of quotient manifold of M. Then M_{r}, 0 ≤ r ≤ k − 1, has dimension (r + 1)n and we have a family of fibrations

π_{r}^{k} : M_{k} ⟶ M_{r} , 0 ≤ r ≤ k − 1 .

Proposition 4.1 J induces an integrable almost tangent structure J_{r} of order r on M_{r}, for all r, 1 ≤ r ≤ k − 1.

Proof. We define J_{r} by

J_{r}X = T π_{r}^{k}(JX̃), X ∈ T_{x_{r}}M_{r}, x_{r} ∈ M_{r} ,  (4)

where X̃ ∈ T_{x}M, x_{r} = π_{r}^{k}(x) and T π_{r}^{k}(X̃) = X.

Now, we shall prove that (4) is well defined. In fact, let X̃, Ỹ ∈ T_{x}M such that T π_{r}^{k}(X̃) = T π_{r}^{k}(Ỹ). Then X̃ − Ỹ ∈ Ker T π_{r}^{k} = Im J^{r+1}, and so J(X̃ − Ỹ) ∈ Im J^{r+2} ⊂ Im J^{r+1} = Ker T π_{r}^{k}, hence T π_{r}^{k}(JX̃) = T π_{r}^{k}(JỸ).

On the other hand, (4) does not depend on the choice of the point x. In fact, it is sufficient to check that Im (L_{X}J) ⊂ Im J^{r+1}, for all X ∈ Im J^{r+1}, which is a matter of routine.

Finally we can directly check that Im J_{r} = Ker J_{r}^{r}, then J_{r}^{r+1} = 0 and J_{r} has constant rank rn. The integrability of J_{r} follows from the integrability of J since the Nijenhuis tensor of J projects onto the Nijenhuis tensor of J_{r}. □

Now, since Ker(T π_{s}^{r}) = Im J_{r}^{s+1}, the quotient space M_{r}/ Im J_{r}^{s+1} may be identified to M/ Im J^{s+1} = M_{s} and then M_{s} possesses a differentiable structure of quotient manifold such that the canonical projection M_{r} → M_{r}/ Im J_{r}^{s+1} is identified to π_{s}^{r}, and so we have a family of fibrations π_{s}^{r} : M_{r} → M_{s} ≡ M_{r}/ Im J_{r}^{s+1}.

Also, the integrable almost tangent structure (J_{r})_{s} of order s induced by J_{r} on M_{r}/Im J_{r}^{s+1} coincides with the almost tangent structure J_{s} of order s induced by J on M/Im J^{s+1}.

Since J_{r} is integrable then there exists a symmetric linear connection ∇_{r} on M_{r} such that ∇_{r}J_{r} = 0. Also, there exists a symmetric linear connection ∇ = ∇_{k} on M = M_{k} such that ∇J = 0.

Finally, we remark that the connection induced by ∇_{r} on the fibres of the fibration π_{r−1}^{r} : M_{r} → M_{r−1} = M_{r}/ Im J_{r}^{r}, 1 ≤ r ≤ k, is flat. In fact, we can define the vertical lift X^{v} of a π_{r−2}^{r−1}–vertical tangent vector X ∈ T_{x_{r−1}}M_{r−1} to T_{x_{r}}M_{r}, π_{r−1}^{r}(x_{r}) = x_{r−1}, by X^{v} = J_{r}X̃, where X̃ ∈ T_{x_{r}}M_{r}, T π_{r−1}^{r}(X̃) = X, and we obtain the analogous to Proposition 3.3.

Theorem 4.1 Suppose that:

(1) each fibre of the fibration π_{r−1}^{r} : M_{r} → M_{r−1}, 1 ≤ r ≤ k, is simply connected and geodesically complete with respect to the flat connection induced on it by ∇_{r};

(2) π_{r}^{k} : M → M_{r}, 0 ≤ r < k, admits a global section. Then we have:

(1) M_{1} is isomorphic as a vector bundle to T M_{0};

(2) M_{r} is diffeomorphic to T^{r}M_{0} for all r, 2 ≤ r ≤ k.

Proof. First, we prove that π_{r−1}^{r} : M → M_{r−1} is an affine bundle modelled on the vector bundle V (M_{r−1}), where V (M_{r−1}) denotes the vertical vector bundle over M_{r−1} defined by the fibration π_{r−2}^{r−1} : M_{r−1} → M_{r−2}. In other words, V (M_{r−1}) = Ker T π_{r−2}^{r−1}. The proof runs in a similar way to that of Proposition 3.4.

Since there exists a global section of π_{0}^{1} : M_{1} → M_{0}, we have (1). Suppose that the theorem holds for the case r − 1. Notice that π_{r−1}^{r} : M_{r} → M_{r−1} admits also a global section and hence M_{r} is diffeomorphic to V (M_{r−1}) in such a way that this diffeomorphism is a fibred mapping over M_{r−1} ≡ T^{r−1}M_{0}.

Proceeding as in Theorem 3.1 we can define a diffeomorphism M_{r} → T^{r}M_{0} given by the following commutative diagram:

[figure: M_{r}; V (M_{r−1}); V (T^{r−1}M_{0}); T^{r}M_{0}; M_{r−1}; T^{r−1}M_{0}; T^{r−2}M_{0}; π_{r−1}^{r}; ρ_{r−1}^{r}; π_{r−2}^{r−1}; ρ_{r−2}^{r−1}]

□

5. Characterization of tangent bundles of order k: an alternative approach

In this section we shall give an alternative approach to the problem of characterizing tangent bundles of second order inspired in the work of Nagano [15] and Filippo et al. [5].

Theorem 5.1 Let M be a (k +1)n–dimensional manifold endowed with an integrable almost tangent structure J of order k and let C be a complete vector field on M satisfying

L_{C} J = −J , J^{k}C = 0 .

Suppose that for each point x ∈ M, there exists lim_{t→−∞}(exp tC)(x), where exp tC denotes the flow of C. Then the set of singular points S of C is an n–dimensional submanifold of M and M is diffeomorphic to T^{k}S. Further M is fibred over S and there exists a diffeomorphism fibred over the identity of S in such a way that it transports the canonical almost tangent structure J_{S} and the canonical vector fields C_{1}, C_{2}, …, C_{k} of T^{k}S to J and C, JC, …, J^{k−1}C, respectively.

Proof. Let (z_{0}^{i}, z_{1}^{i}, …, z_{k}^{i}) be a chart adapted to J defined on an open subset U of M, i.e.,

J (\frac{∂}{∂z_{r}^{i}} ) = \frac{∂}{∂z_{r+1}^{i}} , J (\frac{∂}{∂z_{k}^{i}} ) = 0 , 0 ≤ r ≤ k − 1, 1 ≤ i ≤ n.

Since J^{k}C = 0, C may be locally expressed

C = a_{1}^{i} \frac{∂}{∂z_{1}^{i}} + a_{2}^{i} \frac{∂}{∂z_{2}^{i}} + … + a_{k}^{i} \frac{∂}{∂z_{k}^{i}} .

Now, from L_{C}J = −J, it follows that

a_{r}^{i} = r z_{r}^{i} + ϕ_{r}^{i}(z_{0}^{j}, z_{1}^{j}, …, z_{r−1}^{j}) , 1 ≤ r ≤ k,

where ϕ_{r}^{i} are functions on U satisfying

\frac{∂ϕ_{k}^{i}}{∂z_{s}^{j}} = \frac{∂ϕ_{k−1}^{i}}{∂z_{s−1}^{j}} = … = \frac{∂ϕ_{k−s}^{i}}{∂z_{0}^{j}} , 1 ≤ s ≤ k − 1.  (5)

We put

\begin{matrix}z̄_{0}^{i} = z_{0}^{i}, \\ z̄_{r}^{i} = z_{r}^{i} + ψ_{r}^{i}(z_{0}^{j}, z_{1}^{j}, …, z_{r−1}^{j}), 1 ≤ r ≤ k, \end{matrix}

where

\begin{matrix}ψ_{1}^{i}(z_{0}^{j}) = ϕ_{1}^{i}(z_{0}^{j}), \\ ψ_{r}^{i}(z_{0}^{j}, …, z_{r−1}^{j}) = \frac{1}{r} (ϕ_{r}^{i}(z_{0}^{j}, …, z_{r−1}^{j}) + r − 1 ; ∑ ; s = 1a_{s}^{l} \frac{∂ψ_{r−s}^{i}}{∂z_{0}^{l}} ) , 2 ≤ r ≤ k. \end{matrix}

Obviously, (z̄_{0}^{i}, z̄_{1}^{i}, …, z̄_{k}^{i}) is a coordinate system on U and from equations (5) it follows that it is a chart adapted to J and

C = ∑_{r=1}^{k} r z̄_{r}^{i} \frac{∂}{∂z̄_{r}^{i}} .  (6)

Now, the flow φ_{t} = exp tC of C is locally given by

φ_{t}(z̄_{0}^{i}, z̄_{1}^{i}, …, z̄_{k}^{i}) = (z̄_{0}^{i}, e^{t}z̄_{1}^{i}, …, e^{kt}z̄_{k}^{i})

for sufficiently small t, from which

(φ_{t})_{∗} (\frac{∂}{∂z̄_{r}^{i}} ) = e^{rt} \frac{∂}{∂z̄_{r}^{i}} , φ_{t}^{∗} dz̄_{r}^{i} = e^{rt} dz̄_{r}^{i}.

Also,

(φ_{t})_{∗}J = e^{t}J(φ_{t})_{∗}  (7)

for small t and hence for every t ∈ ℝ.

For each point in the set S of the singular points of C we can consider a neighborhood U with coordinates (z̄_{0}^{i}, z̄_{1}^{i}, …, z̄_{k}^{i}) and so S is locally defined by the equations z̄_{r}^{i} = 0, 1 ≤ r ≤ k, which implies that S is an n–dimensional regular submanifold of M. We can suppose that the image of U by this chart is of the form B × B^{kn}, where B is an open ball in ℝ^{n} and B^{kn} is a zero centred ball in ℝ^{kn} with small radius. Our purpose is to extend this chart to one in which we replace B^{kn} by ℝ^{kn}.

Denote by π : M → M the mapping π(x) = lim_{t→−∞} φ_{t}(x). Clearly, π is continuous and so W = {x ∈ M : π(x) ∈ U} is an open subset of M. We shall transport the chart (z̄_{r}^{i}) along the integral curves of C. We first extend the forms dz̄_{r}^{i} on U to differential 1–forms α_{r}^{i}, 0 ≤ r ≤ k, 1 ≤ i ≤ n, on W. We put

(α_{r}^{i})_{x} = \frac{1}{e^{rt}} (φ_{t})_{x}^{∗}(dz̄_{r}^{i})_{φ_{t}(x)},  (8)

where x ∈ W, φ_{t}(x) ∈ U. Equation (8) is satisfied in a neighborhood of each point of M for some t ∈ ℝ. So, the 1–forms α_{r}^{i} are closed and hence exact by Poincaré Lemma, since W is homotopically equivalent to the open ball B of ℝ^{n}. Then there exist differentiable functions z̃_{r}^{i} on W such that α_{r}^{i} = dz̃_{r}^{i}, for 0 ≤ r ≤ k, 1 ≤ i ≤ n, and we can choose them such that z̃_{r}^{i}_{|U} = z̄_{r}^{i}_{|U}.

From (8) we obtain that α_{0}^{1} ∧ … ∧ α_{0}^{n} ∧ α_{1}^{1} ∧ … ∧ α_{1}^{n} ∧ . . . . . . ∧ α_{k}^{1} ∧ … ∧ α_{k}^{n} is never zero and so the functions z̃_{r}^{i} are independent and define a chart on W.

Moreover, for each x ∈ W, if φ_{t}(x) ∈ U, we have

(\frac{∂}{∂z̃_{r}^{i}} )_{x} = e^{rt} (φ_{−t})_{∗φ_{t}(x)} (\frac{∂}{∂z̄_{r}^{i}} )_{φ_{t}(x)} ,  (9)

and, since C = (φ_{−t})_{∗}C, we get

C_{x} = ∑_{r=1}^{k} r z̃_{r}^{i}(x) (\frac{∂}{∂z̃_{r}^{i}} )_{x}; x ∈ W.

By using (7) and that the chart (z̄_{r}^{i}) is adapted to J, it follows

J (\frac{∂}{∂z̃_{r}^{i}} ) = \frac{∂}{∂z̃_{r+1}^{i}} , J (\frac{∂}{∂z̃_{k}^{i}} ) = 0 , 0 ≤ r ≤ k − 1, 1 ≤ i ≤ n.

The integral curves of C passing through an arbitrary point z ∈ W are contained in W and they are given by t → (z̃_{0}^{i}, e^{t}z̃_{1}^{i}, …, e^{kt}z̃_{k}^{i}). Then, the mapping π is locally given by π(z̃_{0}^{i}, z̃_{1}^{i}, …, z̃_{k}^{i}) = (z̃_{0}^{i}, 0, …, 0) ∈ S, and, in particular, it is a differentiable submersion.

Corresponding to a chart (z̃_{r}^{i}) in a neighborhood of a point of S, we consider the induced chart x_{0}^{i} = z̃_{0}^{i}, i = 1, …, n, on S and the associated chart (x_{0}^{i}, x_{1}^{i}, …, x_{k}^{i}) on T^{k}S. Setting F (z̃_{0}^{i}, z̃_{1}^{i}, …, z̃_{k}^{i}) = (x_{0}^{i}, x_{1}^{i}, …, x_{k}^{i}) we obtain a well defined global diffeomorphism F : M → T^{k}S which is fibred over the identity of S and transports the almost tangent structure J and the vector fields C, JC, …, J^{k−1}C of M to the canonical tangent structure J_{S} and the canonical vector fields C_{1}, C_{2}, …, C_{k} of T^{k}S, respectively. □

6. An application: singular Lagrangians of higher order

Let L : T^{k}Q → ℝ be a Lagrangian of order k over a configuration manifold Q, dim Q = m. We denote by α_{L}, ω_{L} and E_{L} the Poincaré–Cartan 1–form, the Poincare–Cartan 2–form, and the energy function associated to L, respectively, defined by

α_{L} = ∑_{r=1}^{k}(−1)^{r} \frac{1}{r!} d_{T}^{r−1}d_{J_{r}}L, ω_{L} = −dα_{L},

E_{L} = ∑_{r=1}^{k}(−1)^{r} \frac{1}{r!} d_{T}^{r−1}(C_{r}L) − L,

where d_{T} is the total derivative with respect to the time ([12, 19]) and d_{J_{r}} is the derivation induced by J_{r}. Hence they are defined on the phase space of generalized velocities T^{2k−1}Q. The global equation of motion is

ı_{X}ω_{L} = dE_{L}.  (10)

If L is regular, that is ω_{L} is symplectic, then there exists a unique vector field ξ_{L} on T^{2k−1}Q satisfying (10). Furthermore, ξ_{L} is a 2k–th differential equation (i.e., J_{1}ξ_{L} = C_{1}), and its solutions are just the solutions of the Euler–Lagrange equations of L:

∑_{r=0}^{k}(−1)^{r} \frac{d^{r}}{dt^{r}} (\frac{∂L}{∂q_{r}^{i}} ) = 0.

Now, let L : T^{k}Q → ℝ be a singular Lagrangian of order k. Then ω_{L} is not symplectic and thus, in general, the global equation of motion ı_{X}ω_{L} = dE_{L} has not solution. The obstruction to the regularity is given by the characteristic distribution ker ω_{L}, which is called the gauge distribution. In order to avoid it, we need to quotient T^{2k−1}Q by ker ω_{L}. We assume that the following conditions are satisfied: (A1) ω_{L} is presymplectic, i.e., ω_{L} has constant even rank. (A2) J_{r}(ker ω_{L}) = Im J_{r} ∩ ker ω_{L}, 1 ≤ r ≤ 2k − 1.

Hence dim ker ω_{L} = 2ka, where a = dim J_{2k−1}(ker ω_{L}), and ker ω_{L} defines a foliation on T^{2k−1}Q. We denote by

T̂^{2k−1}Q = T^{2k−1}Q/ ker ω_{L}

the corresponding space of leaves, π_{L} : T^{2k−1}Q → T̂^{2k−1}Q being the canonical projection. (A3) π_{L} : T^{2k−1}Q → T̂^{2k−1}Q is a fibration, i.e., T̂^{2k−1}Q has a structure of quotient manifold such that π_{L} is a surjective submersion. (A4) L admits a global dynamics, i.e., there exists a vector field X on T^{2k−1}Q such that ı_{X}ω_{L} = dE_{L}.

In such a case, L admits only primary constraints [7]. From (A2) and (A4) we easily deduce that there exists a 2k–th ODE ξ in T^{2k−1}Q such that ı_{ξ}ω_{L} = dE_{L}. Also, from (A1), (A2) and (A3), we deduce that there exists a unique symplectic form ω̂_{L} on T̂^{2k−1}Q such that π_{L}^{∗}ω̂_{L} = ω_{L}. Moreover, we have that ξ is a projectable vector field with projection ξ̂ and the function E_{L} projects to a function Ê_{L} on T̂^{2k−1}Q. If, in addition, ker ω_{L} is a tangent distribution, i.e., ker ω_{L} is the complete lift of an involutive distribution D on Q, then the canonical almost tangent structure J_{1} of order 2k − 1 and the canonical Liouville vector field C_{1} on T^{2k−1}Q projects onto an almost tangent structure of the same order Ĵ_{1} and a vector field Ĉ_{1} on T̂^{2k−1}Q.

In [9] it is proved that there exists a local regular Lagrangian L̂ defined on some open subset of T̂^{2k−1}Q such that

— L̂ is of order k.

— L and L̂ ∘ π_{L} are gauge equivalent, i.e., L̂ ∘ π_{L} = (L + α̃) ∘ ρ_{k}^{2k−1} + constant, where ρ_{k}^{2k−1} : T^{2k−1}Q → T^{k}Q is the canonical projection and α is a 1–form on T^{k−1}Q.

Hence L and L̂ have the same dynamics. The point now is that T̂^{2k−1}Q is in fact a higher order tangent bundle.

Suppose that L satisfies (A1), (A3), (A4) and ker ω_{L} is a tangent distribution, in which case L satisfies (A2). Then T̂^{2k−1}Q is an almost tangent manifold of order 2k −1 with almost tangent structure Ĵ_{1} and Ĉ_{1} is a vector field on T̂^{2k−1}Q such that

LĈ_{1} Ĵ_{1} = Ĵ_{1},  Ĵ_{1}^{2k−1}Ĉ_{1} = 0.

Since C_{1} is complete, so is Ĉ_{1}, and the flow of Ĉ_{1} is just

(exp tĈ_{1})(π_{L}(x)) = π_{L}((exp tĈ_{1})x), x ∈ T^{2k−1}Q.

Hence there exists lim_{t→−∞}(exp tĈ_{1})(x) and it is unique. Thus, from Theorem 5.1, we deduce the following

Theorem 6.1 The set S of singular points of Ĉ_{1} is an (m − a)–dimensional submanifold of T̂^{2k−1}Q. Further, T̂^{2k−1}Q is fibered over the identity of S and there exists a diffeomorphism T^{2k−1}S → T̂^{2k−1}Q such that it transports the canonical almost structure of higher order (J_{1})_{S} and the canonical Liouville vector field (C_{1})_{S} to Ĵ_{1} and Ĉ_{1}, respectively.

Remark 6.1 ¿From Theorem 3.3 in [14] and Theorem 6.1 (see also [9]) we deduce that L̂ is defined on some open subset of T^{k}S, α is defined on some open subset of T^{k−1}S, and we have

(L̂ + α̃) ∘ (ρ_{k}^{2k−1})_{S} ∘ π_{L} = L ∘ ρ_{k}^{2k−1} + constant,

where (ρ_{k}^{2k−1})_{S} : T^{2k−1}S → T^{k}S is the canonical projection.

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