Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 89–101

ON SOME INTEGRATION METHODS

FOR CONNECTIONS ON FIBERED MANIFOLDS

Olga Krupková and Alexandr Vondra

Abstract. In the paper relations between (Ehresmann) connections on jet prolongations of fibered manifolds are studied. Two indirect integration methods for these connections are proposed, generalizing both the “method of characteristics” from the theory of exterior differential systems and the “method of fields of extremals” (Hamilton-Jacobi integration method) from the classical calculus of variations to (non-variational) first and secondorder partial or ordinary differential equations, explicitly solved with respect to the highest derivatives. Keywords. (Ehresmann) connection, horizontal distribution, integral sections, integration of connections. MS classification. 35A30, 53C05, 70H20.

1. Introduction

The purpose of this paper is to propose some integration methods for differential equations on fibered manifolds. We do not intend to deal with differential equations in general - this is subject of such theories as e.g. the theory of exterior differential systems ([1], [3], [5] and references therein) or the theory of diffieties as submanifolds of ∞-order jet spaces endowed with the Cartan distribution (e.g. [10] and references therein); an interesting approach to the geometry of differential equations can be found also in [25]. We restrict our attention to equations explicitly solved with respect to the highest derivatives i.e. equations for integral sections of Ehresmann connections on fibered manifolds. Equivalently, these equations are represented by horizontal distributions (i.e. by Pfaffian systems of a special kind).

There are several possibilities how to search for solutions to such systems. First of all, direct integration methods should be mentioned. These lead to a direct computation of maximal integral manifolds of the given distribution by finding “complete sets of independent first integrals”, i.e. charts adapted to the integral submanifolds. An interesting method of this kind using symmetries of the corresponding distribution, can be found in [7]. Other methods arise from the fact that any vector distribution on a smooth manifold can be (at least locally) related with the characteristic distribution of a (in general non-unique) closed 2-form. Hence, one can use symmetries of this 2-form to solve the integration problem. In particular, one can find only “a part” of symmetries of the related 2-form and apply the generalized Liouville or Jacobi theorem [16]. Another possibility to solve differential equations is to apply some of the indirect integration methods, transferring the given integration problem to that of solving certain related

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equations to the original ones. A classical example of this procedure is provided by the Hamilton-Jacobi theory of the calculus of variations, where a system of second-order ordinary differential equations, arising from a regular lagrangian, is solved with help of the Hamilton-Jacobi equation, which is a first-order partial differential equation.

In this paper we propose two indirect integration methods for connections. The results are based on relations between various kinds of connections on fibered manifolds. The first of the two methods, explained in Sec. 4, could be referred to as a “method of characteristics for connections”. We introduce to a given connection the characteristic connection, uniquely determined by the original connection and possessing similar properties as do the characteristic systems of general exterior differential systems. Sec. 5 deals with an integration theory for 2-connections, which, in a sense, could be viewed as a generalization of the Hamilton-Jacobi integration method of classical mechanics to arbitrary systems of second-order ordinary or partial differential equations, explicitly solved with respect to second derivatives. Roughly speaking, the classical idea consists in transferring the integration problem for a system of variational equations (which are nothing but equations for integral sections of a 2-connection) to the problem of finding a family of the so-called fields of extremals which are integral sections of a horizontal distribution called a complete integral (see e.g. [2], [8], [14], [15], [20]). These constructions, however, are strongly based on the existence of a lagrangian and therefore cannot be used in the “non-variational” situation. Here we show that they can be generalized to general 2-connections (without any reference to the calculus of variations). We introduce the notion of a field of paths, which is a connection representing a “non-variational field theory parallel” to the field of extremals, and we prove a generalized version of the Van Hove Theorem. We also introduce a concept of an integral of a 2-connection, which again is a connection, now generalizing the notion of a complete integral of the calculus of variations (in the sense of [2] or [14], [15]). We show that locally the integration problem for a 2-connection is equivalent to the problem of looking for an integral of this 2-connection, which, from the point of view of applications, means a reduction of the order of the equations.

Notation and main concepts used throughout the paper are briefly recalled in Sect. 2 and 3. For more details we refer to e.g. [4], [9], [11], [12], [17], [18], [19], [21].

All manifolds and mappings are supposed to be smooth and the summation convention is used as far as possible.

2. Preliminaries

Throughout the paper, π: Y → X denotes a fibered manifold, dim X = n, dim Y = n + m. When speaking of coordinates, we have a fibered chart (V, ψ) with fibered coordinates ψ = (x^{i}, y^{σ}), i = 1, …, n, σ = 1, …, m, in mind.

The first jet prolongation of π will be denoted by J^{1}π (or J^{1}Y if the manifold structure rather than the fibered one is considered). The first jet prolongation of a local section γ of π is denoted by j^{1}γ. Fibered coordinates on J^{1}Y associated with (x^{i}, y^{σ}) are denoted by (x^{i}, y^{σ}, y_{i}^{σ}). Recall that the manifold J^{1}π can be viewed as the total space of two different fibered manifolds; π_{1} : J^{1}Y → X, and π_{1,0} : J^{1}Y → Y. The latter one is an affine bundle modelled on the associated vector bundle V_{π}Y ⊗ π^{∗}(T^{∗}X) → Y, where T^{∗}X is the cotangent bundle of X, V_{π}Y is the π-vertical subbundle of the tangent bundle T Y and π^{∗}(T^{∗}X) is, by definition, the fibered product Y ×_{X} T^{∗}X.

A connection (Ehresmann connection, jet field) on a fibered manifold π: Y → X is a section (possibly defined on an open subset of Y) Γ: Y → J^{1}Y of π_{1,0}. In local coordinates, the equations of Γ are y_{i}^{σ} = Γ_{i}^{σ}(x^{j}, y^{λ}), where the functions Γ_{i}^{σ} are called the components of Γ. A connection Γ on π can be identified (among others) with the horizontal form h_{Γ} : Y → T Y ⊗ π^{∗}(T^{∗}X), which is in an obvious sense a projector on T Y, locally expressed by

h_{Γ} = (\frac{∂}{∂x^{i}} + Γ_{i}^{σ} \frac{∂}{∂y^{σ}} ) ⊗ dx^{i},

and which generates the decomposition T Y = V_{π}Y ⊕ H_{Γ}, where the n-dimensional π-horizontal distribution H_{Γ} = Im h_{Γ} on Y is given by

H_{Γ} = span \{\frac{∂}{∂x^{i}} + Γ_{i}^{σ} \frac{∂}{∂y^{σ}} , 1 ≤ i ≤ n\} = span \{dy^{σ} − Γ_{j}^{σ} dx^{j}, 1 ≤ σ ≤ m\} .

A (local) section γ of π is called a path or an integral section of the connection Γ if j^{1}γ = Γ ∘ γ . This locally means a system of partial differential equations for γ

y_{i}^{σ} ∘ j^{1}γ = Γ_{i}^{σ} ∘ γ,  i.e.  \frac{∂γ^{σ}}{∂x^{i}} = Γ_{i}^{σ}(x^{j}, γ^{λ}).

Hence, from the point of view of applications, connections on π are global counterparts of differential equations (ordinary or partial according to the dimension of X) explicitly solved with respect to the derivatives.

The curvature R_{Γ} of Γ is a vector-valued 2-form R_{Γ} = \frac{1}{2}[h_{Γ}, h_{Γ}]: Y → V_{π}Y ⊗Λ^{2}T^{∗}X, where [ , ] is the Frölicher-Nijenhuis bracket. Recall that a connection Γ is called flat if R_{Γ} ≡ 0.

A connection Γ is called integrable if the distribution H_{Γ} is completely integrable (satisfying the Frobenius theorem). Since the images of integral sections of Γ coincide with integral manifolds of H_{Γ} of maximal dimension, the integrability of Γ means that to each point y ∈ Y there exists a unique maximal integral section of Γ passing through y. The following equivalent assertions are easily proved by a direct computation:

(1) Γ is integrable.

(2) Γ is flat.

(3) The components of Γ satisfy the Frobenius integrability conditions

\frac{∂Γ_{i}^{σ}}{∂x^{j}} + \frac{∂Γ_{i}^{σ}}{∂y^{λ}} Γ_{j}^{λ} = \frac{∂Γ_{j}^{σ}}{∂x^{i}} + \frac{∂Γ_{j}^{σ}}{∂y^{λ}} Γ_{i}^{λ}.

Another integrability condition will be discussed in Sec. 5.

Remark 2.1. Notice a relationship between the horizontal distribution H_{Γ} of Γ and the canonical Cartan distribution C_{π_{1,0}} on J^{1}Y, or more precisely the Cartan distribution C_{π_{1,0}}^{Γ} of the equation Γ(Y) ⊂ J^{1}Y, defined by C_{π_{1,0}}^{Γ} = C_{π_{1,0}} ∩ T Γ(Y). Since C_{π_{1,0}} is spanned by the well-known contact forms dy^{σ} − y_{i}^{σ} dx^{i} for σ = 1, …, m, it holds

C_{π_{1,0}}^{Γ} = span \{\frac{∂}{∂x^{i}} + Γ_{i}^{σ} \frac{∂}{∂y^{σ}} + (\frac{∂Γ_{j}^{σ}}{∂x^{i}} + \frac{∂Γ_{j}^{σ}}{∂y^{λ}} Γ_{i}^{λ}) \frac{∂}{∂y_{j}^{σ}} , 1 ≤ i ≤ n\} .

It can be easily shown that if Γ is an integrable connection on π, then the following diagram commutes for each Γ-horizontal vector field ζ on Y :

J^{1}π ⟶^{J^{1}ζ} C_{π_{1,0}}^{Γ}
Γ↑  ↑^{T Γ}
Y ⟶^{ζ} H_{Γ},

where J^{1}ζ is the first jet prolongation of ζ. This can be expressed in terms of integral sections by the assertion that a section γ of π is an integral mapping of the horizontal distribution H_{Γ} if and only if j^{1}γ is an integral mapping of the distribution C_{π_{1,0}}^{Γ}.

3. Connections on J^{1}Y

Let π: Y → X be a fibered manifold, J^{1}π its first jet prolongation. By J^{2}π we denote the second (holonomic) jet prolongation of π (considered as a fibered manifold π_{2} : J^{2}π → X), and by J^{1}π_{1} the second anholonomic jet prolongation of π, which is, by definition, the first jet prolongation of the fibered manifold π_{1} : J^{1}π → X (repeated jets). Fibered coordinates on J^{2}π and J^{1}π_{1}, associated with fibered coordinates (x^{i}, y^{σ}) on Y, will be denoted by (x^{i}, y^{σ}, y_{i}^{σ}, y_{ij}^{σ}) and (x^{i}, y^{σ}, y_{i}^{σ}, Y_{i}^{σ}, Y_{ij}^{σ}), respectively. Note that y_{ij}^{σ} = y_{ji}^{σ} but, in general, Y_{ij}^{σ} ≠ Y_{ji}^{σ}. The canonical projections of these spaces to J^{1}π will be denoted by π_{2,1} : J^{2}π → J^{1}π and (π_{1})_{1,0} : J^{1}π_{1} → J^{1}π. It is worthwhile to note that there is a canonical inclusion i: J^{2}π → J^{1}π_{1} over J^{1}π, defined by i(j_{x}^{2}γ) = j_{x}^{1}(j^{1}γ), and locally expressed by Y_{i}^{σ} ∘ i = y_{i}^{σ}, Y_{ij}^{σ} ∘ i = y_{ij}^{σ}. Thus we can, for the sake of brevity, identify J^{2}π with the embedded submanifold i(J^{2}π) ⊂ J^{1}π_{1}. Note that, however, there is no natural projection J^{1}π_{1} → J^{2}π.

The second semiholonomic jet prolongation of π will be denoted by Ĵ^{2}π. Equations of this submanifold of J^{1}π_{1} are y_{i}^{σ} = Y_{i}^{σ} and the corresponding canonical projection π̂_{2,1} : Ĵ^{2}π → J^{1}π is the restriction π̂_{2,1} ≡ (π_{1})_{1,0} : Ĵ^{2}π → J^{1}π.

According to the decomposition of the corresponding associated vector bundles, there exists a canonical bundle isomorphism

(3.1)  Ĵ^{2}π ≅ J^{2}π ×_{J^{1}π} [π_{1,0}^{∗}(V_{π}Y ⊗ π^{∗}(Λ^{2}T^{∗}X))] .

This splitting gives rise to affine projections r: Ĵ^{2}π → π_{1,0}^{∗}(V_{π}Y ⊗ π^{∗}(Λ^{2}T^{∗}X)) and s: Ĵ^{2}π → J^{2}π expressing the antisymmetric or symmetric part of every fibered coordinate Y_{ij}^{σ}, respectively.

In keeping with Sec. 2, a connection on π_{1} is a section Σ: J^{1}π → J^{1}π_{1} of (π_{1})_{1,0}. In fibered coordinates, Y_{i}^{σ} ∘ Σ = Σ_{i}^{σ}, Y_{ij}^{σ} ∘ Σ = Σ_{ij}^{σ}, where the components Σ_{ij}^{σ} are not in general symmetric in i, j. Equivalently, Σ is represented by the π_{1}-horizontal n-dimensional distribution H_{Σ} on J^{1}Y

H_{Σ} = span \{\frac{∂}{∂x^{i}} + Σ_{i}^{σ} \frac{∂}{∂y^{σ}} + Σ_{ij}^{σ} \frac{∂}{∂y_{j}^{σ}} , 1 ≤ i ≤ n\}
= span {dy^{σ} − Σ_{j}^{σ} dx^{j}, dy_{i}^{σ} − Σ_{ij}^{σ} dx^{j}, 1 ≤ i ≤ n, 1 ≤ σ ≤ m} .

Using the fact that J^{2}π ≡ i(J^{2}π) ⊂ Ĵ^{2}π ⊂ J^{1}π_{1}, we get the following kinds of connections.

A semiholonomic connection on π_{1} is a section Γ̂^{(2)} : J^{1}π → Ĵ^{2}π of π̂_{2,1}. Thus Γ̂^{(2)} is locally given by Y_{i}^{σ} ∘ Γ̂^{(2)} = y_{i}^{σ} ∘ Γ̂^{(2)} = y_{i}^{σ}, Y_{ij}^{σ} ∘ Γ̂^{(2)} = Γ̂_{ij}^{σ}, where the components Γ̂_{ij}^{σ} again need not be symmetric in i, j. Equivalently, Γ̂^{(2)} is represented by the π_{1}-horizontal n-dimensional distribution on J^{1}Y

HΓ̂^{(2)} = span \{\frac{∂}{∂x^{i}} + y_{i}^{σ} \frac{∂}{∂y^{σ}} + Γ̂_{ij}^{σ} \frac{∂}{∂y_{j}^{σ}} , 1 ≤ i ≤ n\}
= span {dy^{σ} − y_{j}^{σ} dx^{j}, dy_{i}^{σ} − Γ̂_{ij}^{σ} dx^{j}, 1 ≤ i ≤ n, 1 ≤ σ ≤ m} .

Similarly, a 2-connection on π, (a semispray connection on π, or a holonomic connection on π_{1}), is a section Γ^{(2)} : J^{1}π → J^{2}π of π_{2,1}. Locally, the equations of Γ^{(2)} are Y_{i}^{σ} ∘ Γ^{(2)} = y_{i}^{σ} ∘ Γ^{(2)} = y_{i}^{σ}, Y_{ij}^{σ} ∘ Γ^{(2)} = y_{ij}^{σ} ∘ Γ^{(2)} = Γ_{ij}^{σ}, where Γ_{ij}^{σ} = Γ_{ji}^{σ}. The corresponding n-dimensional π_{1}-horizontal distribution H_{Γ^{(2)}} = Im h_{Γ^{(2)}} is given by

H_{Γ^{(2)}} = span \{\frac{∂}{∂x^{i}} + y_{i}^{σ} \frac{∂}{∂y^{σ}} + Γ_{ij}^{σ} \frac{∂}{∂y_{j}^{σ}} , 1 ≤ i ≤ n\}
= span {dy^{σ} − y_{j}^{σ} dx^{j}, dy_{i}^{σ} − Γ_{ij}^{σ} dx^{j}, 1 ≤ i ≤ n, 1 ≤ σ ≤ m} .

Note that both HΓ̂^{(2)} and H_{Γ^{(2)}} are subdistributions of the Cartan distribution C_{π_{1,0}}, which means that the integral sections of these connections are holonomic (i.e. they are prolongations j^{1}γ of sections γ of π). This enables one to define a path or an integral section of a connection Γ^{(2)} on π to be a section γ satisfying j^{2}γ = Γ^{(2)} ∘ j^{1}γ on its domain. Expressing this relation in fiber coordinates we see that 2-connections on π are geometric counterparts of systems of second-order differential equations solved with respect to the highest derivatives of the form

\frac{∂^{2}γ^{σ}}{∂x^{i}∂x^{j}} = Γ_{ij}^{σ} (x^{k}, γ^{λ}, \frac{∂γ^{λ}}{∂x^{k}} ) .

Having in mind the splitting (3.1), the torsion [19] of a semiholonomic connection Γ̂^{(2)} is defined as the mapping TΓ̂^{(2)} = r ∘ Γ̂^{(2)} : J^{1}π → π_{1,0}^{∗} (V_{π}Y ⊗ π^{∗}(Λ^{2}T^{∗}X)). Hence we get that there is a canonical splitting of each semiholonomic connection on π_{1} to its torsion TΓ̂^{(2)} = r ∘Γ̂^{(2)} and the 2-connection Γ^{(2)} = s∘Γ̂^{(2)}. Obviously, Γ̂^{(2)} is holonomic if and only if TΓ̂^{(2)} = 0.

Another fibered manifold under consideration will be the first jet prolongation J^{1}π_{1,0} of π_{1,0} : J^{1}π → Y, i.e. the fibered manifold (π_{1,0})_{1} : J^{1}π_{1,0} → Y. By definition, it is a set of 1-jets of local connections on π. The induced coordinates on J^{1}π_{1,0} are (x^{i}, y^{σ}, y_{i}^{σ}, z_{ij}^{σ}, z_{iλ}^{σ}), where

z_{ij}^{σ}(j_{y}^{1}Γ) = \frac{∂Γ_{i}^{σ}}{∂x^{j}} (y), z_{iλ}^{σ}(j_{y}^{1}Γ) = \frac{∂Γ_{i}^{σ}}{∂y^{λ}} (y).

A connection on π_{1,0} is a (local) section Ξ: J^{1}π → J^{1}π_{1,0} of (π_{1,0})_{1,0}. The distribution H_{Ξ} = Im h_{Ξ} of Ξ is then an (n + m)-dimensional π_{1,0}-horizontal distribution on J^{1}Y, whose integral sections of maximal dimension are local connections on π. It holds

H_{Ξ} = span \{\frac{∂}{∂x^{j}} + Ξ_{ij}^{σ} \frac{∂}{∂y_{i}^{σ}} , \frac{∂}{∂y^{λ}} + Ξ_{iλ}^{σ} \frac{∂}{∂y_{i}^{σ}} , 1 ≤ j ≤ n, 1 ≤ λ ≤ m\}
= span {dy_{i}^{σ} − Ξ_{ij}^{σ} dx^{j} − Ξ_{iλ}^{σ} dy^{λ}, 1 ≤ i ≤ n, 1 ≤ σ ≤ m} ,

where Ξ_{ij}^{σ}, Ξ_{iλ}^{σ} are the components of Ξ. A connection Γ: Y ⊃ V → J^{1}Y is then an integral section of Ξ if and only if j^{1}Γ = Ξ∘Γ, which in fibred coordinates gives a system of (n + m) first-order partial differential equations of the form

\frac{∂Γ_{i}^{σ}}{∂x^{j}} = Ξ_{ij}^{σ}(x^{k}, y^{ν}, Γ_{k}^{ν}), \frac{∂Γ_{i}^{σ}}{∂y^{λ}} = Ξ_{iλ}^{σ}(x^{k}, y^{ν}, Γ_{k}^{ν})

for the components of Γ.

4. Characterizable connections

In this section we will study relations between connections on π_{1,0} and 2-connections on π which lead to the theory of characteristics for connections.

We start with a theorem, proved in [6] using the theory of natural bundles and operators [9], [12].

Theorem 4.1. All natural transformations of J^{1}π_{1,0} into J^{1}π_{1} over the identity of J^{1}π form a 1-parameter family {f_{a}}_{a∈ℝ}, where the mapping f_{a} is in fibered coordinates expressed by

(4.1)  \begin{aligned}Y_{i}^{σ} ∘ f_{a} & = y_{i}^{σ} \\ Y_{ij}^{σ} ∘ f_{a} & = z_{ij}^{σ} + z_{iλ}^{σ}y_{j}^{λ} + a(z_{ij}^{σ} − z_{ji}^{σ} + z_{iλ}^{σ}y_{j}^{λ} − z_{jλ}^{σ}y_{i}^{λ}). \end{aligned}

Note that f_{0} (a = 0 in (4.1)) is nothing but the canonical maping J^{1}π_{1,0} → J^{1}π_{1}, defined for each local connection Γ: Y ⊃ V → J^{1}π on π and each y ∈ V by j_{y}^{1}Γ ↦ j_{x}^{1}(Γ ∘ γ), where γ is a local section of π such that Γ(y) = j_{x}^{1}γ. If γ is an integral section of Γ then obviously j_{x}^{1}(Γ ∘ γ) = j_{x}^{2}γ ∈ J^{2}π.

Consider the mappings s: Ĵ^{2}π → J^{2}π and r: Ĵ^{2}π → π_{1,0}^{∗}(V_{π}Y ⊗π^{∗}(Λ^{2}T^{∗}X)) defined in the previous section and put

S: J^{1}π_{1,0} → J^{2}π, S := s ∘ f_{0}
R: J^{1}π_{1,0} → π_{1,0}^{∗}(V_{π}Y ⊗ π^{∗}(Λ^{2}T^{∗}X)), R := r ∘ f_{0}.

The fibered coordinates expressions of S and R are

S_{ij}^{σ} = \frac{1}{2} (z_{ij}^{σ} + z_{ji}^{σ} + z_{iλ}^{σ}y_{j}^{λ} + z_{jλ}^{σ}y_{i}^{λ}) , R_{ij}^{σ} = \frac{1}{2} (z_{ij}^{σ} − z_{ji}^{σ} + z_{iλ}^{σ}y_{j}^{λ} − z_{jλ}^{σ}y_{i}^{λ}) .

Then we can write (4.1) in the form {f_{b}}_{b∈ℝ} = {Y_{i}^{σ} = y_{i}^{σ}, Y_{ij}^{σ} = S_{ij}^{σ} + b R_{ij}^{σ}}_{b∈ℝ}. Note, that for each a ∈ ℝ we get s ∘ f_{a} = S and R_{a} := r ∘ f_{a} = (1 + 2a)R.

As a direct consequence of Theorem 4.1 we get

Theorem 4.2. (1) All natural transformations transforming connections on π_{1,0} into (in fact, semiholonomic) connections on π_{1} are of the form f_{a} ∘ Ξ.

(2) There is a unique natural transformation transforming connections on π_{1,0} into 2-connections on π; it is of the form S ∘ Ξ.

Theorem 4.2 leads to the following important conclusions:

Corollary 4.1. Let Ξ be a connection on π_{1,0}. There is a unique 2-connection Γ^{(2)} on π naturaly assigned to Ξ. This connection is defined by Γ^{(2)} = S ∘ Ξ. The components of Γ^{(2)} are in fibered coordinates expressed by

Γ_{ij}^{σ} = \frac{1}{2} (Ξ_{ij}^{σ} + Ξ_{ji}^{σ} + Ξ_{iλ}^{σ} y_{j}^{λ} + Ξ_{jλ}^{σ} y_{i}^{λ}) ,

where Ξ_{ij}^{σ}, Ξ_{iλ}^{σ} are the components of Ξ.

Corollary 4.2. Let Ξ be a connection on π_{1,0}. If R ∘ Ξ = 0, then R_{a} ∘ Ξ = 0 for all a ∈ ℝ. Locally, R ∘ Ξ = 0 means

(4.2)  \frac{1}{2} (Ξ_{ij}^{σ} − Ξ_{ji}^{σ} + Ξ_{iλ}^{σ} y_{j}^{λ} − Ξ_{jλ}^{σ} y_{i}^{λ}) = 0.

Clearly, R ∘ Ξ = T_{f_{0}∘Ξ}. As a direct consequence of the properties of the distributions H_{Γ^{(2)}} and H_{Ξ} we get:

Proposition 4.1. Let Ξ be a connection on π_{1,0}, let Γ^{(2)} = S ∘ Ξ. Then H_{Γ^{(2)}} ⊂ H_{Ξ} ⟺ R ∘ Ξ = 0. Consequently, H_{Γ^{(2)}} ⊂ H_{Ξ} ⟺ Γ^{(2)} = f_{0} ∘ Ξ.

The above results enable us to introduce the following concepts:

Definition 4.1. A connection Ξ on π_{1,0} will be called characterizable if R ∘ Ξ = 0. The 2-connection Γ^{(2)} = S ∘ Ξ will be then called the characteristic connection of Ξ. The distribution H_{Γ^{(2)}} of the characteristic connection Γ^{(2)} of Ξ will be called the characteristic distribution of Ξ and the integral manifolds of H_{Γ^{(2)}} of maximal dimension will be called the characteristics of the connection Ξ.

Clearly, the maximal integral manifolds of H_{Ξ} (integral sections of Ξ) are foliated by the maximal integral manifolds of the characteristic distribution (characteristics, 1-jet prolongations of integral sections of the characteristic connection). Accordingly, integral sections of Ξ are “glued together” from the characteristics.

Since Ξ is characterizable if and only if its components satisfy (4.2), the components of the characteristic connection are Γ_{ij}^{σ} = Ξ_{ij}^{σ} +Ξ_{iλ}^{σ} y_{j}^{λ}. The equations for characteristics of a characterizable connection Ξ on π_{1,0} are

\frac{∂^{2}γ^{σ}}{∂x^{i}∂x^{j}} = Ξ_{ij}^{σ}(x^{k}, γ^{ν}, \frac{∂γ^{ν}}{∂x^{k}}) + Ξ_{iλ}^{σ}(x^{k}, γ^{ν}, \frac{∂γ^{ν}}{∂x^{k}}) \frac{∂γ^{λ}}{∂x^{j}} .

It is clear that for the characteristic distribution H_{Γ^{(2)}} of Ξ it holds H_{Γ^{(2)}} ⊂ H_{Ξ}∩C_{π_{1,0}}. Notifying that

H_{Ξ} ∩ C_{π_{1,0}} = span {dy^{σ} − y_{j}^{σ} dx^{j}, dy_{i}^{σ} − Ξ_{ij}^{σ} dx^{j} − Ξ_{iλ}^{σ} dy^{λ}, 1 ≤ i ≤ n, 1 ≤ σ ≤ m}
= span \{\frac{∂}{∂x^{i}} + y_{i}^{σ} \frac{∂}{∂y^{σ}} + (Ξ_{ji}^{σ} + Ξ_{jλ}^{σ} y_{i}^{λ}) \frac{∂}{∂y_{j}^{σ}} , 1 ≤ i ≤ n\} ,

we immediately get the following proposition.

Proposition 4.2. Let Ξ be a characterizable connection on π_{1,0}. The following conditions are equivalent:

(1) H_{Γ^{(2)}} is the characteristic distribution of Ξ.

(2) H_{Γ^{(2)}} = H_{Ξ} ∩ C_{π_{1,0}}.

We will show in Sec. 5 that if Ξ is a characterizable and integrable connection on π_{1,0}, then its characteristic connection is integrable. The following proposition is not much surprising in this respect.

Proposition 4.3. Let Ξ be an integrable connection on π_{1,0}. Then Ξ is characterizable if and only if any integral section Γ of Ξ is integrable.

Proof. Let z be an arbitrary point from the domain of Ξ, let Γ: Y ⊃ V → J^{1}π be the uniquely determined integral section of Ξ passing through z, i.e. Γ(y) = z. Then evidently R ∘ Ξ(z) = R ∘ j^{1}Γ(y) = R_{Γ}(y). □

Remark 4.1. It is easy to see that knowing an integral manifold M of H_{Ξ} of dimension k one can use the characteristics to construct an integral manifold of dimension ≥ k, containing M (“the Cauchy initial problem”).

Remark 4.2. There are some additional interesting results concerning the geometry related to the characterizable connections on π_{1,0}. Roughly speaking, each such a connection splits into the direct sum of its characteristic connection and the so–called reduced connection, represented by a strong horizontal subbundle in T J^{1}Y. The latter one is closely related to the theory of symmetries. We would like to discuss this problem in some of the forthcoming papers.

Example 4.1. Suppose dim X = 1, denote by (t, q^{σ}), 1 ≤ σ ≤ m, (local) fiber coordinates on π. Let Ξ be a connection on π_{1,0} with the components Ξ^{σ}, Ξ_{λ}^{σ}. The distribution H_{Ξ} is locally spanned by (m + 1) vector fields

\frac{∂}{∂t} + Ξ^{σ} \frac{∂}{∂q_{1}^{σ}} , \frac{∂}{∂q^{λ}} + Ξ_{λ}^{σ} \frac{∂}{∂q_{1}^{σ}} , 1 ≤ λ ≤ m,

where (t, q^{σ}, q_{1}^{σ}) are the associated coordinates on J^{1}Y. The equations for integral sections of Ξ (which are local connections on π) are

\frac{∂Γ^{σ}}{∂t} = Ξ^{σ}(t, q^{ν}, Γ^{ν}), \frac{∂Γ^{σ}}{∂q^{λ}} = Ξ_{λ}^{σ}(t, q^{ν}, Γ^{ν})

for the components Γ^{σ} of Γ. Evidently, if dim X = 1 then every connection Ξ on π_{1,0} is characterizable. The components of its characteristic connection Γ^{(2)} are then Γ_{(2)}^{σ} = Ξ^{σ} + Ξ_{λ}^{σ} q_{1}^{λ}, and thus the one-dimensional (consequently integrable) characteristic distribution H_{Γ^{(2)}} is spanned by semisprays

\frac{∂}{∂t} + q_{1}^{σ} \frac{∂}{∂q^{σ}} + (Ξ^{σ} + Ξ_{λ}^{σ}q_{1}^{λ}) \frac{∂}{∂q_{1}^{σ}} .

The equations for characteristics are

\frac{d^{2}γ^{σ}}{dt^{2}} = Ξ^{σ}(t, γ^{ν}, \frac{dγ^{ν}}{dt}) + Ξ_{λ}^{σ}(t, γ^{ν}, \frac{dγ^{ν}}{dt}) \frac{dγ^{λ}}{dt} .

For a more detailed description of this “one-dimensional” case see [22], [24].

Example 4.2. Let π: R × M → R, where M is a smooth manifold of dimension m. Consider a linear connection ∇ on M, denote by Γ_{λρ}^{σ} the components of ∇ in a chart (t, q^{σ}), where t is the canonical global coordinate on R and (q^{σ}) are (local) coordinates on M. Put Ξ^{σ} = 0, Ξ_{λ}^{σ} = Γ_{λρ}^{σ} q_{1}^{ρ}. Then Ξ ≡ (Ξ^{σ}, Ξ_{λ}^{σ}) is a homogeneous dynamical connection on π_{1,0} : R × T M → R × M in the sense of [4], [23]. The equations for integral sections of Ξ are

\frac{∂v^{σ}}{∂t} = 0, \frac{∂v^{σ}}{∂q^{λ}} = Γ_{λρ}^{σ}v^{ρ}

and consequently integral sections of Ξ are local vector fields on M such that ∂v^{σ}/∂q^{λ} − Γ_{λρ}^{σ}v^{ρ} = ∇v = 0 (the covariant derivative of v with respect to ∇ vanishes). The characteristic connection of Ξ can be thus identified with the spray ζ_{Ξ} on R×T M, canonically associated to Ξ, which canonically projects onto the geodesic spray ζ_{∇} of ∇. Locally ζ_{Ξ} = \frac{∂}{∂t} + ζ_{∇}, where

ζ_{∇} = q_{1}^{σ} \frac{∂}{∂q^{σ}} + Γ_{λρ}^{σ} q_{1}^{λ} q_{1}^{ρ} \frac{∂}{∂q_{1}^{σ}} .

Accordingly, we get that the equations of characteristics of Ξ are the equations for the paths of ζ_{Ξ} and thus the equations for the graphs of geodesics of ∇, namely (q_{2}^{σ} − Γ_{λρ}^{σ} q_{1}^{λ} q_{1}^{ρ}) ∘ j^{2}γ = 0.

5. Integration of 2-connections

Let Γ be a connection on π, considered now as a fibered morphism over X. Then its jet prolongation is a map J^{1}(Γ; id_{X}): J^{1}π → J^{1}π_{1} which has the coordinate form

(x^{i}, y^{σ}, y_{i}^{σ}) ↦ (x^{i}, y^{σ}, Γ_{i}^{σ}, y_{i}^{σ}, \frac{∂Γ_{i}^{σ}}{∂x^{j}} + \frac{∂Γ_{i}^{σ}}{∂y^{λ}} y_{j}^{λ}) .

It follows that J^{1}(Γ; id_{X}) is a section of the projection J^{1}(π_{1,0}; id_{X}) and the composition J^{1}(Γ; id_{X}) ∘ Γ has the values in Ĵ^{2}π. Note that this construction gives another characterization of the integrability of Γ: namely, a connection Γ on π is integrable if and only if the composition J^{1}(Γ; id_{X}) ∘ Γ has the values in J^{2}π. Evidently for the curvature of Γ it holds R_{Γ} = −r ∘ J^{1}(Γ; id_{X}) ∘ Γ.

The above considerations allow us to introduce the following concept.

Definition 5.1. Let Γ^{(2)} be a 2-connection on π. A connection Γ: Y ⊃ V → J^{1}π on π will be called a field of paths of Γ^{(2)} if on V

(5.1)  J^{1}(Γ, id_{X}) ∘ Γ = Γ^{(2)} ∘ Γ.

It is clear that an arbitrary field of paths of Γ^{(2)} is integrable. The following two propositions show the geometric meaning of this notion (cf. Remark 2.1).

Proposition 5.1. A connection Γ: Y ⊃ V → J^{1}π is a field of paths of Γ^{(2)} if and only if Γ(V) is foliated by first jets of integral sections of Γ^{(2)}, i.e. if

(5.2)  H_{Γ^{(2)}}|_{Γ(V)} ≡ C_{π_{1,0}}^{Γ}.

In fiber coordinates, (5.1) and (5.2) have the form

Γ_{ij}^{σ} ∘ Γ = \frac{∂Γ_{i}^{σ}}{∂x^{j}} + \frac{Γ_{i}^{σ}}{∂y^{λ}} Γ_{j}^{λ}.

As an expected as the most important property of fields of paths is the next one.

Proposition 5.2. Let Γ be a field of paths of Γ^{(2)} and γ an integral section of Γ. Then γ is an integral section (a path) of Γ^{(2)}.

Proof. Γ^{(2)} ∘ j^{1}γ = Γ^{(2)} ∘ Γ ∘ γ = J^{1}(Γ, id_{X}) ∘ Γ ∘ γ = j^{1}(Γ ∘ γ) = j^{1}(j^{1}γ) = j^{2}γ. □

Corollary 5.1. Let Γ: Y ⊃ V → J^{1}π be a field of paths of Γ^{(2)}. Then H_{Γ} defines a foliation of V each leaf of which is the image of an integral section of Γ^{(2)}.

Proposition 5.3. Let Ξ be a characterizable connection on π_{1,0} and Γ^{(2)} its characteristic 2-connection on π. Then each integral section Γ of Ξ is a field of paths of Γ^{(2)}.

Proof. Let Γ be an integral section of Ξ . Then Γ^{(2)} ∘ Γ = S ∘ Ξ ∘ Γ = S ∘ j^{1}Γ. Since S ∘ j^{1}Γ = J^{1}(Γ, id_{X}) ∘ Γ for any integrable connection Γ on π, we have (5.2). □

The following Corollary generalizes to our “non-variational field theory situation” the Van Hove Theorem of the classical calculus of variations (saying that every extremal can be locally embedded in a field of extremals).

Corollary 5.2. (Embedding Theorem). Let Ξ be an integrable characterizable connection on π_{1,0} and Γ^{(2)} its characteristic 2-connection on π. Then Γ^{(2)} is integrable, and each integral section of Γ^{(2)} is locally embedded in a field of paths Γ, which is an integral section of Ξ.

By the above results, the problem of finding all the integral sections of a given integrable 2-connection Γ^{(2)} (i.e. the problem of finding the foliation defined by H_{Γ^{(2)}}) is equivalent to the problem of solving all fields of paths of Γ^{(2)}. In what follows, we will show that, under some assumptions, it is sufficient to solve a certain “minimal” family of fields of paths of Γ^{(2)} to obtain a complete local solution of Γ^{(2)}. In this respect, characteristic 2-connections have the same properties as possess variational equations of classical mechanics (cf. [14], and [15] for higher-order equations).

Definition 5.2. Let Γ^{(2)} be an integrable 2-connection on π. A (local) connection Ξ on π_{1,0} will be called an integral of Γ^{(2)} if Ξ is integrable and Γ^{(2)} is its characteristic connection.

We shall show that the existence of integrals of an integrable 2-connection is a direct consequence of the integrability property, and that one can construct an integral of Γ^{(2)} by means of a set of independent first integrals of H_{Γ^{(2)}}. To this purpose recall that if Δ is a distribution on a manifold M then a function f on an open subset V of M is said to be a first integral of Δ on V if the form df belongs to the anihilator of Δ. A family f^{1}, …, f^{k} of functions on V is called a complete set of independent first integrals of Δ if the forms df^{1}, …, df^{k} are linearly independent at each point of V and it holds Δ = span{df^{1}, …, df^{k}} on V.

Theorem 5.1. Let Γ^{(2)} be an integrable 2-connection on π, let {a^{1}, …, a^{nm}} be a set of independent first integrals of Γ^{(2)}, defined on an open subset W ⊂ J^{1}π. Suppose that W ⊂ π_{1,0}^{−1}(V), where (V, ψ) is a fiber chart on Y. If

(5.3)  det (\frac{∂a^{K}}{∂y_{j}^{σ}} ) ≠ 0

on W, then by

(5.4)  H_{Ξ} = span{da^{1}, …, da^{nm}}

an integral Ξ: J^{1}π ⊃ W → J^{1}π_{1,0} of Γ^{(2)} is defined.

Proof. Consider the distribution (5.4) where a^{1}, …, a^{nm} are independent first integrals of Γ^{(2)}. This distribution is obviously integrable. Denote by (Λ_{jK}^{σ}), where σ, j label rows and K label columns, the matrix inverse to (∂a^{K}/∂y_{j}^{σ}). Computing the vector fields spanning H_{Ξ} on W we get

H_{Ξ} = span \{\frac{∂}{∂x^{i}} + Ξ_{ji}^{σ} \frac{∂}{∂y_{j}^{σ}} , \frac{∂}{∂y^{λ}} + Ξ_{jλ}^{σ} \frac{∂}{∂y_{j}^{σ}} , 1 ≤ i ≤ n, 1 ≤ λ ≤ m\} ,

where

Ξ_{ji}^{σ} = −Λ_{jK}^{σ} \frac{∂a^{K}}{∂x^{i}} , Ξ_{jλ}^{σ} = −Λ_{jK}^{σ} \frac{∂a^{K}}{∂y^{λ}} .

Further denote by Γ_{ij}^{σ} the components of Γ^{(2)} in the chart (V, ψ). It holds

Ξ_{ij}^{σ} − Ξ_{ji}^{σ} + Ξ_{iλ}^{σ} y_{j}^{λ} − Ξ_{jλ}^{σ} y_{i}^{λ} = −Λ_{iK}^{σ} \frac{∂a^{K}}{∂x^{j}} + Λ_{jK}^{σ} \frac{∂a^{K}}{∂x^{i}} − Λ_{iK}^{σ} \frac{∂a^{K}}{∂y^{λ}} y_{j}^{λ} + Λ_{jK}^{σ} \frac{∂a^{K}}{∂y^{λ}} y_{i}^{λ} =

= −Λ_{iK}^{σ} (\frac{∂a^{K}}{∂x^{j}} + \frac{∂a^{K}}{∂y^{λ}} y_{j}^{λ}) + Λ_{jK}^{σ} (\frac{∂a^{K}}{∂x^{i}} + \frac{∂a^{K}}{∂y^{λ}} y_{i}^{λ}) =

= Λ_{iK}^{σ} \frac{∂a^{K}}{∂y_{ℓ}^{λ}} Γ_{ℓj}^{λ} − Λ_{jK}^{σ} \frac{∂a^{K}}{∂y_{ℓ}^{λ}} Γ_{ℓi}^{λ} = Γ_{ij}^{σ} − Γ_{ji}^{σ} = 0.

Since

Ξ_{ij}^{σ} + Ξ_{iλ}^{σ} y_{j}^{λ} = −Λ_{iK}^{σ} \frac{∂a^{K}}{∂x^{j}} − Λ_{iK}^{σ} \frac{∂a^{K}}{∂y^{λ}} y_{j}^{λ} = Λ_{iK}^{σ} \frac{∂a^{K}}{∂y_{ℓ}^{λ}} Γ_{ℓj}^{λ} = Γ_{ij}^{σ},

the proof is completed. □

Notice that if Γ^{(2)} is integrable then the existence of a set of independent first integrals satisfying the condition (5.3) is ensured by definition (since H_{Γ^{(2)}} is horizontal).

Theorem 5.1 suggests an integration method (parallel to the “method of fields of extremals” from the classical calculus of variations), transferring the problem of solving second-order equations related to Γ^{(2)} to the problem of solving a family of first-order equations for integral sections of Ξ:

Let Ξ be an integral of Γ^{(2)}, defined on an open subset of J^{1}π and let H_{Ξ} = span{da^{1}, …, da^{nm}}. Then a^{1}, …, a^{nm} are independent first integrals of Γ^{(2)}. Let x ∈ dom Ξ be a point, (V, ψ), ψ = (x^{i}, y^{σ}) a fiber chart on Y such that x ∈ π_{1,0}^{−1}(V). Since Ξ is a connection on π_{1,0}, the condition (5.3) is satisfied. Hence, there is a chart (W, χ) at x such that W ⊂ π_{1,0}^{−1}(V), χ = (x^{i}, y^{σ}, a^{K}), 1 ≤ K ≤ nm. Consider the nontrivial components y_{i}^{σ}(x^{j}, y^{ν}, a^{K}) of the overlap mapping and put for each fixed a = (a^{K}), Γ_{(a)i}^{σ} = y_{i}^{σ}(x^{j}, y^{ν}, a^{K}). Then for each a, Γ_{a} : π_{1,0}(W) → W, Γ_{a}(x^{i}, y^{σ}) = (x^{i}, y^{σ}, Γ_{(a)i}^{σ}) is a connection on π. We will say that {Γ_{a}} is an nm-parametric system of connections. Clearly, for each a, Γ_{a} is an integral section of Ξ. Consequently, a section γ: U → Y such that j^{1}γ(U) ⊂ W is an integral section of Γ^{(2)} if and only if γ is an integral section of Γ_{a} for some a. This means that the problem of solving Γ^{(2)} on W is equivalent to the problem of solving an nm-parametric system of first-order equations of the form

(5.5)  \frac{∂γ^{σ}}{∂x^{i}} = Γ_{(a)i}^{σ}(x^{i}, y^{ν})

where Γ_{(a)i}^{σ} are the components of the connection Γ_{a} defined by

H_{Γ_{a}} = span \{\frac{∂}{∂x^{i}} + y_{i}^{σ}(x^{j}, y^{ν}, a^{K})\frac{∂}{∂y^{σ}} \} , a = const.

Example 5.1. Classical mechanics. Since the above integration method works for any connection (in mechanics spanned by a semispray = “a second-order differential equation”), it is applicable to general systems of regular equations of classical mechanics (without any reference to the existence of lagrangians). We shall illustrate briefly that in fact it is a generalization of some techniques and notions known from the first-order Lagrangean dynamics (cf. e.g. [4], [8], [11], [13] - [15]).

Let π: Y → X be a fibered manifold over one-dimensional base X. Consider a regular lagrangian λ on J^{1}Y and the corresponding Euler-Lagrange form E_{λ} on J^{2}Y; in a fiber chart (V, ψ), ψ = (x^{i}, y^{σ}), we get λ = Ldt, E_{λ} = (A_{σ} + B_{σν}q_{2}^{ν})dq^{σ} ∧ dt, where the functions A_{σ}, B_{σν} depend on (t, q^{λ}, q_{1}^{λ}) only, (B_{σν}) is a regular matrix at each point of π_{2,0}^{−1}(V), and A_{σ}, B_{σν} are expressed by

B_{σν} = −\frac{∂^{2}L}{∂q_{1}^{σ}∂q_{1}^{ν}} , A_{σ} = \frac{∂L}{∂q^{σ}} − \frac{∂^{2}L}{∂t∂q_{1}^{σ}} − \frac{∂^{2}L}{∂q^{ν}∂q_{1}^{σ}} q_{1}^{ν}.

The Euler-Lagrange equations are equations for sections γ of π of the form E_{λ} ∘j^{2}γ = 0, or equivalently j^{1}γ^{∗}i_{J^{1}ζ}dΘ_{λ} = 0 for each π-projectable vector field on Y, where Θ_{λ} is the Cartan form of λ. Equivalently, they are equations for integral sections of the 2connection Γ^{(2)} on π defined by Γ^{(2)∗}E_{λ} = 0. The distribution H_{Γ^{(2)}} of Γ^{(2)} is spanned by semisprays

\frac{∂}{∂t} + q_{1}^{σ} \frac{∂}{∂q^{σ}} − B^{σν}A_{ν} \frac{∂}{∂q_{1}^{σ}} ,

where (B^{σν}) is the inverse matrix to (B_{σν}), or equivalently, by the Pfaffian system i_{ζ}dΘ_{λ}, where ζ runs over all vector fields on J^{1}Y. Hence, H_{Γ^{(2)}} is the characteristic distribution of the closed 2-form dΘ_{λ}. A section φ: Y → J^{1}Y is called a field of extremals of λ if it is an integral section of dΘ_{λ}, i.e. if φ^{∗}dΘ_{λ} = 0; the latter equation is called the eikonal equation (Hamilton-Jacobi equation). It is easy to see that a field of extremals satisfies the condition (5.1), i.e. it is a field of paths of Γ^{(2)}. (In fact, the condition it satisfies is stronger, expressing that it is related to a given lagrangian). If a function S, depending on time t, position (q^{1}, …, q^{m}) and m parameters (a^{1}, …, a^{m}) is a solution of the Hamilton-Jacobi equation, then the distribution spanned by da^{1}, …, da^{m} is an integral of H_{Γ^{(2)}}, and the functions b^{i} = \frac{∂S}{∂a^{i}}, i = 1, …, m form a complete set of independent first integrals of the corresponding m-parametric system of first order equations for extremals (5.5).

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Olga Krupková Department of Mathematics, Silesian University at Opava Bezručovo nám. 13 746 01 Opava Czechoslovakia

Alexandr Vondra Department of Mathematics, Military Academy in Brno PS 13, 612 00 Brno Czechoslovakia

Received 28 August 1992