Lobachevskii Journal of Mathematics Vol. 3, 1999, 39–71 \begin{matrix}c \\ ◯ \end{matrix}B. Doubrov and B. Komrakov and T. Morimoto

B. Doubrov and B. Komrakov and T. Morimoto

EQUIVALENCE OF HOLONOMIC DIFFERENTIAL EQUATIONS

(submitted by B.N.Shapukov)

1. Equivalence of differential equations

Roughly speaking, two differential equations are called equivalent if one can be transformed into another by a certain change of variables. In particular, this change of variables transforms solutions of one equation into solutions of another. Moreover, most methods of solving differential equations consist of finding a change of variables that will transform a given equation to a simplier one, for which solutions are known. Thus, equivalence theory of differential equations underlies most branches in the theory of differential equations.

In this paper we show that for a large class of differential equations (which, in particular, includes arbitrary systems of ordinary differential equations) it is possible to answer constructively to the question, whether two given differential equations are equivalent. This is achieved by means of transforming this problem into the equivalence problem for specific geometric structures on smooth manifolds and then applying powerful techniques of modern differential geometry.

In general, the problem of equivalence of differential equations consists in determining whether two differential equations are equivalent up to a given class of transformations. We examine differential equations up to contact transformations. The reason for this is that contact transformations constitute the most general class of transformations of differential equations which is in agreement with the geometric notion of differential equations. This problem involves a number of related problems such as the construction of a transformation establishing the equivalence of the equations, description of invariants, computation of symmetry groups (algebras) of the differential equations, etc. Sophus Lie was the first to use this approach to the problem of equivalence of differential equations. In [18], he carried out a group classification for ordinary differential equations and obtained some partial results for second-order equations. That problem was later on completely solved by Tresse [36, 37].

1991 Mathematical Subject Classification. Primary: 34A26, 58A30. Secondary:

17B66, 53C30.

Key words and phrases. Cartan connection, filtered manifold, graded Lie algebra, contact distribution.

The next important advance in the development of Sophus Lie’s ideas was a series of brilliant papers by E. Cartan, who generalized affine connections to what now is known as Cartan connections and established an intimate relation between differential geometry and differential equations [4]. In particular, E. Cartan showed that for each second-order ordinary differential equation of the form

y′′ = A(y′)^{3} + B(y′)^{2} + C(y′) + D  (1)

there is associated a unique normal projective connection whose geodesic submanifolds are described by this equation. He also showed that one can, in a similar way, associate a projective connection to an arbitrary second-order ordinary differential equation which is solved for the highest-order derivative. In this way Cartan discovered that two second-order differential equations are equivalent if and only if so are the corresponding connections, which belong to the realm of differential geometry. In one of his last works [5], Cartan carried out similar constructions for third-order ordinary differential equations, which he viewed up to the group of point transformations. In effect, E. Cartan introduced a fundamentally new approach to the problem of equivalence of differential equations. The idea of this approach is to associate with every differential equation a uniquely defined geometric structure of a certain form, which is then studied with the help of differential-geometric methods. In what follows, we shall call this approach geometric, as opposed to the analytical approach of Sophus Lie. This approach was further developed by Chern to study third-order ordinary differential equations [7, 8].

Now in applying this geometric approach to the problem of equivalence of differential equations, one has first to formulate differential equations as some geometric structures (geometrization). Then geometric structures may be treated as G-structures, for which the theory of equivalence problem was developed by many authors after Cartan ([27, 29, 30]). This approach was used, for example, in the papers [11, 12, 13] to study second-order ordinary differential equations.

However, the usual method of G-structure is not always effective for detailed geometric studies. For more effective approach, another more refined geometric method is needed, that is, to associate canonically with each differential equation a unique normal Cartan connection, which is singled out from the rest of the Cartan connections associated with the equation by imposing certain conditions on the curvature tensor. A Cartan connection is said to be associated with a differential equation if the equation determines all of its geodesic submanifolds or their generalizations.

In the present paper we shall show that with every holonomic system of differential equations (see Section 3), viewed up to contact transformations, one can naturally associate a Cartan connection. In this way we reduce the problem of equivalence of holonomic systems of differential equations to that of equivalence of Cartan connections.

Cartan connections appeared implicitly in the works of E. Cartan himself, while a modern formulation may be found in [14].

41 EQUIVALENCE OF HOLONOMIC DIFFERENTIAL EQUATIONS

To give an example of the application of Cartan connections to the problem of equivalence of second-order differential equations, we cite the paper [25], in which invariant conditions are provided that single out those of the equations (1) whose symmetry algebra has dimension ≥ 3. In [2], invariant conditions determining equations with 6- and 7-dimensional symmetry algebra are described using the Cartan connection associated with a third-order ordinary differential equation. In [26] third order ordinary differential equation are studied on the basis of Tanaka’s theory of normal Cartan connection and the structure equation is concretely described.

It should be mentioned that the class of transformations up to which this or that class of differential equations is to be studied depends on the origins of the problem and may vary greatly from problem to problem. For instance, in the papers [28] ordinary differential equations are examined up to point transformations in a context similar to that of the present paper. The papers [13] are concerned with the study of systems of ordinary differential equations up to the point transformations preserving the independent variable.

The classical work [41] is devoted to the problem of equivalence of systems of linear differential equations up to the transformations preserving the class of linear equations. Of other approaches to the theory of local equivalence of differential equations, the paper [3] should be mentioned, in which constructive solutions of certain special problems are given based on the compatibility analysis for systems of partial differential equations.

Though not published fully, we should mention the work of Tanaka on geometric study of ordinary differential equations developed in 1980’s. As announced in [33] and reported in [35], he made a rigorous formulation of geometric study of ordinary differential equations (including holonomic system of differential equations) by introducing the notion of pseudo-projective structure. Based on his theory of Cartan connection [32], he obtained a normal Cartan connection associated with a system of ordinary differential equations in the ”simple” case, that is, the case when the model space is a homogeneous space of a simple Lie group. Then he, in particular, carried out detailed studies of second order systems. He stated that it was also possible to construct a Cartan connection in the other non-simple case. Meanwhile Morimoto remarked that the construction of Cartan connection could be achieved systematically by using his general criterion of Cartan connection [21]. A detailed monograph on geometric study of ordinary differential equation was intended by Tanaka and should have included the above results.

Independently far from Japan, the first and second named authors were studying ordinary differential equations starting from Cartan’s view point, and recently has begun our joint work with the opening of the cooperation between ISLC and Japanese mathematicians. Since constructing a Cartan connection associated with differential equations seems a fundamental issue for further geometric studies of differential equations, we dare to publish the present article which, we hope, will be replaced by a more complete monograph of Tanaka waited for.

The outline of the paper is as follows. In Section 2 we consider the geometric notion of a differential equation based on the geometry of jet spaces.

In Section 3 we introduce the notion of a holonomic differential equation and construct a pair of completely integrable distributions associated with any holonomic equation. In Sections 4 and 5 we study this geometric structure by methods of nilpotent differential geometry and find the symbol of this structure. In Section 6 we prove that with any holonomic differential equation there is naturally associated Cartan connection. This allows to reduce the equivalence problem of holonomic differential equations to the well-known equivalence problem for absolute parallelism structures. As an example we consider in Section 7 systems of second order ODE’s and third order ODE’s and compute explicitly their fundamental invariants. In particular, we find conditions, when these equations are contactly equivalent to the trivial ones.

2. Geometrization of differential equations

In this section, we give a geometric interpretation of differential equations which will serve as a basis for the rest of the presentation in this paper. The key feature of this interpretation, which is based on the book [17], is the structure of jet spaces and related geometric objects.

2.1. Jet spaces. Let M be a smooth manifold of dimension n + m. We say that two n-dimensional submanifolds L and \overline{L} passing through a point p ∈ M have contact of order k ⊂ ℕ at p if, relative to some local coordinate system (x_{1}, …, x_{n}, y_{1}, …, y_{m}) with origin at p, these two submanifolds are graphs of some functions

f: ℝ^{n} → ℝ^{m}, (x_{1}, …, x_{n}) ↦ (f_{1}(x), …, f_{m}(x))

and

f̄: ℝ^{n} → ℝ^{m}, (x_{1}, …, x_{n}) ↦ (f̄_{1}(x), …, f̄_{m}(x))

respectively such that \frac{∂f_{i}^{|α|}}{∂x^{α}} (0) = \frac{∂f̄_{i}^{|α|}}{∂x^{α}} (0) for all i = 1, …, m and for all multiindices α = (α_{1}, …, α_{n}) with |α| = α_{1} + ⋯ + α_{n} ≤ k. It can be easily verified that this definition is independent of the choice of local coordinates. Two submanifolds L and \overline{L} that meet at a point p are said to have contact of order 0 at that point.

It is clear that the relation of having k-order contact at a point p is an equivalence relation on the set of all n-dimensional submanifolds passing through p. The equivalence class determined by a submanifold L is called the k-jet of the manifold L at the point p and is denoted [L]_{p}^{k}. In addition, let J_{p}^{k}(M, n) denote the set of all equivalence classes.

The union ⋃_{p∈M} J_{p}^{k}(M, n), denoted by J^{k}(M, n), is called the space of kjets of n-dimensional submanifolds in M. Let (x_{1}, …, x_{n}, y_{1}, …, y_{m}) be a local coordinate system in a domain U ⊂ M. Denote by U^{k} the subset of J^{k}(M, n) consisting of all points [L]_{p}^{k} such that p ∈ U and L = Γ_{f} is the graph of a smooth function

f: ℝ^{n} → ℝ^{m}, (x_{1}, …, x_{n}) ↦ (f_{1}(x), …, f_{m}(x)).  (2)

Then the functions x_{i}, i = 1, …, n; y_{j}^{α} = \frac{∂^{|α|}f_{j}}{∂x^{α}}, j = 1, …, m, |α| ≤ k define a coordinate system on U^{k}. The set of all such coordinate systems provides J^{k}(M, n) with a smooth manifold structure of dimension n + m\binom{n+k}{n}.

For any two natural numbers k > l, denote by π_{k,l} the natural projection

π_{k,l}: J^{k}(M, n) → J^{l}(M, n), [L]_{p}^{k} ↦ [L]_{p}^{l}.

We also make the convention to identify J^{0}(M, n) with M (by letting [L]_{p}^{0} ≡ p for all [L]_{p}^{0} ∈ J^{0}(M, n)) and denote by π_{k,0} the natural projection J^{k}(M, n) → M.

2.2. Definition of differential equations. Given an n-dimensional submanifold L ⊂ M, we define its k-th lift L^{(k)} as the submanifold in J^{k}(M, n) of the form

L^{(k)} = {[L]_{p}^{k} | p ∈ L}.

If, relative to a local coordinate system (x_{1}, …, x_{n}, y_{1}, …, y_{m}), the submanifold L is the graph of some function f of the form (2), then, relative to the corresponding local coordinate system in J^{k}(M, n), its k-th lift L^{(k)} has the form

L^{(k)} = \{(x_{1}, …, x_{n}, y_{j}^{α} = \frac{∂f_{i}^{|α|}}{∂x^{α}} ) | x_{1}, …, x_{n} ∈ ℝ\} .

We are now ready to give a geometric definition for the notion of differential equation.

Definition. By a k-order differential equation (with n independent and m dependent variables) we mean a submanifold E in J^{k}(M, n), where dim M = n + m. By a solution of the equation E we mean an n-dimensional submanifold L ⊂ M such that L^{(k)} ⊂ E.

The following table shows the correspondence between the geometric and the classical interpretations of differential equations.

Classical interpretation  Geometric interpretation
Dependent and independent  Local coordinate system on
variables  a manifold M of dimension n + m

x_{1}, …, x_{n} and y_{1}, …, y_{m}

The set of all possible

derivatives up to the order k  The jet space J^{k}(M, n)

inclusive

A function f: ℝ^{n} → ℝ^{m}  An n-dimensional submanifold L ⊂ M
The derivatives of f up to the The k-th lift L^{(k)} ⊂ J^{k}(M, n)
order k inclusive  of the submanifold L
A system of k-order differential A submanifold E ⊂ J^{k}(M, n)

equations

Local coordinate systems (x_{1}, …, x_{n}, y_{1}, …, y_{m}) on M provide a link between the two columns in the table; in particular, they determine local coordinate systems on [L]_{p}^{k}.

These notions of jet spaces and differential equations are not the only ones. For example, the structure of jet spaces of mappings of smooth manifolds [1] or, more generally, of sections of (nonlinear) fiber bundles [17] is used widely. These jet spaces can be regarded in the obvious way as open subsets of submanifold jet spaces discussed above. Finally, a purely geometric definition of jet spaces which is independent of the choice of local coordinates and is based on contact geometry can be found in the works of Yamaguchi [38, 39].

2.3. Contact distribution. Jet spaces admit a nontrivial geometry corresponding to the so-called contact distribution, which can be defined naturally on any jet space J^{k}(M, n).

We define the contact distribution C^{k} on the space J^{k} = J^{k}(M, n) to be the smallest among those subbundles of the tangent bundle T J^{k} for which all L^{(k)} with L ⊂ M are integral manifolds. To be more precise, let q be an arbitrary point of the submanifold J^{k}. Then C_{q}^{k} ⊂ T_{q}J^{k} is generated by all subspaces of the form T_{q}L^{(k)}, where L is an arbitrary n-dimensional submanifold in M such that q is contained in L^{(k)}.

We shall now give the coordinate expression for the contact distribution. To do this, we fix a local coordinate system (x_{1}, …, x_{n}, y_{1}, …, y_{m}) on M and consider the corresponding local coordinate system (x_{i}, y_{j}^{α}) on J^{k}. Relative to these coordinates, the contact distribution is defined by the following collection of differential forms:

ω_{j}^{α} = dy_{j}^{α} − ∑_{i=1}^{n} y_{j}^{α,i}dx_{i}, j = 1, …, m, |α| ≤ k − 1,

where the multi-index α, i is obtained by adding 1 to the i-th component of α.

It can be easily verified that the distribution C^{k} is generated by the following vector fields:

D_{i} = \frac{∂}{∂x_{i}} + ∑_{j=1}^{m} ∑_{|α|≤k−1} y_{j}^{α,i} \frac{∂}{∂y_{j}^{α}} , i = 1, …, n

and

E_{j}^{α} = \frac{∂}{∂y_{j}^{α}} , j = 1, …, m, |α| = k.

In particular, its dimension is equal to n + m\binom{n+k−1}{k}.

We define the vertical distribution V^{k} on the space J^{k} as the distribution that is tangent to the fibers of the projection π_{k,k−1} : J^{k} → J^{k−1}. Note that, in the local coordinates, this distribution is generated by the vector fields E_{j}^{α} with j = 1, …, m, |α| = k. This implies, in particular, that the vertical distribution lies in the contact distribution and has codimension n.

Using the contact and vertical distributions on the space J^{k}, we shall now give a different interpretation of the elements of J^{k+1}. Let L_{1}, L_{2} be two ndimensional submanifolds in M such that [L_{1}]_{p}^{k} = [L_{2}]_{p}^{k} = q. It is obvious that L_{1} and L_{2} have contact of order k + 1 at the point p if and only if the tangent spaces T_{q}L_{1}^{k} and T_{q}L_{2}^{k} coincide. From the coordinate expression of L_{1}^{(k)} and L_{2}^{(k)} it follows that these tangent spaces can be arbitrary complements of V_{q}^{k} in C_{q}^{k}. Thus for any point q ∈ J_{k}, there is a natural mapping

l_{q}^{k}: π_{k+1,k}^{−1}(q) → Gr_{n}(C_{q}^{k})

whose image coincides with the set of all n-dimensional complements of V_{q}^{k} in C_{q}^{k}. This mapping allows to introduce a structure of affine bundle on J^{k+1}, which plays a key part in determining whether a differential equation is involutive.

2.4. Solutions of differential equations. By the definition of the contact distribution C^{k} on the space J^{k} = J^{k}(M, n), all k-th lifts L^{(k)} of submanifolds L ⊂ M are integral manifolds of C^{k}. Yet there exist other integral manifolds of the contact distribution. For example, all fibers of the projection π_{k,k−1} are integral manifolds. The complete description of all maximal integral manifolds of the contact distribution C^{k} can be found in [17]. In particular, an integral manifold N of the contact distribution has the form L^{(k)} if and only if N is projected diffeomorphically onto its image in M under the projection π_{k,0}. Thus, to solve a given equation E ⊂ J^{k} is equivalent to describing all integral manifolds of the contact distribution that lie entirely in E and are projected diffeomorphically onto M.

By a quasi-solution of E we shall mean any integral manifold of the contact distribution that lies entirely in E. Thus, the k-th lift of any solution of E is a quasi-solution of E, and in order that a quasi-solution determine an ordinary solution it is sufficient that it admit a diffeomorphic projection onto M.

2.5. Transformations of equations. Now we shall discuss the analog of a change of variables in our geometric model of differential equations. Let ϕ: M_{1} → M_{2} be an arbitrary diffeomorphism of smooth manifolds of dimension n + m. It determines the diffeomorphism of groups:

ϕ^{(k)} : J^{k}(M_{1}, n) → J^{k}(M_{2}, k), [L]_{p}^{k} ↦ [ϕ(L)]_{ϕ(p)}^{k},

which is called the k-th prolongation of ϕ. Below we list a number of obvious properties of prolongations.

1. If ϕ_{1} : M_{1} → M_{2} and ϕ_{2} : M_{2} → M_{3} are two diffeomorphisms of manifolds, then (ϕ_{1} ∘ ϕ_{2})^{(k)} = ϕ_{1}^{(k)} ∘ ϕ_{2}^{(k)}. In particular, for any manifold M we get the injective homomorphism

Diff(M) → Diff(J^{k}(M, n)), ϕ ↦ ϕ^{(k)}.

2. For any two natural numbers k > l, the following diagram is commutative:

J^{k}(M_{1}, n) ⟶^{ϕ^{(k)}} J^{k}(M_{2}, n)
π_{k,l}↓  ^{π_{k,l}}↓
J^{l}(M_{2}, n) ⟶^{ϕ^{(l)}} J^{l}(M_{2}, n)

3. For any n-dimensional submanifold L ⊂ M_{1}, we have ϕ^{(k)}(L^{(k)}) = (ϕ(L))^{(k)}.

4. Every diffeomorphism of the form ϕ^{(k)} preserves the contact distribution, i.e., its differential takes the contact distribution on J^{k}(M_{1}, n) to the contact distribution on J^{k}(M_{2}, n).

5. For any equation E ⊂ J^{k}(M_{1}, n), the diffeomorphism ϕ (respectively, ϕ^{(k)}) takes solutions (respectively, quasi-solutions) of this equation to solutions (respectively, quasi-solutions) of the equation ϕ^{(k)}(E).

Transformations ϕ^{(k)} of differential equations, where ϕ are arbitrary diffeomorphisms of the underlying manifolds, are called point transformations. A larger class of transformations of differential equations is formed by the socalled contact transformations.

A diffeomorphism ψ: J^{k}(M_{1}, n) → J^{k}(M_{2}, n) is called a contact transformation if it preserves the contact distribution. It is immediate from this definition that a contact diffeomorphism ψ takes quasi-solutions of an equation E ⊂ J^{k}(M_{1}, n) to quasi-solutions of the equation ψ(E).

Contact diffeomorphisms have been thoroughly studied (see for example [17]). Their description is based on the following result, proved by Sophus Lie (see also [17, 38]):

Lemma 2.1. Suppose that m, k are natural numbers such that (m, k) ≠ (1, 1). Then

1. Every contact transformation ψ: J^{k}(M_{1}, n) → J^{k}(M_{2}, n), where dim M_{i} = n + m, preserves the vertical distribution V^{k} on the jet spaces.

2. There exists a unique contact transformation ϕ: J^{k−1}(M_{1}, n) → J^{k−1}(M_{2}, n) such that the following diagram is commutative:

J^{k}(M_{1}, n) ⟶^{ψ} J^{k}(M_{2}, n)
π_{k,k−1}↓  ^{π_{k,k−1}}↓
J^{k−1}(M_{1}, n) ⟶^{ϕ} J^{k−1}(M_{2}, n)

Consider the exceptional case m = k = 1. In this case the dimension of J^{1}(M, n) is equal to 2n+1, while the contact distribution C^{1} is 2n-dimensional and defines the classical contact structure on J^{1}(M, n). Relative to the local coordinate system

(x_{1}, …, x_{n}, y = y_{1}, z_{1} = y_{1}^{(1,0...,0)}, …, z_{n} = y_{1}^{(0,...,0,1)}),  (3)

the contact distribution C^{1} is given by a single differential form, namely ω = dy − z_{1} dx_{1} − ⋯ − z_{n} dx_{n}. It is well known that the symmetry group of the classical contact structure is primitive and hence preserves no foliations.

Conversely, if (m, k) ≠ (1, 1), any contact transformation ψ: J^{k}(M_{1}, n) → J^{k}(M_{2}, n) uniquely determines a contact transformation ψ^{(1)} : J^{k+1}(M_{1}, n) → J^{k+1}(M_{2}, n), which is called the (first) prolongation of ψ. Indeed, let q_{k+1} ∈ J^{k+1}(M_{1}, n) and q_{k} = π_{k+1,k}(q_{k+1}). Recall that to the point q_{k+1} corresponds the subspace W = l_{q_{k}}^{k}(q_{k+1}) in C_{q_{k}}^{k} complementary to V_{q_{k}}^{k}. Then the point ψ^{(1)}(q_{k+1}) is determined by the subspace d_{q_{k}}ψ(W), which lies in C_{ψ(q_{k})}^{k} and, according to Lemma 2.1, forms a complement to V_{ψ(q_{k})}^{k}. Note that in the case m = k = 1, the transformation ψ^{(1)} can be defined in the same way only in some open domain of J^{2}(M, n). For an arbitrary i ≥ 2, let ψ^{(i)} = (ψ^{(i−1)})^{(1)}.

Thus, for m ≥ 2 any contact transformation ψ: J^{k}(M_{1}, n) → J^{k}(M_{2}, n) has the form ϕ^{(k)} for some diffeomorphism ϕ: M_{1} → M_{2} and hence is a point transformation. For m = 1, any contact transformation ψ of two spaces of k-th jets locally has the form ϕ^{(k−1)}, where ϕ: J^{1}(M_{1}, n) → J^{1}(M_{2}, n) is a contact transformation of spaces of first order jets.

Infinitesimal contact transformations can be described in a similar way. Namely, a vector field X on the jets space J^{k}(M, n) is called contact if the corresponding one-parameter transformation group ϕ_{t} consists of contact transformations. In this case, for any l ∈ N the family ϕ_{t}^{(l)} forms a one-parameter group of contact transformations on the space J^{k+l}(M, n). Its generator is a contact vector field, which is called the l-th prolongation of X and is denoted X^{(l)}.

Knowing the structure of contact transformations, we obtain the following result:

Lemma 2.2. Let dim M = n + m and k ∈ ℕ. Then

1) For m ≥ 2, the mapping X ↦ X^{(k)} is an isomorphism of the Lie algebra of vector fields on M onto the Lie algebra of contact vector fields on J^{k}(M, n).

2) For m = 1, the mapping X ↦ X^{(k−1)} is an isomorphism of the Lie algebra of vector fields on J^{1}(M, n) onto the Lie algebra of contact vector fields on J^{k}(M, n).

Now we describe the coordinate representation of contact vector fields, which we shall need later on. In view of Lemma 2.2, it will suffice for us to describe how, given a contact vector field X on J^{k}(M, n) with k ≥ 0, one can compute its first prolongation X^{(1)}, and to find the coordinate expression for all contact vector fields on J^{1}(M, n) in the case m = 1.

Suppose that

X = ∑_{i=1}^{n} A_{i} \frac{∂}{∂x_{i}} + ∑_{j=1}^{m} ∑_{|α|≤k} B_{j}^{α} \frac{∂}{∂y_{j}^{α}}

is the coordinate representation of a contact vector field X on J^{k}(M, n). We now describe the coordinate representation for the vector field X^{(1)}. Recall that the contact distribution C^{k+1} on the space J^{k+1}(M, n) is determined by the differential forms

ω_{j}^{α} = dy_{j}^{α} − ∑_{i=1}^{n} y_{j}^{α,i} dx_{i}, j = 1, …, m, |α| ≤ k.

Hence L_{X^{(1)}}ω_{j}^{α} lies in the ideal generated by these differential forms. Using this fact, after elementary manipulations we get that the vector field X^{(1)} has the form

X^{(1)} = ∑_{i=1}^{n} A_{i} \frac{∂}{∂x_{i}} + ∑_{j=1}^{m} ∑_{|α|≤k+1} B_{j}^{α} \frac{∂}{∂y_{j}^{α}} ,  (4)

where

B_{j}^{α,i} = D_{i}B_{j}^{α} − ∑_{s=1}^{n} y_{j}^{α,s}D_{i}A_{s} for |α| = k.

These formulas allow to find the coordinate representation of X^{(k)} from the coordinate representation of X for any number k ∈ ℕ.

Consider the case m = 1 and the space J^{1}(M, n). In the local coordinate system (3), the contact distribution C^{1} is given by the differential form ω = dy−z_{1} dx_{1}−⋯−z_{n} dx_{n}, and any contact vector field X on J^{1}(M, n) is uniquely determined by the function f = ω(X) and has the form

X_{f} = − ∑_{i=1}^{n} \frac{∂f}{∂z_{i}} \frac{∂}{∂x_{i}} + (f − ∑_{i=1}^{n} z_{i} \frac{∂f}{∂z_{i}} ) \frac{∂}{∂y} + ∑_{i=1}^{n} (\frac{∂f}{∂x_{i}} + z_{i} \frac{∂f}{∂y} ) \frac{∂}{∂z_{i}} .

The correspondence f ↦ X_{f} allows to introduce a Lie algebra structure on the set of all smooth functions on ℝ^{2n+1}:

(f, g) ↦ {f, g} = ω([X_{f}, X_{g}]).  (5)

This completes the coordinate description of all contact vector fields on jet spaces.

3. Holonomic differential equations

3.1. Definition of holonomic differential equations. Holonomic differential equations form an extension of the class of ordinary differential equations. Roughly speaking, a holonomic differential equation is a system of (ordinary or partial) differential equations whose solutions can be parametrized by finitely many parameters. In this case, there is a number k ∈ ℕ such that all derivatives of the unknown functions of order k + 1 and higher can be expressed in terms of the lower-order derivatives. This gives rise to the following definition of holonomic differential equations:

Definition. A differential equation E ⊂ J^{k+1}(M, n) of order k + 1 is called holonomic if the projection π = π_{k+1,k}|_{E}: E → J^{k}(M, n) is a diffeomorphism.

In this case, the mapping s = π^{−1} : J^{k}(M, n) → J^{k+1}(M, n) is a section of the projection π_{k+1,k}, and the equation E itself is the image of this section. Thus, there is a one-to-one correspondence between holonomic differential equations of order k + 1 and sections of the projection π_{k+1,k}.

In local coordinates, this means that the submanifold E is given by a system of equations of the form

y_{j}^{α} = F_{j}^{α}(x_{i}, y_{l}^{β}), |α| = k + 1, j = 1, …, m,  (6)

where |β| ≤ k, and F_{j}^{α} are smooth functions, or in the classical notation:

\frac{∂^{k+1}y_{j}}{∂x^{α}} = F_{j}^{α}(x_{i}, \frac{∂^{|β|}y_{j}}{∂x^{β}} ), |α| = k + 1, j = 1, …, m.  (7)

In particular, a typical example of a holonomic differential equation is a system of ordinary differential equations solved for the highest derivatives.

3.2. Solutions of holonomic differential equations. For any point q ∈ E, we define a subspace E_{q} ⊂ T_{q}E by letting

E_{q} = T_{q}E ∩ C_{q}^{k+1}.

Lemma 3.1.

1. The subspaces E_{q} form an n-dimensional distribution on E.

2. The system (7) is compatible if and only if the distribution E is completely integrable.

3. Quasi-solutions of E are precisely the integral manifolds of the distribution E.

4. Locally, any quasi-solution can be expressed as L^{(k+1)}, where L is a solution of E.

Proof. 1. Since π: E → J^{k} is a diffeomorphism, we have T_{q}E ⊕ V_{q}^{k+1} = T_{q}J^{k+1} for any q ∈ E, where, as before, V^{k+1} denotes the vertical foliation on J^{K+1}. Since C^{k+1} ⊃ V^{k+1} and codim_{C^{k+1}} V^{k+1} = n, it now follows immediately that dim E_{q} = n for all q ∈ E.

2. Consider a local coordinate system on J^{k+1}, in which the equation E is given by the formulas (6). Then the functions (x_{i}, y_{j}^{α}), where i = 1, …, n, j = 1, …, m, |α| ≤ k, form a local coordinate system on E. Relative to this coordinate system, the distribution E is generated by the following vector fields:

X_{i} = D_{i} + ∑_{j=1}^{m} ∑_{|α|=k} F_{j}^{α} \frac{∂}{∂y_{j}^{α}} , i = 1, …, n,  (8)

where, as before, D_{i} = \frac{∂}{∂x_{i}} + ∑_{j=1}^{m} ∑_{|α|<k} y_{j}^{α,i} \frac{∂}{∂y_{j}^{α}}.

Since the vector fields D_{i}, i = 1, …, n commute with each other, the distribution E is completely integrable if and only if the vector fields X_{i}, i = 1, …, n commute with each other. This condition is in turn equivalent to the conditions

D_{r}(F_{j}^{α,s}) = D_{s}(F_{j}^{α,r}) for all 1 ≤ r, s ≤ n, j = 1, …, m, |α| = k.

But these are precisely the compatibility conditions for the system (7).

3. This statement follows immediately from the definition of quasi-solutions of differential equations.

4. Let N be a quasi-solution of the equation E. Show that it can be projected onto M without singularities, i.e., the projection π_{k,0}|_{N}: N → M is an immersion. Actually, it suffices to show that T_{q}N does not intersect the kernel of the differential d_{q}π_{k,0}. But T_{q}N = E_{q}, and in local coordinates, it is generated by the tangent vectors (8). It now follows immediately that T_{q}N∩ker d_{q}π_{k,0} = {0}, and locally, N has the form L^{(k+1)} for some n-dimensional submanifold L in M.

In what follows, we assume that all holonomic differential equations E to be considered are compatible, i.e., that the corresponding distributions E are completely integrable.

3.3. Holonomic equations as geometric structures. Let E ⊂ J^{k+1}(M, n) be a holonomic differential equation, and let s: J^{k}(M, n) → J^{k+1}(M, n) be the corresponding section of the projection π_{k+1,k} whose image coincides with E.

To every point q ∈ J^{k} = J^{k}(M, n), the section s assigns a point s(q) ⊂ J^{k+1}, to which, in turn, corresponds a subspace E_{q}^{k} in C_{q}^{k} complementary to V_{q}^{k}. Therefore, choosing s is equivalent to choosing a subdistribution E^{k} in the contact distribution that complements the vertical distribution. It is easy to verify that the differential s_{∗} of the section s takes E^{k} precisely to E.

Thus, we have the following three equivalent definitions: a holonomic differential equation is A. a submanifold in J^{k+1} which is diffeomorphically projected onto J^{k}; B. a section s: J^{k} → J^{k+1} of the projection π_{k+1,k}; C. a subdistribution E^{k} of the contact distribution C^{k} that complements the

vertical distribution. In the sequel, by a holonomic differential equation we shall mainly mean a

submanifold E ⊂ J^{k+1}. In addition to the distribution E, which determines the solutions of E, we define two more distributions V and C on E to be the images of the distributions V^{k} and C^{k} on J^{k}(M, n) under s. Therefore,

C = s_{∗}(C^{k}) = {X ∈ T E | π_{∗}(X) ∈ C^{k}}
V = s_{∗}(V^{k}) = {X ∈ T E | π_{∗}(X) ∈ V^{k}},

where, as before, π denotes the projection π_{k+1,k}|_{E}.

Thus, every holonomic differential equation E ⊂ J^{k+1}(M, n) admits the following naturally defined geometric structure: a pair of nonintersecting completely integrable distributions (E, V) of dimension n and m\binom{n+k−1}{k}, respectively. The following result demonstrates the importance of this structure.

Theorem 1. Let E_{i} ⊂ J^{k+1}(M_{i}, n), i = 1, 2 be two holonomic differential equations, and let (E_{i}, V_{i}) be the corresponding pairs of distributions. Assume that m ≥ 2, k ≥ 1 or m = 1, k ≥ 2. Then the equations E_{1} and E_{2} are contactly equivalent if and only if so are the corresponding pairs of distributions.

Proof. Let ψ: J^{k+1}(M_{1}, n) → J^{k+1}(M_{2}, n) be a contact transformation establishing the equivalence of the equations E_{1} and E_{2}. Then the differential of the mapping ϕ = ψ|_{E_{1}} : E_{1} → E_{2} takes the pair (E_{1}, V_{1}) to the pair (E_{2}, V_{2}).

Conversely, let ϕ: E_{1} → E_{2} be a diffeomorphism establishing the equivalence of the pairs (E_{1}, V_{1}) and (E_{2}, V_{2}). Let ψ: J^{k}(M_{1}, n) → J^{k}(M_{2}, n) be a diffeomorphism for which the following diagram is commutative:

\begin{matrix}E_{1} & ⟶^{ϕ} & E_{2} \\ ^{π_{1}}↓ & & ^{π_{2}}↓ \\ J^{k}(M_{1}, n) & ⟶^{ψ} & J^{k}(M_{2}, n) \end{matrix}  (9)

where π_{i} denotes the diffeomorphism π_{k+1,k}|_{E_{i}}: E_{i} → J^{k}(M_{i}, k), i = 1, 2. Since the differential of the mapping π_{i} takes the sum of the distributions E_{i} + V_{i} to the contact distribution on the space J^{k}(M_{i}, n), i = 1, 2, we see that ψ is a contact diffeomorphism. We now show that its first prolongation ψ^{(1)} : J^{k+1}(M_{1}, n) → J^{k+1}(M_{2}, n) takes the equation E_{1} to the equation E_{2}, and that moreover its restriction to E_{1} coincides with the mapping ϕ. Indeed, to every point q ∈ E_{1} corresponds the subspace W_{q} = (π_{1})_{∗}(E_{1})_{q} of the contact distribution on J^{k}(M_{1}, q), which is a complement of the vertical distribution. From the diagram (9) and from the fact that ϕ_{∗} takes E_{1} to E_{2}, it follows that ψ_{∗} takes W_{q} to the subspace W_{ϕ(q)} = (π_{2})_{∗}(E_{2})_{ϕ(q)} corresponding to the point ϕ(q). It now easily follows that ψ^{(1)}(q) = ϕ(q) for all q ∈ E_{1}, as was to be proved.

Thus the problem of contact equivalence of holonomic differential equations is equivalent to the problem of equivalence of the corresponding pairs of completely integrable distributions defined on these equations. The latter problem, to which the rest of this paper is devoted, is a classical problem in modern differential geometry.

4. Methods of nilpotent differential geometry

Geometric structures containing distributions as their component parts can be conveniently studied within the framework of nilpotent differential geometry, which was developed in the works of Tanaka [31, 32] and Morimoto [22]. This theory generalizes the theory of G-structures to the case in which the tangent bundle of the manifold is supplied with a filtration, which is usually determined by the geometric structure itself.

We shall now briefly discuss the basic constructions and definitions of nilpotent differential geometry (see [22]). A filtered manifold is a smooth manifold M supplied with an increasing filtration of the tangent bundle T M:

T M = T^{−μ}M ⊂ T^{−μ+1}M ⊂ ⋯ ⊂ T^{−1}M ⊂ T^{0}M = 0,

such that [\underline{T^{i}M}, \underline{T^{j}M}] ⊂ \underline{T^{i+j}M} for all −μ ≤ i, j ≤ 0. (Here for any subbundle D of the tangent bundle, by \underline{D} we denote its sheaf of local cross-sections.) The role of the tangent bundle for filtered manifolds is played by the graded vector bundle associated with the given filtration:

gr T M = ∑_{i=1}^{μ} gr_{−i} T M, where gr_{−i} T M = T^{−i}M/T^{−i+1}M.

Let X ⊂ \underline{T^{i}M}, Y ⊂ \underline{T^{j}M} (−μ ≤ i, j < 0). Then it is easy to see that for any point p ∈ M the element [X, Y]_{p} + T^{i+j+1}M ⊂ gr_{i+j} T_{p}M depends only on X_{p} + T^{i+1}M and Y_{p} + T^{j+1}M. Hence, the Lie bracket of vector fields induces a natural structure of a graded nilpotent Lie algebra on each fiber of the vector bundle gr T M.

Example 4.1. Let D be a smooth distribution on a smooth manifold M. We can then define the derived series of subsheaves of the sheaf \underline{T M}:

D^{−1} = \underline{D},
D^{−p−1} = D^{−p} + [D^{−p}, D^{−1}].

The distribution D is called regular if for any p > 0 there exists a distribution D^{−p} such that D^{−p} = \underline{D}^{−p}. Suppose that there exists an integer number μ such that D^{−μ} = T M. Then the decreasing sequence of subbundles

T M = D^{−μ} ⊂ D^{−μ+1} ⊂ … D^{−1} ⊂ D_{0} = 0

defines a filtered manifold structure on M.

Let 𝔪 = ⨁_{i≥1} 𝔤_{−i} be an arbitrary graded finite-dimensional nilpotent Lie algebra such that 𝔪 is generated by 𝔤_{−1}. (In [31] such Lie algebras are called fundamental.) We say that a filtered manifold M is of type 𝔪 if the graded Lie algebra gr T_{p}M is isomorphic to 𝔪 for all points p ∈ M

Let G be a Lie group with the Lie algebra 𝔪. Then 𝔤_{−1} defines a left-invariant distribution D on G, which is called the standard distribution of type 𝔪. Then its derived distributions D^{−k} are also left-invariant and correspond to the subspaces ∑_{i=1}^{k} 𝔤_{−i} in 𝔪. Thus, G becomes a filtered manifold and is called the standard filtered manifold of type 𝔪.

Let 𝔪 = ⊕_{i<0}𝔤_{i} be an arbitrary fundamental graded Lie algebra. As follows from [31] it uniquely defines the graded Lie algebra 𝔤(𝔪) = ⊕_{p∈ℤ}𝔤_{p}(𝔪), which satisfies the conditions

1. 𝔤_{p}(𝔪) = 𝔤_{p} for all p < 0;

2. from [X, 𝔤_{−1}] = {0} for X ∈ 𝔤_{p}(𝔪), p ≥ 0, it follows that X = 0;

3. 𝔤(𝔪) is the largest graded Lie algebra satisfying the first two conditions (i.e., any other graded Lie algebra satisfying these two conditions can be embedded into 𝔤(𝔪)).

The Lie algebra 𝔤(𝔪) is called the universal prolongation of the fundamental algebra 𝔪. It has the following geometric meaning.

Let G be a connected Lie group with 𝔪 as its Lie algebra, and let D be the standard distribution on G of type 𝔪. Now let A be the Lie algebra of all germs of infinitesimal symmetries of D at the identity element e ∈ G. Define A_{0} (resp., A_{−1}) to be the subspace of A that consists of germs of infinitesimal symmetries X such that X_{e} = 0 (resp. X_{e} ∈ D_{e}). Now let

A_{p} = [A_{p+1}, A_{−1}] + A_{p+1} for all p ≤ −2;
A_{p} = {ξ ∈ A_{p−1} | [ξ, A_{−1}] ⊂ A_{p−1}} for all p ≥ 1.

The family of subspaces (A_{p})_{p∈ℤ} thus defined forms a decreasing filtration of the Lie algebra A. Let A be the corresponding graded Lie algebra: A_{p} = A_{p}/A_{p+1} for all p ∈ ℤ. As shown in the work [31], there is a natural isomorphism between the graded Lie algebras 𝔤(𝔪) and A.

For any filtered manifold of type 𝔪, we can construct an analog of the frame bundle. Namely, by a frame at a point p of a filtered manifold M of type 𝔪 we mean an isomorphism ϕ_{p}: 𝔪 → gr T_{p}M. Let R be the set of all frames ϕ_{p} at all points p ∈ M. Then it is easy to see that R can be naturally supplied with the structure of a principal G_{0}(𝔪)-bundle over M, where G_{0}(𝔪) is the group of all automorphisms of 𝔪 preserving the grading. As for usual manifolds, by a G_{0}-structure (of type 𝔪) on M we mean a reduction of the frame bundle R to the subgroup G_{0} ⊂ G_{0}(𝔪). The equivalence of two G_{0}-structures can be defined in a natural way, so we get the category of G_{0}-structures of type 𝔪, in which morphisms are precisely the equivalence mappings.

Let 𝔤_{0} ⊂ 𝔤_{0}(𝔪) be the subalgebra corresponding to the subgroup G_{0}. Its prolongation is defined in [31] to be the largest graded subalgebra 𝔤(𝔪, 𝔤_{0}) in 𝔤(𝔪) satisfying the conditions

1. ∑_{p<0} 𝔤_{p}(𝔪, 𝔤_{0}) = 𝔪;

2. 𝔤_{0}(𝔪, 𝔤_{0}) = 𝔤_{0}. It can be described constructively as

𝔤_{p+1}(𝔪, 𝔤_{0}) = {X ∈ 𝔤_{p+1}(𝔪) | [X, 𝔪_{−1}] ∈ 𝔤_{p}(𝔪, 𝔤_{0})} for p ≥ 0. (10)

As we shall see later, the Lie algebra 𝔤(𝔪, 𝔤_{0}) plays an important part in the study of G_{0}-structures of type 𝔪.

5. Holonomic equations as structures on filtered manifolds

In Subsection 3.3, we showed that on each holonomic differential equation E ⊂ J^{k+1}(M, n), dim M = n + m, we can naturally define a pair of completely integrable distributions (E, V), and that if m ≥ 2, k ≥ 1 or m = 1, k ≥ 2, two holonomic differential equations are contactly equivalent if and only if the corresponding pairs of distributions are equivalent. Thus, the problem of contact equivalence of holonomic differential equations reduces to the problem of equivalence of the corresponding pairs of distributions.

Note that despite both distributions E and V are completely integrable, their sum C = E⊕V is not, because it is equivalent to the contact distribution C^{k} on the space J^{k}(M, n). This kind of structures are known in nilpotent differential geometry as pseudo-product structures [33].

It can be easily verified that the distribution C is regular and determines a filtration on the tangent bundle of E (see example 4.1). Now, following the principles of differential geometry discussed in the previous section, we must

1. find the nilpotent graded Lie algebra 𝔪 that determines the type of the filtered manifold E;

2. describe the universal prolongation 𝔤(𝔪) of the Lie algebra 𝔪;

3. find a subgroup G_{0} ⊂ G_{0}(E) such that to the pair of completely integrable distributions (E, V) corresponds some G_{0}-structure of type 𝔪 on E;

4. compute the prolongation 𝔤(𝔪, 𝔤_{0}) of the subalgebra 𝔤_{0} ⊂ 𝔤_{0}(𝔪) corre-

sponding to the subgroup G_{0}. Having carried out all stages of this algorithm, we shall show in the end of this section that the geometric structure corresponding to a holonomic differential equation E ⊂ J^{k+1}(M, n) with dim M = n + m and (m, k) ≠ (1, 1) is of finite type. In particular, the algebra of contact symmetries of such an equation is always finite dimensional.

5.1. Type of contact distribution. Consider now the contact distribution C^{k} on the manifold J^{k}(M, n). Let us fix the graded nilpotent Lie algebra

𝔠(n, m; k) = ⊕_{i=1}^{k+1}𝔤_{−i} = ⟨ h_{1}, …, h_{n}, e_{j}^{α} | 1 ≤ i ≤ n, 1 ≤ j ≤ m, |α| ≤ k⟩,

where all non-zero multiplications of basis vectors are exhausted by

[e_{j}^{α,i}, h_{i}] = e_{j}^{α}, 0 ≤ |α| ≤ k − 1, i = 1, …, n, j = 1, …, m,

and the grading of 𝔠(k, m) has the form

𝔤_{−1} = ⟨h_{i}, e_{j}^{α} | |α| = k⟩,
𝔤_{−p} = ⟨e_{j}^{α} | |α| = k − p + 1⟩, 2 ≤ p ≤ k + 1.

Lemma 5.1. The contact distribution C^{k} on J^{k}(M, n) (dim M = n + m) is locally equivalent to the standard distribution of type 𝔠(n, m; k).

Proof. Fix a local coordinate system (x_{i}, y_{j}^{α}), 0 ≤ i ≤ n, 1 ≤ j ≤ m, |α| ≤ k on J^{k}(M, n). Then it is easy to see that the distribution (C^{k})^{−p} (p ≥ 1) is generated by the vector fields of the form

(C^{k})^{−p} = ⟨D_{i} = \frac{∂}{∂x}_{i} + ∑_{j=1}^{m} ∑_{|α|<k} y_{j}^{α,i} \frac{∂}{∂y_{j}^{α}} , E_{j}^{α} = \frac{∂}{∂y_{j}^{α}} | \begin{matrix}k + 1 − p ≤ |α| ≤ k, \\ 1 ≤ j ≤ m \end{matrix} ⟩ .

But the vector fields D_{i}, E_{j}^{α} (1 ≤ i ≤ m, 1 ≤ j ≤ m, |α| ≤ k) locally parallelize the manifold J^{k}(M) and form a basis of a finite-dimensional Lie algebra whose structure constants are the same as those of the Lie algebra 𝔠(n, m; k). This proves the lemma.

5.2. Universal prolongation of 𝔠(n, m; k). Let us describe the universal prolongation of the fundamental Lie algebra 𝔠 = 𝔠(n, m; k). Though it can be computed algebraically (See [20]), we will explain it by means of the geometric interpretation of 𝔤(𝔠) given in the previous section. Namely, 𝔤(𝔠) can be identified with the graded Lie algebra associated with the filtered Lie algebra of all germs of infinitesimal symmetries of the standard distribution of type 𝔠.

Lemma 5.2. Let 𝔤(𝔠) be the universal prolongation of the fundamental Lie algebra 𝔠 = 𝔠(n, m; k).

1. If m ≥ 2, k ≥ 1, then 𝔤(𝔠) is isomorphic to the Lie algebra of polynomial vector fields on ℝ^{n}×ℝ^{m} with coordinates x_{1}, …, x_{n}, y_{1}, …, y_{m}. Moreover, 𝔤_{p}(𝔠) consists of all homogeneous polynomials of degree p in x_{i}, y_{j}, \frac{∂}{∂x_{i}} , \frac{∂}{∂y_{j}}, where we assume that these expressions have degree 1, k + 1, −1, −k − 1 respectively (i = 1, …, n, j = 1 …, m).

2. If m = 1, 𝔤(𝔠) is isomorphic to the Lie algebra of all polynomials in x_{1}, …, x_{n}, y, z_{1}, …, z_{n} with the Lie bracket (5); 𝔤_{p}(𝔠) consists of all homogeneous polynomials of degree k + 1 + p, where x_{i}, y, z_{j} have degree 1, k + 1, k respectively (i, j = 1, …, n).

Proof. Since the contact distribution C^{k} on J^{k}(ℝ^{n+m}, n) is locally equivalent to the standard distribution of type 𝔠 = 𝔠(n, m; k), in the above, instead of the Lie group with Lie algebra 𝔠 and the identity element, we can consider the jet space J^{k}(ℝ^{n+m}, n), the contact distribution on it, and an arbitrary point e ∈ J^{k}(ℝ^{m+1}). Fix a local coordinate system (x_{i}, y_{j}^{α}) (i = 1, …, n, j = 1, …, m, |α| ≤ k) on J^{k}(ℝ^{n+m}) and let e = 0.

Consider the case m ≥ 2. Then by Lemma 5.1 all infinitesimal symmetries of the contact distribution C^{k} have the form X^{(k)}, where X is a vector field on J^{0}(ℝ^{n+m}) ≡ ℝ^{n+m}. Locally, in a neighborhood of the point π_{k}(e), the vector field X can be written in the form

X = A_{1} \frac{∂}{∂x_{1}} + ⋯ + A_{n} \frac{∂}{∂x_{n}} + B_{1} \frac{∂}{∂y_{1}} + ⋯ + B_{m} \frac{∂}{∂y_{m}} ,

where A_{1}, …, A_{n}, B_{1}, …, B_{m} are smooth functions of x_{1}, …, x_{n}, y_{1}, …, y_{m}. The germ of X^{(k)} at the point e is uniquely determined by the germ of X at π_{k}(e) = 0. Thus the Lie algebra A of all germs of contact vector fields at e may be identified with the algebra of germs of smooth vector fields on ℝ^{n+m} at the origin.

From the formulas (4) for prolongations of vector fields it immediately follows that X_{e}^{(k)} = 0 if and only if the functions A_{1}, …, A_{n}, \frac{∂^{|}α|B_{j}}{∂x^{α}} (0 ≤ |α| ≤ k, 1 ≤ j ≤ m) vanish at the point 0. Similarly, X_{e}^{(k)} ⊂ C_{e}^{k} if and only if the functions \frac{∂^{|}α|B_{j}}{∂x^{α}} (0 ≤ |α| ≤ k − 1, 1 ≤ j ≤ m) vanish at the origin.

We develop X into a Taylor series in x_{1}, …, x_{n}, y_{1}, …, y_{m} (i.e., consider the Tailor series of A_{1}, …, A_{n}, B_{1}, …, B_{m}) and rearrange its terms in the form ∑_{p∈ℤ} X_{p}, where X_{p} is the homogeneous term of degree p, assuming that x_{i}, y_{j}, \frac{∂}{∂x_{i}} and \frac{∂}{∂y_{j}} have the degrees specified in the theorem. Then the condi-

tions X_{e}^{(k)} = 0 and X_{e}^{(k)} ∈ C_{p}^{k} are equivalent to the conditions X_{p} = 0 for all p < 0 and X_{p} = 0 for all p < −1, respectively. Note also that if Y_{p} and Z_{q} are homogeneous vector fields of degree p and q respectively, then the vector field [Y_{p}, Z_{q}] has the degree p + q. It follows that \underline{X}_{0} ∈ A_{p} if and only if X_{r} ≠ 0 for all r ≤ p − 1. Hence, the quotient space A_{p}/A_{p+1} can then be in a natural way identified with the space of all vector fields of degree p, which proves the assertion of the theorem.

The proof in case m = 1 is similar.

Thus we obtain another representation of the fundamental Lie algebra 𝔠 = 𝔠(n, m; k). Indeed 𝔠 can be identified with the negative part of the universal prolongation 𝔤(𝔠). Hence, by Lemma 5.2 we have

m ≥ 2:

𝔠_{−p} = 𝔤_{−p}(𝔠) = ⟨x^{α} \frac{∂}{∂y_{i}} | i = 1, …, m, |α| = k + 1 − p⟩ for p = 2, …, k + 1;
𝔠_{−1} = 𝔤_{−1}(𝔠) = ⟨x^{α} \frac{∂}{∂y_{i}} , \frac{∂}{∂x_{j}} | i = 1, …, m, j = 1, …, n, |α| = k⟩ .

m = 1:

𝔠_{−p} = 𝔤_{−p}(𝔠) = ⟨x^{α} | |α| = k + 1 − p⟩ for p = 2, …, k + 1;
𝔠_{−1} = 𝔤_{−1}(𝔠) = ⟨x^{α}, z_{i} | i = 1, …, n, |α| = k⟩ .

In the following we shell assume that the fundamental Lie algebra 𝔠(n, m; k) is given by these formulas.

5.3. Holonomic equations as G_{0}-structures of type 𝔠(n, m; k). Let us take a holonomic differential equation E ⊂ J^{k+1}(M, n). As it was shown in the previous subsection, E is naturally supplied with a structure of a filtered manifold of type 𝔠 = 𝔠(n, m; k). We shall show that the pair (E, V) of completely integrable distributions on E determines a G_{0}-structure of type 𝔠 on E for some G_{0} ⊂ G_{0}(𝔠).

Lemma 5.3. For m ≥ 2, the automorphism group of the graded algebra 𝔠 = 𝔠(n, m; k) is induced by polynomial diffeomorphisms of ℝ^{n} × ℝ^{m} of the form

(x_{1}, …, x_{n}, y_{1}, …, y_{m}) ↦ (x̄_{1}, …, x̄_{n}, ȳ_{1}, …, ȳ_{m}), where  (11)

\begin{aligned}x̄_{i} & = ∑_{j=1}^{n} α_{ij}x_{j}, i = 1, …, n; \\ ȳ_{i} & = ∑_{j=1}^{m} β_{ij}y_{j} + ∑_{|α|=k+1} γ_{i,α}x^{α}, i = 1, …, m. \end{aligned}  (12)

Proof. Let ϕ: 𝔠 → 𝔠 be an automorphism of 𝔠 that preserves the grading. Recalling that

𝔠_{−1} = ⟨\frac{∂}{∂x_{i}} , x^{α} \frac{∂}{∂y_{j}} | i = 1, …, n, j = 1, …, m, |α| = k⟩ ,

we have the direct sum decomposition:

𝔠_{−1} = 𝔞 ⊕ 𝔟,

where we put

𝔞 = ⟨x^{α} \frac{∂}{∂y_{j}} | i = 1, …, n, j = 1 …, m, |α| = k⟩
𝔟 = ⟨\frac{∂}{∂x_{i}} | i = 1, …, n⟩ .

Since [𝔞, 𝔠_{p}] = 0 for all p ≤ −2, we have ϕ(𝔞) = 𝔞. Hence

ϕ(\frac{∂}{∂x_{i}} ) = ∑_{j=1}^{n} a_{ij} \frac{∂}{∂x_{j}} mod 𝔞, i = 1, …, n.  (13)

Similarly, since 𝔠_{−k−1} = ⟨\frac{∂}{∂y_{i}} | i = 1, …, m⟩, we see that

ϕ(\frac{∂}{∂y_{i}} ) = ∑_{j=1}^{n} b_{ij} \frac{∂}{∂y_{j}} , j = 1, …, m.  (14)

Consider the automorphusm ϕ_{0} of 𝔠 that is induced by the diffeomorphism (11), where (α_{ij}) = ^{t}(a_{ij})^{−1}, (β_{ij}) = ^{t}(b_{ij})^{−1} and γ_{i,α} = 0. Then it is easy to see that the automorphism ψ = ϕ_{0}^{−1} ∘ ϕ satisfies the following conditions:

ψ(\frac{∂}{∂x_{i}} ) = \frac{∂}{∂x_{i}} + ∑_{j=1}^{m} f_{ij} \frac{∂}{∂y_{j}} , i = 1, …, n;

ψ(\frac{∂}{∂y_{i}} ) = \frac{∂}{∂y_{i}} i = 1, …, m,

where f_{ij} are some homogeneous polynomials of degree k in x_{1}, …, x_{n}. Then for all i, j = 1, …, n we have

0 = ψ([\frac{∂}{∂x_{i}} , \frac{∂}{∂x_{j}} ]) = ∑_{l=1}^{m} (\frac{∂f_{jl}}{∂x_{i}} − \frac{∂f_{il}}{∂x_{j}} ) \frac{∂}{∂y_{l}} .

Hence, \frac{∂f_{jl}}{∂x_{i}} = \frac{∂f_{il}}{∂x_{j}} for all i, j = 1, …, n, l = 1, …, m. Therefore, for any l = 1, …, m there exist a unique polynomial g_{l} of of degree k + 1 in x_{1}, …, x_{n} such that f_{il} = \frac{∂g_{l}}{∂x_{i}}. Then it easy to see that the automorphism ψ is induced by the diffeomorphism (11), where the matrices (α_{ij}) and β_{ij} are identical and g_{i} = ∑_{α} γ_{i,α}x^{α} for all i = 1, …, m.

Since ϕ = ϕ_{0} ∘ ψ, the diffeomorphism ϕ is also induced by a certain diffeomorphism of form (11).

Fixation of a completely integrable distribution E ⊂ C complementary to V determines a reduction of the G_{0}(𝔠)-bundle of frames R on E to the subgroup

G_{0} = {ϕ ∈ G_{0}(𝔠) | ϕ(𝔟) = 𝔟},

namely,

G_{0} = {γ_{i,α} = 0} = {x̄_{i} = ∑_{j=1}^{n} α_{ij}x_{j}, ȳ_{i} = ∑_{j=1}^{m} β_{ij}y_{j}} ≅ GL(n, ℝ) × GL(m, ℝ). (15)

Indeed, if such a subdistribution is given, then the frames ϕ: 𝔠 → gr T E such that

ϕ (⟨\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}} ⟩) = E

form the desired subbundle with the structure group G_{0}. The corresponding subalgebra 𝔤_{0} in 𝔤_{0}(𝔠) has the form

𝔤_{0} = ⟨x_{i} \frac{∂}{∂x_{j}} , y_{r} \frac{∂}{∂y_{s}} | i, j = 1, …, n, r, s = 1, …, m⟩.

Let us describe the prolongation of 𝔤_{0} ⊂ 𝔤_{0}(𝔠) that can be found by means of (10). Simple computations show that 𝔤(𝔠, 𝔤_{0}) has the form:

For k = 1:

𝔤_{1} = ⟨y_{j} \frac{∂}{∂x_{i}} , x_{i}E_{1} | i = 1, …, n, j = 1, …, m⟩;
𝔤_{2} = ⟨y_{j}E_{1} | j = 1, …, m⟩;
𝔤_{p} = {0} for p ≥ 3,

where E_{s} denotes the vector field x_{1} \frac{∂}{∂x_{1}} + … x_{n} \frac{∂}{∂x_{n}} + s(y_{1} \frac{∂}{∂y_{1}} + … y_{n} \frac{∂}{∂y_{n}} ).

For k ≥ 2:
𝔤_{1} = ⟨x_{i}E_{k} | i = 1, …, n⟩;
𝔤_{p} = {0} for p ≥ 2.

Notice that for k = 1 the Lie algebra 𝔤(𝔠, 𝔤_{0}) is simple and can be identified with 𝔰𝔩(n + m + 1, ℝ) with the following grading (see [40]):

𝔤_{−2} = \{(\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ X & 0 & 0 \end{matrix}) | X ∈ ℝ^{m}\};

𝔤_{−1} = \{(\begin{matrix}0 & 0 & 0 \\ X & 0 & 0 \\ 0 & Y & 0 \end{matrix}) | X ∈ ℝ^{n}, Y ∈ Mat_{m×n}(ℝ)\};

𝔤_{0} = \{(\begin{matrix}x & 0 & 0 \\ 0 & Y & 0 \\ 0 & 0 & Z \end{matrix}) | x ∈ ℝ, Y ∈ 𝔤𝔩(n, ℝ), Z ∈ 𝔤𝔩(m, ℝ), x + tr Y + tr Z = 0\};
𝔤_{i} = {^{t}X | X ∈ 𝔤_{−i}} for i > 0.

For m = 1, k ≥ 2 similarly to Lemma 5.3 we can prove that all automorphisms of 𝔠 = 𝔠(n, 1; k) are induced by polynomial contact diffeomorphisms of ℝ^{2n+1} of the form

(x_{1}, …, x_{n}, y, z_{1}, …, z_{n}) ↦ (x̄_{1}, …, x̄_{n}, ȳ, z̄_{1}, …, z̄_{n}), where (16)

\begin{aligned}x̄_{i} & = ∑_{j=1}^{n} α_{ij}x_{j}, i = 1, …, n; \\ ȳ & = λy + ∑_{|α|=k+1} μ_{α}x^{α}; \\ z̄_{i} & = ∑_{j=1}^{n} β_{ij}z_{j} + ∑_{|α|=k} γ_{i,α}x^{α}, i = 1, …, m, \end{aligned}  (17)

where the product of the matrices (α_{ij}) and (β_{ij}) is equal to the scalar matrix λE_{n}, and the parameters γ_{i,α} are uniquely determined by parameters μ_{α} from the equality

d ( ∑_{|α|=k+1} μ_{α}x^{α}) = ∑_{i=1}^{n} ∑_{|α|=k} γ_{i,α}x^{α}dx_{i}.

Here the choice of a completely integrable distribution E ⊂ C determines a reduction of the group G_{0}(𝔠) to the subgroup

G_{0} = {x̄_{i} = ∑_{j=1}^{n} α_{ij}x_{j}, ȳ = λy, z̄_{i} = ∑_{j=1}^{n} β_{ij}z_{j}} ≅

≅ GL(n, ℝ) × ℝ^{∗} = GL(n, ℝ) × GL(1, ℝ). (18)

The corresponding subalgebra 𝔤_{0} in 𝔤_{0}(𝔠) has the form

𝔤_{0} = ⟨x_{i}z_{j}, y | i, j = 1, …, n⟩.

By formula (10) we get that the prolongation 𝔤(𝔠, 𝔤_{0}) can be described as follows:

For k = 2:
𝔤_{1} = ⟨z_{i}z_{j}, x_{i}y | i, j = 1, …, n⟩;
𝔤_{2} = ⟨yz_{i} | i = 1, …, n⟩;
𝔤_{3} = ⟨y^{2}⟩;
𝔤_{p} = {0} for p ≥ 4.
For k ≥ 3:
𝔤_{1} = ⟨x_{i}y | i = 1, …, n⟩;
𝔤_{p} = {0} for p ≥ 2.

Notice that for k = 2 the Lie algebra 𝔤(𝔠, 𝔤_{0}) is simple and is isomorphic to 𝔰𝔭(2(n + 1), ℝ), where the grading has the form (see [40]):

𝔤_{−3} = \{(\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ x & 0 & 0 & 0 \end{matrix}) | x ∈ ℝ\};

𝔤_{−2} = \{(\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ X & 0 & 0 & 0 \\ 0 & ^{t}X & 0 & 0 \end{matrix}) | X ∈ ℝ^{n}\};

𝔤_{−1} = \{(\begin{matrix}0 & 0 & 0 & 0 \\ X & 0 & 0 & 0 \\ 0 & Y & 0 & 0 \\ 0 & 0 & ^{t}X & 0 \end{matrix}) | X ∈ ℝ^{n}, Y ∈ 𝔤𝔩(n, ℝ), Y = Y′\};

𝔤_{0} = \{(\begin{matrix}x & 0 & 0 & 0 \\ 0 & Y & 0 & 0 \\ 0 & 0 & −Y′ & 0 \\ 0 & 0 & 0 & −x \end{matrix}) | x ∈ ℝ, Y ∈ 𝔤𝔩(n, ℝ)\};

𝔤_{i} = {^{t}X | X ∈ 𝔤_{−i}} for i > 0,

where X′ denotes the transpose of X ∈ 𝔤𝔩(n, ℝ) with respect to the secondary diagonal.

6. Cartan connections associated with holonomic differential equations

6.1. The notion of Cartan connection. Cartan connections may be viewed as a way of approximating a smooth manifold by a given homogeneous space of the same dimension. Namely, let G/H be a homogeneous space of a Lie group G, where H is a Lie subgroup in G. Let 𝔤 denote the Lie algebra of G, and let 𝔥 be the subalgebra of 𝔤 corresponding to the subgroup H.

Suppose M is a smooth manifold of dimension dim G/H = codim_{𝔤} 𝔥, and π: P → M is a principal H-bundle over M. Given an element X ∈ 𝔥, let X^{∗} denote the fundamental vector field on P corresponding to X. For g ∈ H, let R_{g} denote the right shift in P by the element g:

R_{g}: P → P, p ↦ pg, p ∈ P.

Definition. A Cartan connection of type (G, H) on P is a 𝔤-valued 1-form ω on the principal H-bundle P over M such that 1^{∘}. ω(X^{∗}) = X for all X ∈ 𝔥; 2^{∘}. R_{g}^{∗}ω = (Ad g^{−1})ω for all g ∈ H; 3^{∘}. ω_{p}: T_{p}P → 𝔤 is a vector space isomorphism for all p ∈ P.

Example 6.1. Let M = G/H. Then P = G may be regarded as a principal H-bundle over M. Suppose ω is the canonical left-invariant Maurer–Cartan form on G. It is easy to verify that ω satisfies the requirements of the above definition, so that ω is a Cartan connection. From now on, such connections will be called standard Cartan connections.

Two Cartan connections ω_{1} and ω_{2} of type (G, H) on principal H-bundles P_{1} and P_{2} are said to be (locally) equivalent if there exists a (local) isomorphism ϕ: P_{1} → P_{2} of these principal bundles such that ϕ^{∗}ω_{2} = ω_{1}. Thus, all Cartan connections with a given model form a category with respect to equivalence mappings.

Let s_{α} be a section of the fiber bundle π: P → M defined on an open subset U_{α} ⊂ M. It allows to identify π^{−1}(U_{α}) with U_{α} × H as follows:

ϕ_{α}: U_{α} × H → π^{−1}(U_{α}), (x, g) ↦ s_{α}(x)g.

Suppose that we are given another section s_{β}: U_{β} → P such that the intersection U_{α} ∩ U_{β} is not empty. Then there is a uniquely defined transition function ψ_{αβ} on U_{α} ∩ U_{β} with values in H such that

ϕ_{β} ∘ ϕ_{α}^{−1} : (U_{α} ∩ U_{β}) × H → (U_{α} ∩ U_{β}) × H, (x, g) ↦ (x, ψ_{αβ}g).

The fiber bundle P is uniquely determined by a covering {U_{α}}_{α∈I} of the manifold M with given transition functions ψ_{αβ}: U_{α} ∩ U_{β} → H satisfying the obvious condition that ψ_{αγ} = ψ_{αβ}ψ_{βγ} for all α, β, γ ∈ I.

Let ω be a Cartan connection on P. The section s_{α}: U_{α} → P defines a 𝔤-valued 1-form ω_{α} = s_{α}^{∗}ω on U_{α}. It is easily verified that for a different section s_{β}: U_{β} → P, the form ω_{β} = s_{β}^{∗}ω satisfies the following relation on the intersection U_{α} ∩ U_{β}:

ω_{β} = Ad (ψ_{αβ}^{−1})ω_{α} + ψ_{αβ}^{∗}θ,  (19)

where θ is the canonical left-invariant Maurer–Cartan form on H.

Remark 6.1. Formula (19) coincides with a similar formula for ordinary connections of principal fiber bundles, except that the forms ω_{α}, ω_{β} are 𝔤-valued and not 𝔥-valued as in the case of ordinary connections.

Conversely, suppose that on each submanifold U_{α} with α ∈ I, there is a 𝔤-valued 1-form ω_{α} such that 1^{∘}. relation (19) is satisfied for all α, β ∈ I; 2^{∘}. for any α ∈ I and any point x ∈ U_{α}, the mapping

T_{x}M → 𝔤/𝔥, v ↦ (ω_{α})_{x}(v) + 𝔥

is a vector space isomorphism. It is then easy to show that there exists a unique Cartan connection ω on P such that ω_{α} = s_{α}^{∗}ω for all α ∈ I.

If ω is a Cartan connection on P, the 𝔤-valued 2-form

Ω = dω + 1/2[ω, ω]

on P is called the curvature form of ω.

From the definitions of a Cartan connection and its curvature form, it follows immediately that the curvature form Ω satisfies the following conditions: 1^{∘}. R_{g}^{∗}Ω = (Ad g^{−1})Ω for all g ∈ H; 2^{∘}. Ω(X, Y) = 0 if at least one of the tangent vectors X, Y is vertical. 3^{∘}. dΩ = [Ω, ω] (structure equation). Note that the canonical Maurer–Cartan form θ on an arbitrary Lie group satisfies the equation dθ + 1/2[θ, θ] = 0. Thus, the standard Cartan connection ω of any homogeneous space G/H has zero curvature. Conversely, using the Frobenius theorem, it is easy to show that a Cartan connection with zero curvature on a principal H-bundle P is locally isomorphic to the standard Cartan connection.

Let ω be a Cartan connection on a principal H-bundle π: P → M. For any element X ∈ 𝔤, we define a vector field X^{∗} on P by letting

(X^{∗})_{p} = (ω_{p})^{−1}(X).

This definition is in agreement with the definition of X^{∗} for X ∈ 𝔥. Moreover, the field X^{∗} is vertical if and only if X ∈ 𝔥. Consider how the vector filed X^{∗} changes along the fibers of the bundle π: P → M. It follows from the definition of a Cartan connection that

ω_{pg} ∘ d_{p}R_{g} = Ad g^{−1} ∘ ω_{p} ∀p ∈ P, g ∈ H.

If we apply both sides of this equality to the tangent vector X_{p}^{∗} for some X ∈ 𝔤, we get

ω_{pg}(d_{p}R_{g}(X_{p}^{∗})) = Ad g^{−1}(X).

Hence

( Ad g^{−1}(X))_{pg}^{∗} = d_{p}R_{g}(X_{p}^{∗}) ∀p ∈ P, g ∈ H,

or in short dR_{g}(X^{∗}) = (Ad g^{−1}(X))^{∗} for all g ∈ H. Thus, the right shift R_{g} takes the fundamental vector field X^{∗} to the fundamental vector field corresponding to the vector (Ad g^{−1})X.

The function c: P → L(∧^{2}𝔤/𝔥, 𝔤) defined by

c(p): (X + 𝔥, Y + 𝔥) ↦ Ω_{p}(X^{∗}, Y^{∗})

is called the structure function of the Cartan connection ω.

Recall the natural action of H on the space L(∧^{2}𝔤/𝔥, 𝔤):

(g.φ)(X + 𝔥, Y + 𝔥) = (Ad g)φ( Ad g^{−1}(X + 𝔥), Ad g^{−1}(Y + 𝔥))

for all X, Y ∈ 𝔤. Then it is easy to see that

c(pg) = (g^{−1}).c(p) for all p ∈ P, g ∈ H.

6.2. Cartan connections associated with structures on filtered manifolds. Cartan connections are a special case of the concept of absolute parallelism. The problem of equivalence for absolute parallelisms is studied in detail in [29] and can be thought of to be solved completely. For example, all local invariants of an absolute parallelism are precisely its structure functions and their consecutive covariant derivatives. A finite number of these invariants uniquely (up to local equivalence) determines the absolute parallelism itself. If applied to Cartan connections, this means that all local invariants of a given Cartan connection can be derived from its structure function and from the covariant derivatives of this function.

Therefore, Cartan connections are a convenient object to study and, together with ordinary connections, play an important part in investigating geometric structures. Our immediate goal is to prove that there is a unique Cartan connection naturally associated with each holonomic differential equation. Moreover, we wish to construct a full functor from the category of holonomic differential equations with contact transformations as morphisms to the category of Cartan connections in which the morphisms are equivalence mappings. This will enable us to reduce the problem of equivalence of holonomic differential equations to the well-studied problem of equivalence of absolute parallelisms.

To this end, we shall use a general criterion for the existence of Cartan connections that are naturally associated with G_{0}-structures of type 𝔪 on filtered manifolds. The criterion is due to Morimoto [22].

Theorem 2 ([22]). Let 𝔪 be a fundamental graded Lie algebra. Let G_{0} be a subgroup in G_{0}(𝔪). Let 𝔤_{0} be the subalgebra of 𝔤_{0}(𝔪) corresponding to G_{0}, 𝔤 = 𝔤(𝔪, 𝔤_{0}) the prolongation of 𝔤_{0}, and 𝔥 = ⊕_{p≥0}𝔤_{p} its non-negative part.

Assume that the prolongation 𝔤 is finite-dimensional and that there exist a positive definite bilinear form

(,): 𝔤 × 𝔤 → ℝ,

a mapping τ: 𝔥 → 𝔤 and a mapping τ_{0} : G_{0} → G_{0} such that 1. (𝔤_{p}, 𝔤_{q}) = 0 for p ≠ q;

2. τ(𝔤_{p}) ⊂ 𝔤_{−p} for p ≥ 0, and ([A, x], y) = (x, [τ(A), y]) ∀x, y ∈ 𝔤, A ∈ 𝔥;

3. (ax, y) = (x, τ_{0}(a)y) for x, y ∈ 𝔤, a ∈ G_{0}. Then there exists a full functor from the category of G_{0}-structures of type 𝔪 to the category of Cartan connections of type (G, H), where G and H are the Lie groups with Lie algebras 𝔤 and 𝔥 respectivelywhich are determined from G_{0} in some natural manner .

Let us briefly describe how this functor can be constructed. Suppose we are under conditions of Theorem 2. Then the quotient space 𝔤/𝔥 can be identified with 𝔪 = ⊕_{i<0}𝔤_{i}. The structure function c of a Cartan connection of type G/H takes values in the space Hom(∧^{2}𝔪, 𝔤), which can be viewed as the subspace of all 2-cochains in the cohomology complex C(𝔪, 𝔤) (here 𝔤 is viewed as a natural 𝔪-module). We can extend the bilinear form (,) on 𝔤 to the positive definite bilinear form on the complex C(𝔪, 𝔤) and define the mapping ∂^{∗}: C^{k+1}(𝔪, 𝔤) → C^{k}(𝔪, 𝔤) adjoint to the cochain operator ∂: C^{k}(𝔪, 𝔤) → C^{k+1}(𝔪, 𝔤). Then Morimoto proves in [22] that for a given G_{0}-structure of type 𝔪 on the filtered manifold M, there exists a naturally associated principal H-bundle π: P → M, which can be obtained by a series of prolongations and reductions of the G_{0}- structure, and a unique 𝔤-valued 1-form ω on P such that

(a) ω is a Cartan connection of type (G, H) on P;

(b) its structure function c takes values in the subspace

W = ker ∂^{∗} ∩ F^{1} Hom(∧^{2}𝔪, 𝔤),

where

F^{1} Hom(∧^{2}𝔪, 𝔤) = {α ∈ Hom(∧^{2}𝔪, 𝔤) | α(𝔤_{i}, 𝔤_{j}) ⊂ ⊕_{k>i+j}𝔤_{k} ∀i, j < 0} .

In the case when 𝔤 is semisimple, these conditions determine the normal Cartan connection, as described in [14, 32].

6.3. Fundamental invariants. As was pointed out before, all local invariants of any Cartan connection ω can be derived from its structure function by means of covariant differentiation. In the case of Cartan connections associated with G_{0}-structures of type 𝔪, described by Theorem 2, it turns out that in order to construct all local invariants it suffices to consider only the harmonic component of the structure function of ω. For semisimple algebras, this result was proved in [32], and it can, with almost no changes, be carried over to the case when the conditions of Theorem 2 are satisfied.

Let 𝔪 be a fundamental graded Lie algebra and let G_{0} be a subgroup in G_{0}(𝔪) such that the conditions of Theorem 2 are fulfilled. Then any G_{0}- structure of type 𝔪 determines a unique (up to the equivalence) Cartan connection ω on some principal H-bundle P such that its structure function c satisfies the condition (b) above.

The space C^{2}(𝔪, 𝔤) is naturally supplied with a ℤ-grading:

C^{2}(𝔪, 𝔤) = ∑_{k∈ℤ}C_{k}^{2}(𝔪, 𝔤),

where

C_{k}^{2}(𝔪, 𝔤) = {α ∈ C^{2}(𝔪, 𝔤) | α(𝔤_{i}, 𝔤_{j}) ⊂ 𝔤_{i+j+k}}.

Corresponding to this grading, the structure function c can be decomposed as c = ∑_{k∈ℤ} c_{k}, where c_{k} is a C_{k}^{2}(𝔪, 𝔤)-valued function. Then the condition (b) on connection ω can be rewritten in the following way:

\begin{matrix}c_{k} = 0 for k ≤ 0; \\ ∂^{∗}c_{k} = 0 for k > 0. \end{matrix}  (20)

Let us define the Laplace operator Δ = ∂ ∘ ∂^{∗} + ∂^{∗} ∘ ∂ on C^{2}(𝔪, 𝔤). Then Δ preserves the grading of C^{2}(𝔪, 𝔤) and it is well-known that

C^{2}(𝔪, 𝔤) = im ∂ ⊕ im ∂^{∗} ⊕ ker Δ,

where ker Δ is isomorphic to the cohomology space H^{2}(𝔪, 𝔤). According to this decomposition the structure function c can be represented as c = H(c)+c̃, where c̃ = Δc, and H(c) = c − c̃ is called the harmonic component of c.

Let V be an arbitrary finite-dimensional vector space and f a smooth function f: P → V. Denote by L_{0}(f) the space of all functions of the form ⟨f, v^{∗}⟩, where v^{∗} ∈ V^{∗} and by L(f) the algebra generated by elements from L_{0}(f) and all their covariant derivatives. For example, the algebra L(c) consists of local invariants of the connection ω which uniquely (up to the local equivalence) determine ω.

The following result was proved by Tanaka [32] in case, when 𝔤 is semisimple, and can be extended without changes to an arbitrary algebra 𝔤 = 𝔤(𝔪, 𝔤_{0}) that satisfies conditions of Theorem 2.

Theorem 3. We have L(c) = L(H(c)). In particular, the structure function c vanishes if and only if its harmonic part vanishes.

Following Cartan [6] we call any basis of L_{0}(H(c)) the fundamental system of invariants of the connection ω.

Example 6.2. The spaces H^{2}(𝔪, 𝔤) for simple graded Lie algebras 𝔤 were computed by Kostant [16] (see also [40]). Based on these results let us describe explicitely the positive part ∑_{k>0} H_{p}^{2}(𝔪, 𝔤) of the cohomology space H^{2}(𝔪, 𝔤) in the case when n = 1, m ≥ 2, k = 1. In this case the Lie algebra 𝔤 = 𝔤(𝔠, 𝔤_{0}) can be identified with 𝔰𝔩(n + 2, ℝ), where the subalgebra 𝔥 = ∑_{p≥0} 𝔤_{p} is a parabolic subalgebra of the form:

𝔥 = \{(\begin{matrix}x & ∗ & ∗ \\ 0 & y & ∗ \\ 0 & 0 & Z \end{matrix}) | x, y ∈ ℝ, Z ∈ 𝔤𝔩(m, ℝ), x + y + tr Z = 0\} .

From [40] it follows that the positive part of the cohomology space H^{2}(𝔠, 𝔤), considered as 𝔤_{0}-module, has two irreducible components W_{1} and W_{2} that can be explicitely described as follows.

The 𝔤_{0}-module W_{1} consists of all 2-cochains α ∈ C^{2}(𝔠, 𝔤) of the following form:

α: (\begin{matrix}0 & 0 & 0 \\ x_{1} & 0 & 0 \\ Y_{1} & Z_{1} & 0 \end{matrix}) ∧ (\begin{matrix}0 & 0 & 0 \\ x_{2} & 0 & 0 \\ Y_{2} & Z_{2} & 0 \end{matrix}) ↦ (\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & x_{1}ϕ(Y_{2}) − x_{2}ϕ(Y_{1}) & 0 \end{matrix}) ,

where ϕ is an arbitrary mapping from ℝ^{m} to ℝ^{m} with zero trace.

Similarly, the 𝔤_{0}-module W_{1} consists of all 2-cochains α of the form:

α: (\begin{matrix}0 & 0 & 0 \\ x_{1} & 0 & 0 \\ Y_{1} & Z_{1} & 0 \end{matrix}) ∧ (\begin{matrix}0 & 0 & 0 \\ x_{2} & 0 & 0 \\ Y_{2} & Z_{2} & 0 \end{matrix}) ↦ (\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & ψ(Y_{1}, Z_{2}) − ψ(Y_{2}, Z_{1}) & 0 \end{matrix}) ,

where ψ is a mapping from ℝ^{m} × ℝ^{m} to 𝔰𝔩(m, ℝ) such that the mapping

ℝ^{m} × ℝ^{m} × ℝ^{m} → ℝ^{m}, (X, Y, Z) ↦ ψ(X, Y)Z

is symmetric.

6.4. Characteristic Cartan connections for holonomic differential equations. Recall that for each holonomic differential equation E ⊂ J^{k+1}(M, n), dim M = n + m with (k, m) ≠ (1, 1), there is a naturally associated G_{0}- structure of type 𝔠 = 𝔠(n, m; k), where the subgroup G_{0} ⊂ G_{0}(𝔠) is defined by formula (15) for m ≥ 2 and by formula (18) for m = 1.

Theorem 4. The fundamental graded Lie algebra 𝔠 = 𝔠(n, m; k) and the the subgroup G_{0} ⊂ G_{0}(𝔠) corresponding to a holonomic differential equation E ⊂ J^{k}(M, n) satisfy the condition of Theorem 2 provided that m ≥ 2, k ≥ 1 or m = 1, k ≥ 2.

Proof. In cases k = 1, m ≥ 2 and k = 2, m = 1 the Lie algebra 𝔤(𝔠, 𝔤_{0}) is simple, and the construction of the required scalar product and the mappings τ, τ_{0} is well-known (see [15, 32]). In all other cases the structure of the graded Lie algebra 𝔤(𝔠, 𝔤_{0}) can be described as follows.

Let V_{1} = ℝ, V_{2} = ℝ^{n}, V = V_{1} ⨁ V_{2}, W = ℝ^{m}, which we regard as ℤ-graded vector spaces by setting deg V_{1} = 0, deg V_{2} = −1, and deg W = −k − 1. Then the Lie algebras 𝔤𝔩(W) × 𝔰𝔩(V)(⊂ W ⨂ W^{∗} ⨁ V ⨂ V^{∗}) can be considered as a graded Lie algebra with the induced ℤ-grading; deg V_{2}^{∗} = 1, deg(V_{1} ⨂ V_{2}^{∗}) = 1,etc. Let S^{k}(V^{∗}) be the k-th symmetric power of V^{∗}, then W ⨂ S^{k}(V^{∗}) is a graded 𝔤𝔩(W) × 𝔰𝔩(V) module with the natural actions of 𝔤𝔩(W) on W^{∗} and 𝔰𝔩(V) on S^{k}(V^{∗}). Taking the semi-direct product (𝔰𝔩(V) × 𝔤𝔩(W)) ⋌ (S^{k}(V) ⊗ W^{∗}), we have a graded Lie algera, and it is this graded Lie algebra that 𝔤(𝔠, 𝔤_{0}) is identified with. In fact, the identification is given as follows:

For m ≥ 2, k ≥ 2 we set

𝔰𝔩(V) = ⟨\frac{∂}{∂x_{i}} | i = 1, …, n⟩ ⊕ ⟨x_{i} \frac{∂}{∂x_{j}} | i, j = 1, …, n⟩ ⊕ ⟨x_{i}E_{k} | i = 1, …, n⟩;

𝔤𝔩(W) = ⟨y_{i} \frac{∂}{∂y_{j}} | i, j = 1, …, m⟩;

S^{k}(V) ⊗ W^{∗} = ⟨x^{α} \frac{∂}{∂y_{i}} | i = 1, …, m, |α| ≤ k⟩ .

For m = 1, k ≥ 3 we set

𝔰𝔩(V) = ⟨z_{i} | i = 1, …, n⟩ ⊕ ⟨x_{i}z_{j} | i, j = 1, …, n⟩ ⊕ ⟨x_{i}y | i = 1, …, n⟩; 𝔤𝔩(W) = ⟨y⟩; S^{k}(V) ⊗ W^{∗} = ⟨x^{α} | |α| ≤ k⟩.

Let us define the scalar products on V and W as standard scalar products on ℝ^{n+1} and ℝ^{m} and naturally extend them to the scalar product on S^{k}(V)⊗W^{∗}. We also define the scalar products on 𝔰𝔩(V) and 𝔤𝔩(W) by:

(X, Y) = tr^{t}XY, where X, Y ∈ 𝔰𝔩(V) or X, Y ∈ 𝔤𝔩(W).

Assuming that the subspaces 𝔰𝔩(V), 𝔤𝔩(W) and S^{k}(V) ⊗ W^{∗} are mutually orthogonal we obtain the scalar product on the Lie algebra 𝔤(𝔠, 𝔤_{0}).

Since 𝔥 = 𝔤_{0} ⊕ 𝔤_{1} lies in 𝔰𝔩(V) × 𝔤𝔩(W), we can put τ(X) = ^{t}X for X ∈ 𝔥. Then τ(𝔤_{0}) = 𝔤_{0} and τ(𝔤_{1}) ⊂ 𝔤_{−1}. Finally, the group G_{0} can be naturally identified with GL(n, ℝ) × GL(m, ℝ) and we let τ_{0}(X) = ^{t}X for all X ∈ G_{0}.

It is easy to check that the scalar product on 𝔤(𝔠, 𝔤_{0}) and the mappings τ, τ_{0} thus defined satisfy the conditions of Theorem 2.

Corollary. If (k, m) ≠ (1, 1), then there exists a full functor from the category of holonomic differential equations of order k + 1 with n independent and m dependent variables to the category of Cartan connections of type (G, H).

Proof. Indeed, by the results of Section 5 we have the full functor from the category of ordinary differential equations to the category of G_{0}-structures of type 𝔠(k, m), and by the Theorems 2 and 4 it can be extended to the category of Cartan connections of type (G, H).

7. Examples

In this section we consider two cases of systems of ordinary differential equations when the Lie algebra 𝔤(𝔠, 𝔤_{0}) is simple. As a demonstration of our approach we explicitly compute fundamental invariants in these cases, which gives us, in particular, explicit criterion under which these systems of differential equations are contactly equivalent to trivial ones.

7.1. Systems of second-order ODE’s. Consider a system of second-order ordinary differential equations of the form

y′′_{i} = f_{i}(x, y_{1}, …, y_{m}, y′_{1}, …, y′_{m}), i = 1, … m  (21)

with m ≥ 2. This system determines a holonomic differential equation E ⊂ J^{2}(ℝ^{m+1}, 1) whose solutions are 1-dimensional submanifolds in ℝ^{m+1}. Thus, we can use Theorems 2 and 4 to study this system, where k = 1, n = 1.

The functions x, y_{1}, …, y_{m}, p_{1} = y_{1}^{1}, …, p_{m} = y_{m}^{1} form a coordinate system on the equation E, and relative to this coordinate system, the distributions E, V have the form

E = ⟨\frac{∂}{∂x} + p_{1} \frac{∂}{∂y_{1}} + ⋯ + p_{m} \frac{∂}{∂y_{m}} ⟩;
V = ⟨\frac{∂}{∂p_{1}} , …, \frac{∂}{∂p_{m}} ⟩ .

We choose the following co-frame on E:

τ = dx;
θ^{i} = dy_{i} − p_{i} dx, i = 1, …, m;
π^{i} = dp_{i} − f_{i} dx, i = 1, …, m.

Then for any function f ∈ C^{∞}(E), we have

df = \frac{df}{dx}ω + ∑_{i=1}^{m} \frac{∂f}{∂y_{i}} θ^{i} + ∑_{i=1}^{m} \frac{∂f}{∂p_{i}} π^{i};

where

\frac{df}{dx} = \frac{∂f}{∂x} + ∑_{i=1}^{m} p_{i} \frac{∂f}{∂y_{i}} + ∑_{i=1}^{m} f_{i} \frac{∂f}{∂p_{i}} .

By Theorem 2, there is a Cartan connection on E with model G/H that is naturally associate with the pair (E, V), where

\begin{matrix}G = SL(m + 2, ℝ), \\ H = \{(\begin{matrix}a & ∗ & ∗ \\ 0 & b & ∗ \\ 0 & 0 & C \end{matrix}) | a, b ∈ ℝ^{∗}, C ∈ GL(m, ℝ), ab det C = 1\} . \end{matrix}

We shall now outline how this connection can be constructed explicitly and compute the fundamental invariants for this particular case. Let ω: P → 𝔰𝔩(m+2, ℝ) be the Cartan connection that we are looking for, where π: P → E is a principal H-bundle. We shall write ω as follows:

ω = (\begin{matrix}ω_{−1}^{−1} & ω_{0}^{−1} & ω_{1}^{−1} & … & ω_{m}^{−1} \\ ω_{−1}^{0} & ω_{0}^{0} & ω_{1}^{0} & … & ω_{m}^{0} \\ ω_{−1}^{1} & ω_{0}^{1} & ω_{1}^{1} & … & ω_{m}^{1} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ ω_{−1}^{m} & ω_{0}^{m} & ω_{1}^{m} & … & ω_{m}^{m} \end{matrix})

We fix the section s: E → P uniquely defined by the conditions

s^{∗}ω_{−1}^{i} = θ^{i} = dy_{i} − p_{i} dx, i = 1, …, m;
s^{∗}ω_{−1}^{0} = τ = dx;
s^{∗}(ω_{0}^{0} − ω_{−1}^{−1}) ⊂ ⟨π^{1}, …, π^{m}⟩.

Let Ω be the curvature tensor of ω, and let ω̄ = s^{∗}ω, \overline{Ω} = s^{∗}Ω.

Now, from the condition (20) on the structure function of ω we obtain

ω̄_{0}^{i} = π^{i} − \frac{1}{2} ∑_{j=1}^{m} \frac{∂f_{i}}{∂p_{j}} θ^{j}, i = 1, …, m;
ω̄_{0}^{0} − ω̄_{−1}^{−1} = 0;
ω̄_{j}^{i} − δ_{j}^{i}ω̄_{−1}^{−1} = −\frac{1}{2} (\frac{∂f_{i}}{∂p_{j}} τ + ∑_{k=1}^{m} \frac{∂^{2}f_{i}}{∂p_{j}∂p_{k}} ) , i, j = 1, …, m;

ω̄_{0}^{−1} = aτ + ∑_{i=1}^{m} b_{i}θ^{i};

ω̄_{i}^{0} = ∑_{j=1}^{m} c_{ij}θ^{j}, i = 1, …, m.

where a, b_{i}, c_{ij} are certain functions on E such that c_{ij} = c_{ji} for all i, j = 1, …, m.

A straightforward computation shows that

\overline{Ω}_{0}^{i} = ∑_{j=1}^{m} Â_{j}^{i}τ ∧ θ^{j} + ∑_{1≤k<l≤m} B̂_{kl}^{i}θ^{k} ∧ θ^{l},

where

Â_{j}^{i} = A_{j}^{i} − δ_{j}^{i}a,
A_{j}^{i} = \frac{∂f_{i}}{∂y_{j}} − \frac{1}{2} \frac{d}{dx} (\frac{∂f_{i}}{∂p_{j}} ) + \frac{1}{4} ∑_{k=1}^{m} \frac{∂f_{i}}{∂p_{k}} \frac{∂f_{k}}{∂p_{j}};
B̂_{kl}^{i} = B_{kl}^{i} − B_{lk}^{i} + δ_{k}^{i}b_{l} − δ_{l}^{i}b_{k},
B_{kl}^{i} = −\frac{1}{2} \frac{∂^{2}f_{i}}{∂p_{k}∂p_{l}} + \frac{1}{4} ∑_{j=1}^{m} \frac{∂^{2}f_{i}}{∂p_{j}∂p_{k}} \frac{∂f_{j}}{∂p_{l}} .

From (20) it follows that the tensors Â_{j}^{i} and B̂_{jk}^{i} have zero trace. Hence

a = \frac{1}{m} ∑_{k=1}^{m} A_{k}^{k};

b_{i} = \frac{1}{1 − m} ∑_{k=1}^{m}(B_{ki}^{k} − B_{ik}^{k}), i = 1, …, m.

The tensor Â_{j}^{i} of trace zero is one of the two fundamental invariants that we are looking for, while the tensor B̂_{kl}^{i} can be obtained from Â_{j}^{i} by means of covariant differentiation.

In a similar manner we find that the second fundamental invariant has the form

Ĉ_{jkl}^{i} = C_{jkl}^{i} + δ_{l}^{i}c_{jk} + δ_{k}^{i}c_{jl} + δ_{j}^{i}c_{kl},

where

C_{jkl}^{i} = −\frac{1}{2} \frac{∂^{3}f_{i}}{∂p_{j}∂p_{k}∂p_{l}} ,

while the functions c_{ij} are uniquely determined by the condition that the tensor Ĉ_{jkl}^{i} be of zero trace. It follows that

c_{ij} = \frac{1}{2(m + 2)} ∑_{k=1}^{n} \frac{∂^{3}f_{k}}{∂p_{i}∂p_{j}∂p_{k}} .

In particular, we find that the system of equations (21) is contactly equivalent to the trivial system of equations if and only if Â_{j}^{i} = Ĉ_{jkl}^{i} = 0 for all i, j, k, l = 1, …, m (cp. [9, 34]).

7.2. Third-order ODE’s. Consider a third-order ordinary differential equation of the form

y′′′ = f(x, y, y′, y′′).  (22)

Relative to the coordinate system (x, y, y^{(1)}, y^{(2)}, y^{(3)}) on J^{3}(ℝ^{2}, 1) corresponding to the standard coordinates (x, y) on the plane, the corresponding hypersurface E ⊂ J^{3}(ℝ^{2}, 1) is given by the equation y^{(3)} − f(x, y, y^{(1)}, y^{(2)}) = 0.

The functions x, y, p = y^{(1)}, q = y^{(2)}, restricted to E, form a coordinate system. In these coordinates, the completely integrable distributions E, V on E have the form

E = ⟨\frac{∂}{∂x} + p\frac{∂}{∂y} + q \frac{∂}{∂p} + f(x, y, p, q)\frac{∂}{∂q} ⟩ ,
V = ⟨\frac{∂}{∂q} ⟩ .

Then, by Theorems 2 and 4, there exist a principal H-bundle π: P → E and a Cartan connection ω of type (G, H) on P, naturally associated with equation (22), where G = SP (4, ℝ) and H is a Borel subgroup in SP (4, ℝ). The corresponding pair of Lie algebras (𝔤, 𝔥) has the form

𝔤 = 𝔰𝔭(4, ℝ) = \{(\begin{matrix}x_{5} & x_{7} & x_{9} & x_{10} \\ x_{1} & x_{6} & x_{8} & x_{9} \\ x_{3} & x_{2} & −x_{6} & x_{9} \\ x_{4} & x_{3} & −x_{1} & −x_{5} \end{matrix})\} , 𝔤_{0} = \{(\begin{matrix}x_{5} & x_{7} & x_{9} & x_{10} \\ 0 & x_{6} & x_{8} & x_{9} \\ 0 & 0 & −x_{6} & x_{9} \\ 0 & 0 & 0 & −x_{5} \end{matrix})\} .

Denote by e_{1}, …, e_{10} the corresponding basis of the Lie algebra 𝔰𝔭(4, ℝ). From (20) it follows that the curvature form of ω has the form Ω = ∑_{i=1}^{10} Ω_{i}e_{i}, where

Ω_{1} = Ω_{3} = Ω_{4} = Ω_{5} = 0;
Ω_{2} = I_{1}ω_{1} ∧ ω_{4};
Ω_{6} = Aω_{1} ∧ ω_{4};
Ω_{7} = ω_{1} ∧ (Bω_{3} + Cω_{4});
Ω_{8} = I_{2}ω_{2} ∧ ω_{3} mod⟨ω_{4}⟩.

The functions I_{1}, I_{2} are fundamental invariants of ω, and with a suitable choice of the section s: E → P, their inverse images have the form

s^{∗}I_{1} = −\frac{1}{2}f_{y} − \frac{1}{6}f_{p}f_{q} − \frac{1}{27}f_{q}^{3} + \frac{1}{4}f_{px} + \frac{1}{6}f_{q}f_{qx} − \frac{1}{2}f_{qxx};
s^{∗}I_{2} = −\frac{4}{3}f_{qqqq}.

Here f_{y}, f_{p}, f_{q} denote the partial derivatives of f with respect to y, p, q respectively, while f_{x} denotes the total derivative with respect to x:

f_{x} = \frac{df}{dx} = \frac{∂f}{∂x} + p\frac{∂f}{∂y} + q \frac{∂f}{∂p} + f \frac{∂f}{∂q} .

In particular, we find that an arbitrary third-order ordinary differential equation 22 is contactly equivalent to the trivial equation y′′′ = 0 if and only if the function f is cubic in q and the invariant I_{1} is identically equal to zero (cp. [3, 8]).

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International Sophus Lie Centre E-mail address: Doubrov@islc.minsk.by, Boris.Komrakov@wis.kuleuven.ac.be

Received May 19, 1999