Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 23, 2006, 151–181 \begin{matrix}c \\ ◯ \end{matrix} Alexei Kushner

Alexei Kushner

ALMOST PRODUCT STRUCTURES AND MONGE-AMPÈRE EQUATIONS

(submitted by V.V. Lychagin)

Abstract. Tensor invariants of an almost product structure are constructed. We apply them to solving the problem of contact equivalence and the problem of contact linearization for Monge-Ampère equations.

In this paper we study almost product structures. By this structure we mean an ordered set P of real or complex distributions on a smooth manifold N such that the tangent space T_{a}N (or its complexification T_{a}N^{ℂ}) splits into the direct sum of the subspaces from P at each point a ∈ N.

Almost product structures in above sense arise in various forms: as a fields of semi-simple endomorphisms, as non-holonomic webs, and (what is most important for us) as Monge-Ampère equations.

An interpretation of a Monge-Ampère equation as an almost product structure allows us to solve the problems of contact linearization and the problem of contact equivalence for Monge-Ampère equations.

The solution of the first problem for non-degenerated Monge-Ampère equations was annonced by the author in [9]. Here we give a complete proof.

In the series of papers (see [3], [4], [5], [6], [7], [8]) for generic symplectic Monge-Ampère equations was constructed an e-structure (absolute parallelism). In this paper we solve similar problem in contact case. After this result the problem of contact equivalence of Monge-Ampère equations becomes trivial.

An history (not only!) of classification problem for Monge-Ampère equations can be founded by reader in [10].

Moreover, we suppose that the results obtained in the paper for general almost product structure are interesting without their applications to Monge-Ampère equations. For example, Monge-Ampère structure considered in the first section is no else than a non-integrable CR-structure (Cauchy-Riemann structure) in an elliptic case or a non-integrable para-CR-structure in a hyperbolic case on a 5-dimensional contact smooth manifold.

A few words about the structure of the paper.

In the first section we recall a notion of an almost product structure and give some important examples.

The tensor invariants of an almost product structure are constructed in the second section. To this end we consider a decomposition of the de Rham complex to the generated by an almost product structure direct sum. The main result of this section is based on differential’s structure of differential graded algebra. We explain a geometrical sense of the constructed in the previous section tensors and prove an analog of the Frobenius theorem for subdistributions of an almost product structure.

Our main results in theory of Monge-Ampère equations are presented in the last section.

In the first and second subsections of the fourth section we introduce a reader to V. Lychagin’s theory of Monge-Ampère equations and recall some definitions and notations.

In the third one we construct contact tensor invariants of hyperbolic and elliptic Monge-Ampère equations and calculate (as an example) the tensors for the non-linear wave equation.

A solution of the linearization problem for non-degenerate equations we present in the fourth subsection. As an example of applications of the result we consider the generalized Hunter-Saxton equation. This equation arises in the theory of a director field of a liquid crystal and in the geometry of Einstein-Weil spaces.

At last, in the fifth section we construct an e-structure for generic Monge-Ampère equations. We introduce the non-holonomic de Rham complex and construct the set of relative and absolute contact invariants of equations.

1. Almost product structures: definition and examples

Let N be a (real) smooth manifold and let P = (P_{1}, …, P_{r}) be an ordered set of real or complex distributions on N, i.e.

P_{i} : N ∋ a ↦ P_{i} (a) ⊂ T_{a}N ,

or

P_{i} : N ∋ a ↦ P_{i} (a) ⊂ T_{a}N^{ℂ} = T_{a}N ⊗ ℂ,

i = 1, …, r.

The set P is called an almost product structure (AP-structure) on N if at each point a ∈ N the tangent space T_{a}N (for real distributions) or its complexification T_{a}N^{ℂ} (for complex ones) splits in the direct sum of the subspaces P_{1} (a) , …, P_{r} (a), i.e.

⨁_{i=1}^{r}P_{i} (a) = T_{a}N (or T_{a}N^{ℂ}).

Let D(N) and Ω^{s} (N) be the modules of vector fields and differential s-forms on N respectively. A submodule of vector fields from the distribution P_{i} we denote by D(P_{i}):

D(P_{i}) =^{def} {X ∈ D(P)|X_{a} ∈ P (a)∀a ∈ M} .

For the distribution P_{i} we define a submodule of vanishing on the distributions P_{j} (j = 1, …, r; j ≠ i) differential s-forms:

Ω^{s}(P_{i}) =^{def} {α ∈ Ω^{s} (N)| X⌋α = 0 ∀X ∈ D(P_{j}), j = 1, …, r; j ≠ i}.

Let us consider some examples.

Example 1 (Field of semi-simple endomorphisms). Let A be a field of endomorphisms on a smooth n-dimensional manifold N. Suppose that at each point a ∈ N the linear operator A_{a} : T_{a}N → T_{a}N is semi-simple.

Let λ_{1}, …, λ_{r} be eigenvalues of A and let p_{i} is a multiplicity of the eigenvalue λ_{i} (i = 1, …, r; p_{1} + ⋯ + p_{r} = n). Then T_{a}N^{ℂ} splits into the direct sum of eigensubspaces P_{1} (a) , …, P_{r} (a)^{1} of the operator A_{a}: T_{a}N^{ℂ} = ⊕_{i=1}^{r}P_{i} (a). Here dim P_{i} (a) = p_{i} (i = 1, …, r). Suppose also that p_{1}, …, p_{r} are constant. Then the maps P_{i} : a ↦ P_{i} (a) (i = 1, …, r) are complex distributions on N and the set P = (P_{1}, …, P_{r}) is a complex AP-structure.

Suppose n = 2k. If A^{2} = −1 or A^{2} = 1, then A is a classical almost complex structure (AC-structure) or classical almost product structure respectively.

1P_{i} corresponds to λ_{i}.

Example 2 (𝔣-structure). A field of endomorphisms 𝔣 on a smooth manifold N is called 𝔣-structure if 𝔣^{3} +ε𝔣 = 0, where ε = ±1 (see [16], [2]). An 𝔣-structure is called hyperbolic or elliptic if ε = −1 or ε = 1 respectively. At each point a ∈ N the tangent space T_{a}N splits into the direct sum

T_{a}N = 𝔐_{a} ⊕ 𝔏_{a},

where 𝔐_{a} = ker 𝔣_{a} and 𝔏_{a} = Im 𝔣_{a}, dim 𝔐_{a} = m, dim 𝔏_{a} = 2n.

For hyperbolic 𝔣-structure the tangent space T_{a}N at each point a ∈ N splits into the direct sum of real eigensubspaces of 𝔣_{a} that are corresponding to eigenvalues 0, +1 and −1:

T_{a}N = 𝔐_{a} ⊕ 𝔏_{a}^{+} ⊕ 𝔏_{a}^{−},

For elliptic 𝔣-structure the complexification of T_{a}N splits into the direct sum of complex eigensubspaces of 𝔣_{a} that are corresponding to eigenvalues 0, ι = \sqrt{−1} and −ι:

T_{a}N^{ℂ} = 𝔐_{a}^{ℂ} ⊕ 𝔏_{a}^{+} ⊕ 𝔏_{a}^{−}.

The splitting generates an almost product structures on N.

Example 3 (Almost contact structures). The triplet (η, ξ, Φ), where η is a contact differential 1-form, ξ is a vector field, and Φ is a field of endomorphism on a smooth manifold N is called an almost contact structure if the following conditions hold:

(1) η (ξ) = 1,

(2) η ∘ Φ = 0,

(3) Φ (ξ) = 0,

(4) Φ^{2} = −ε + η ⊗ ξ, where ε = ±1 (see [2]). Similar to the previous example the almost

contact structure is called hyperbolic or elliptic if ε = −1 or ε = 1 respectively. An almost contact structure generates an almost product structure with three distributions: one of them (the one-dimensional distribution) is generated by the vector field ξ and other two are generated by eigensubspaces of the restriction Φ_{a}|_{ker η_{a}}, a ∈ N.

The following example is main for us.

Example 4 (Monge-Ampère structure on a 5-dimensional manifold). Let N be a 5-dimensional smooth manifold which is endowed with a contact structure C : a ⟼ C(a) ⊂ T_{a}N and let J be a ”non-holonomic” field of endomorphisms^{2} J, J_{a} : C(a) → C(a), J^{2} = ε, where ε = ±1.

2i.e. J is a section of the vector bundle π : N ∋ a → α_{a} ⊗ X_{a} ∈ Λ^{1} (C(a)^{∗}) ⊗ C(a).

Suppose that the distribution C is generated by the differential 1-form U on N: C(a) = ker U_{a} for each a ∈ N^{3}. Let Ω_{a} is a restriction of dU to C(a): Ω_{a} = dU|_{C(a)}. Then Ω_{a} is a symplectic structure on C(a). Assume that J_{a} is symmetric with respect to Ω_{a},i.e.

Ω_{a} (J_{a}X, Y) = Ω_{a} (X, J_{a}Y)

∀X, Y ∈ C(a), ∀a ∈ N.

The pair (C, J) is called a Monge-Ampère structure (MA-structure) on N. This structure is called hyperbolic or elliptic if ε = 1 or ε = −1 respectively.

A Monge-Ampère structure generates an almost-product structure P = (C_{+,}l, C_{−}) on N, where the 2-dimensional distributions^{4} C_{±} : a ↦ C_{±} (a) are generated by the eigensubspaces C_{±}(a) of J and 1-dimensional distribution l is generated by the intersection of the first derivatives^{5} C_{+}^{(1)}, C_{−}^{(1)} of the distributions C_{+} and C_{−}: l(a) = C_{+}^{(1)}(a) ∩ C_{−}^{(1)}(a).

Indeed (see [12]), C_{+}(a) and C_{−}(a) are skew-orthogonal with respect to Ω_{a} or its complexification Ω_{a}^{ℂ}. Moreover, Ω_{a} is non-degenerate on C_{+}(a) and C_{−}(a). Then for hyperbolic MA-structure

U ([X_{±}, Y_{±}]) = −dU (X_{±}, Y_{±}) ≠ 0

for any X_{±}, Y_{±} ∈ D(C_{±}). Similarly, U^{ℂ} ([X_{±}, Y_{±}]) ≠ 0 for elliptic one. This means that the tangent space T_{a}N (for a hyperbolic MA-structure) or its complexification T_{a}N^{ℂ} (for an elliptic one) splits into the direct sum

T_{a}N (or T_{a}N^{ℂ}) = C_{+}(a) ⊕ l(a) ⊕ C_{−}(a).

Note that in the elliptic case the complex distribution l is generated by a real vector field. Indeed, since the operator J_{a} is real, the subspaces C_{+} (a) and C_{−} (a) are complex conjugate: C_{+} (a) = \overline{C_{−} (a)}. Then the subspaces C_{+}^{(1)} (a) and C_{−}^{(1)} (a) are complex conjugated also, and its intersection is complex conjugate to itself: l (a) = \overline{l (a)}. Therefore this complex line is generated by a real vector Z: l (a) = ℂZ_{a}, Z ∈ T_{a} (N) (see [10]).

The previous example is a partial case of the non-integrable (=nonholonomic) Cauchy-Riemann or para-Cauchy-Riemann structure.

3The differential 1-form is defined up to multiplication by non-vanishing function.

4dim_{ℝ} C_{±} = 2 for hyperbolic MA-structures and dim_{ℂ} C_{±} = 2 for elliptic ones.

5The first derivative P^{(1)} of a distribution P is the distribution which is generated by the vector fields from P and by its all possible sorts of commutators.

Example 5 (CR- and para-CR-structures). A smooth manifold N is called a Cauchy-Riemann manifold or CR-manifold if there exists a distribution P on N such that at each point a ∈ N the vector space P (a) ⊂ T_{a}N endowed with a complex structure J_{a}, J_{a}^{2} = −1 and J depends on a smoothly. Analogously N is called para-CR-manifold if J_{a}^{2} = 1.

In this case P (a)^{ℂ} (or P (a) in case of para-CR-structure) splits into two eigensubspaces of the operator J_{a} and we obtain two (complex for the CR-structure and real for the para-CR-structure) subdistributions P_{1} and P_{2} of the distribution P. A (para)-CR-structure is called integrable if the distributions P_{1} and P_{2} are completely integrable, otherwise it is called non-integrable or non-holonomic.

Let us consider a non-holonomic (para)-CR-structure. Let P_{3}(a) =^{def} P_{1}^{(1)}(a)∩P_{2}^{(2)}(a) be an intersection of the first derivatives of the distributions P_{1} and P_{2} at a point a ∈ N. Suppose that P_{3} : a ↦ P_{3}(a) is a distribution. If by some reason P_{3}(a) is a compliment of P (a) to the tangent space T_{a}N, the we obtain an almost product structure P = (P_{1}, P_{2}, P_{3})on N.

2. Algebra and geometry of almost product structures

2.1. A structure of a differential graded algebra. Let A = ⊕_{k}A^{k} be a differential r-graded algebra over a field of characteristic 0 with a differential d, i.e.

d (a · b) = da · b + (−1)^{a} a · db.

Here k is a multi-index, k = (k_{1}, …, k_{r}), k_{i} ∈ {0, 1, …, n_{i}}, n_{i} ∈ ℕ are some numbers, |k| = k_{1} + ⋯ + k_{r}, and (−1)^{a}=(−1)^{deg a} where deg a =^{def} s if a ∈ A_{s} =^{def} ⊕_{|k|=s}A^{k}. We assume also that A is a super-commutative algebra, i.e.

a · b = (−1)^{ab}b · a,

and the differential and multiplication are compatible with grad:

A^{k} · A^{t} ⊂ A^{k+t}

dA_{s} ⊂ A_{s+1}

The differential d splits in the following direct sum:

d = ⨁_{|σ|=1}d_{σ} = (⨁^{r} ; i=1d_{i}) ⊕ (⨁_{|t|=1}d_{t}) ,

where σ = (σ_{1}, …, σ_{r}), σ_{j} ∈ I_{j} = {z ∈ ℤ| |z| ≤ n_{j}}, d_{σ} : A^{k} → A^{k+σ} (|σ| = 1), d_{i} = d_{(0...1_{i}...0)}^{6} and t has negative components.

Lemma 1. The operators d_{i} and d_{t} satisfy the Leibniz rule:

d_{i} (a · b) = d_{i}a · b + (−1)^{a} a · d_{i}b

and

d_{t} (a · b) = d_{t}a · b + (−1)^{a} a · d_{t}b

In particular, d_{t} is an A_{0}-homomorphism, i.e.

d_{t} (a · b) = a · d_{t}b

for any a_{0} ∈ A_{0} and for any a, b ∈ A.

Proof. We prove the second part of the Lemma only. For a ∈ A_{0} = A^{(0,...,0)} and b ∈ A^{k} we get:

d(a · b) = ∑_{|σ|=1}d_{σ}(a · b) = ∑_{i=1}^{r} d_{i}(a · b) + ∑_{|t|=1} d_{t}(a · b),

On the other hand,

d(a · b) = da · b + (−1)^{a} a · db

= (∑_{i=1}^{r} d_{i}a + ∑_{|t|=1} d_{t}a) · b + (−1)^{a} a · (∑_{i=1}^{r} d_{i}b + ∑_{|t|=1} d_{t}b) .

Therefore d_{i}a · b ∉ A^{k+t} for each i = 1, …, r and one gets that d_{t}(a · b) = a · d_{t}b. □

Example 6. Let us consider the case r = 2 and n_{1,2} = 2, i.e. k = (k_{1}, k_{2}), where k_{i} ∈ {0, 1, 2}. Then

A^{s} (N) = ⨁_{p+q=s}A^{p,q} (N) ,

s = 0, 1, 2, 4 and

d = d_{1,0} ⊕ d_{0,1} ⊕ d_{2,−1} ⊕ d_{−1,2},

where d_{2,−1}, d_{−1,2} are A_{0}-homomorphisms and d_{1,0}, d_{0,1} are differentiations (see diagram below).

61 is only on ith place.

\xymatrix{ &  & A^{2,0} \ar[dr]^{d_{0,1}} \ar[dddr]^{d_{−1,2}} \\  & A^{1,0} \ar[ur]^{d_{1,0}} \ar[dr]^{d_{0,1}} \ar[dddr]_{d_{−1,2}} &  & A^{2,1} \ar[dr]^{d_{1,0}} \\ A^{0,0} \ar[ur]^{d_{1,0}} \ar[dr]_{d_{0,1}} &  & A^{1,1} \ar[ur]^{d_{1,0}} \ar[dr]_{d_{0,1}} &  & A^{2,2} \\  & A^{0,1} \ar[uuur]^{d_{2,−1}} \ar[dr]_{d_{0,1}} \ar[ur]_{d_{1,0}} &  & A^{1,2} \ar[ur]_{d_{1,0}} \\  &  & A^{0,2} \ar[ur]_{d_{1,0}} \ar[uuur]_{d_{2,−1}}}

2.2. Tensor invariants for almost product structures. Using Lemma 1 one can construct tensor invariants of almost product structures.

First, we consider a real almost product structures on N. The vector space Λ^{s} (T_{a}^{∗}N) of exterior s-forms on T_{a}N falls into direct sum

Λ^{s} (T_{a}^{∗}N) = ⨁_{|k|=s}Λ^{k} (T_{a}^{∗}N) ,

where k is a multi-index, k = (k_{1}, …, k_{r}), k_{i} ∈ {0, 1, …, dim P_{i}} and

Λ^{k} (T_{a}^{∗}N) = ⨂_{i=1}^{r}Λ^{k_{i}}(P_{i} (a)).

Here

Λ^{k_{i}}(P_{i}(a)) =^{def} { α ∈ Λ^{k_{i}} (T_{a}^{∗}N)| X⌋α = 0

∀X ∈ P_{j}(a), j = 1, …, r; j ≠ i}.

This gives us a decomposition of the de Rham complex: the C^{∞}(N) - modules of differential s-forms Ω^{s} (N) split in the direct sum

Ω^{s} (N) = ⨁_{|k|=s}Ω^{k},  (1)

where

Ω^{k} =^{def} ⨂_{i=1}^{r}Ω^{k_{i}}(P_{i}).

In the case of complex almost product structures we have to consider the complexification Ω^{s} (N)^{ℂ} of the module Ω^{s} (N).

Then de Rham differential d splits into the following direct sum:

d = ⨁_{|σ|=1}d_{σ},

where

σ_{j} ∈ I_{j} =^{def} {z ∈ ℤ| |z| ≤ dim P_{j}}

and

d_{σ} : Ω^{k} → Ω^{k+σ}.

From Lemma 1 it follows that if one of the component t_{i} of a multiindex t is negative, then operator d_{t} is a tensor which acts from Ω^{k} to Ω^{k+t}.

The tensor d_{t} is a sum of the tensors of the type θ ⊗ X, where θ is a differential s-form and X is a (s − 1)-vector field on N.

Recall that the tensor θ ⊗ X acts on a differential (s − 1)-form α and on an s-vector field Y as

(θ ⊗ X) (α) = (X⌋α) θ,
(θ ⊗ X) (Y) = (Y ⌋θ) X

respectively. Therefore a tensor θ ⊗ X for θ ∈ Ω^{s}(P_{i}) and X ∈ D^{s−1} (P_{j}) can be regarded as a map from D^{s} (P_{i}) to D^{s−1} (P_{j}). Here D^{s} (P) is a module of s-vector fields from the distribution P.

Example 7 (Classical AP- and AC-structures on ℝ^{4}). Let us consider a classical AP-structure (or a classical AC-structure) J on ℝ^{4}. The tangent space T_{a}ℝ^{4} (or the complexification (T_{a}ℝ^{4})^{ℂ} for AC-structure) splits into the direct sum of eigensubspaces P_{1} (a) and P_{2} (a) of the operator J_{a} Here dim_{ℝ} P_{i} (a) = 2 (or dim_{ℂ} P_{i} (a) = 2 for AC-structure), i = 1, 2. The module Ω^{s} (ℝ^{4}) (or its complexification for AC-structure) falls into the direct sum of Ω^{p,q}, where p + q = s, and

Ω^{p,q} =^{def} Ω^{p}(P_{1}) ⊗ Ω^{q}(P_{2}).

(see Example 6). Moreover, we get the following decomposition of the exterior differential d : Ω^{s} (ℝ^{4}) → Ω^{s+1} (ℝ^{4}):

d = d_{1,0} ⊕ d_{0,1} ⊕ d_{2,−1} ⊕ d_{−1,2}.

The components d_{2,−1} and d_{−1,2} in this sum are tensors.

Note that for AC-structure the tensors d_{−1,2} and d_{2,−1} are complex conjugated: \overline{d_{−1,2}} = d_{2,−1}. In case d_{−1,2} = d_{2,−1} = 0 we obtain the usual Dolbeault complex.

Example 8 (Monge-Ampère structure on ℝ^{5}). Let (C, J) be a Monge-Ampère structure on ℝ^{5}. We suppose that this structure is hyperbolic, i.e. J^{2} = 1. The corresponding AP-structure is P = (C_{1,0}, l, C_{0,1}), where C_{1,0} = C_{+} and C_{0,1} = C_{−}. We get (see diagram below) the decompositions of the modules of exterior differential forms:

Ω^{0} (ℝ^{5}) = C^{∞} (ℝ^{5}) ,
Ω^{1} (ℝ^{5}) = Ω^{1,0,0} ⊕ Ω^{0,1,0} ⊕ Ω^{0,0,1},
Ω^{2} (ℝ^{5}) = Ω^{2,0,0} ⊕ Ω^{1,1,0} ⊕ Ω^{1,0,1} ⊕ Ω^{0,1,1} ⊕ Ω^{0,0,2},
Ω^{3} (ℝ^{5}) = Ω^{2,1,0} ⊕ Ω^{2,0,1} ⊕ Ω^{1,1,1} ⊕ Ω^{1,0,2} ⊕ Ω^{0,1,2},
Ω^{4} (ℝ^{5}) = Ω^{2,1,1} ⊕ Ω^{2,0,2} ⊕ Ω^{1,1,2},
Ω^{5} (ℝ^{5}) = Ω^{2,1,2},

and the decomposition of exterior differential:

\xymatrix{ &  & Ω^{2,0,0} ⋯ \\  & Ω^{1,0,0} \ar[ur]^{d_{1,0,0}} \ar[r]^{d_{0,1,0}} \ar[dr]^{d_{0,0,1}} \ar[ddr]^{d_{−1,1,1}} \ar[dddr]^{d_{−1,0,2}} & Ω^{1,1,0} ⋯ \\ Ω^{0,0,0} \ar[ur]^{d_{1,0,0}} \ar[r]^{d_{0,1,0}} \ar[dr]_{d_{0,0,1}} & Ω^{0,1,0} \ar[uur]^{d_{2,−1,0}} \ar[ur]^{d_{1,0,0}} \ar[r]^{d_{1,−1,1}} \ar[dr]^{d_{0,0,1}} \ar[ddr]_{d_{0,−1,2}} & Ω^{1,0,1} ⋯ \\  & Ω^{0,0,1} \ar[uuur]_{d_{2,0,−1}} \ar[uur]_{d_{1,1,−1}} \ar[ur]^{d_{1,0,0}} \ar[r]^{d_{0,1,0}} \ar[dr]_{d_{0,0,1}} & Ω^{0,1,1} ⋯ \\  &  & Ω^{0,0,2} ⋯}

We have the following tensors: d_{−1,1,1}, d_{1,1,−1}, d_{1,−1,1}, d_{0,−1,2}, d_{2,−1,0}, d_{2,1,−2} and d_{−2,1,2}.

Since the distributions C_{1,0} and C_{0,1} are skew-orthogonal, we get d_{1,−1,1} = 0. It is not hard to prove that the tensors d_{2,1,−2}, d_{−2,1,2}, d_{2,0,−1}, and d_{−1,0,2} are zero also. We have four two-covariant and one-contravariant tensors: d_{−1,1,1}, d_{1,1,−1}, d_{0,−1,2}, and d_{2,−1,0}.

Any tensor q_{j,k}^{s} =^{def} d_{1_{j}+1_{k}−1_{s}} : Ω^{1_{s}} → Ω^{1_{j}+1_{k}} (s ≠ k, j) can be regarded as a map

q_{j,k}^{s} : D (P_{j}) × D(P_{k}) → D(P_{s}).

Extend an action of q_{j,k}^{s} to the module D(N) × D(N) by the formula:

q_{j,k}^{s} (X, Y) =^{def} q_{j,k}^{s} (P_{j}X, P_{k}Y) ,

where P_{s} is the projector to the distribution P_{s}.

For any almost product structure P we can define a tensor field Q_{P} on N:

Q_{P} =^{def} ∑_{s,j,k (s≠j,k)} q_{j,k}^{s}.

It is not hard to see that

Q_{P} (X, Y) = − ∑_{s,j,k (s≠j,k)} P_{s} [P_{j}X, P_{k}Y] .

2.3. Subdistributions. Let

P_{I} =^{def} ⨁_{i∈I}P_{i},  (2)

where I ⊂ {1, 2, …, r}. In this Section we study the following problem: when the distribution P_{I} is completely integrable?

It is convenient to formulate answer to this question in terms of multiindices. Let Ann (P) be an annihilator of a distribution P:

Ann (P) =^{def} {α ∈ Ω^{1} (N)| α (X) = 0 ∀X ∈ D(P)}.

Let us introduce multi-indices of length r: 1 =^{def} (1, …, 1), 1_{i} =^{def} (0, …, 0, 1_{i}, 0, …, 0)^{7}, k =^{def} ∑_{i∈I} 1_{i}, and \overline{k}, where \overline{k} + k = 1, and put

(a, b) =^{def} a_{1}b_{1} + ⋯ a_{r}b_{r} for a = (a_{1}, …, a_{r}), b = (b_{1}, …, b_{r}).

71 is in the ith place only.

The distribution P_{I} describes by the index k uniquely, therefore we will denote P_{I} by P_{k} also. Then

Ann (P_{k}) = ⨁_{(1_{i},\overline{k})=1}Ω^{1_{i}},

Theorem 1. The distribution P_{k} is completely integrable if and only if the tensors d_{t} = 0 for all multi-indices t such that (t,\overline{k}) = −1.

Proof. Without loss of generality we can suppose that I = {l + 1, …, r}. Then

\overline{k} = (\underbrace{1, …, 1}_{l}, \underbrace{0 … 0})._{r−l}

Then

Ann (P_{k}) = ⟨α_{1}, …, α_{p_{1}}, α_{p_{1}+1}, …, α_{p_{1}+p_{2}}, …, α_{p_{1}+···+p_{l}}⟩ ,

where p_{i} = dim P_{i}, (α_{1}, …, α_{p_{1}}) is a free basis of Ω^{1_{1}}, (α_{p_{1}+1}, …, α_{p_{1}+p_{2}}) is a free basis of Ω^{1_{2}}, etc.

Let α_{j} ∈ Ω^{1_{j}} ⊂ Ann (P_{k}) and t = (t_{1}, …, t_{r}), where t_{j} = −1. If t_{s} = 0 for s ∈ {1, …, ĵ, …, l}, then

α_{1} ∧ ⋯ ∧ α_{p_{1}+···+p_{l}} ∧ d_{t}α_{j} = 0 ⇔ d_{t} = 0.

If t_{s} ≠ 0 for s ∈ {1, …, ĵ, …, l}, then α_{1} ∧⋯∧α_{p_{1}+···+p_{l}} ∧d_{t}α_{j} = 0. So, α_{1} ∧⋯∧α_{p_{1}+···+p_{l}} ∧dα_{j} = 0 ⇔ d_{t} = 0 for all t such that (t,\overline{k}) = −1. □

3. Monge-Ampère equations

3.1. A geometric point of view. A differential-geometric structures that generated by Monge-Ampère equations was described by V. Lychagin. In this section we recall his ideas and some his results [11], [12]. We restrict our consideration by Monge-Ampère equations with two independent variables only.

Let M be a 2-dimensional smooth manifold and let J^{1}M be the manifold of 1-jets of smooth functions on M. The manifold J^{1}M is endowed by the natural contact structure (Cartan’s distribution)

C : a ∈ J^{1}M → C(a) ⊂ T_{a}(J^{1}M)

given by the universal differential one-form U ∈ Ω^{1} (J^{1}M) (Cartan’s form): C(a) =^{def} ker U_{a}. Naturally, the form U is defined up to multiplication by non-vanishing smooth function on J^{1}M.

At each point a ∈ J^{1}M the 2-form

Ω_{a} =^{def} dU|_{C(a)} ∈ Λ^{2} (C^{∗}(a))

is the standard symplectic structure on C(a). We get a so-called nonholonomic symplectic structure

Ω : J^{1}M ∋ a ⟼ Ω_{a} ∈ Λ^{2} (C^{∗}(a))

on J^{1}M.

With any differential 2-form ω on J^{1}M we can associate a differential operator Δ_{ω} : C^{∞}(M) → Ω^{2}(M), which acts as

Δ_{ω}(v) = ω|_{j^{1}(v)(M)}.  (3)

Here v ∈ C^{∞} (M) is a smooth function and j_{1} (v) (M) ⊂ J^{1}M is the graph of 1-jet j_{1} (v), and ω|_{j^{1}(v)(M)} is the restriction of ω to j_{1} (v) (M).

The equation

E_{ω} =^{def} {Δ_{ω}(v) = 0} ⊂ J^{2}M

is called a Monge-Ampère equation.

But constructed map ”differential 2-forms” → ”differential operators” is not a one-to-one map. In order to set one-to-one map we should restrict a class of differential 2-forms and consider only effective differential 2forms.

Recall the notion of an effective form.

Differential forms on J^{1}M vanishing on any integral manifold of the Cartan distribution, and therefore producing zero differential operators, form a graded ideal of the exterior algebra Ω^{∗} (J^{1}M). We denote this ideal by

I^{∗} = ⨁_{s≥0}I^{s},

I^{s} ⊂ Ω^{s} (J^{1}M). The ideal I^{∗} is generated by forms

U ∧ α + dU ∧ β,  (4)

where α and β are some differential forms. Note also, that I^{0} = 0 and I^{s} = Ω^{s} (J^{1}M) for s ≥ 3.

Elements of the quotient module Ω^{s} (J^{1}M) /I^{s} we call effective s-forms (s ≤ 2):

Ω_{ε}^{s} (J^{1}M) ≅ Ω^{s} (J^{1}M) /I^{s}.  (5)

For each element of Ω_{ε}^{2} (J^{1}M) one can choice a unique representative ω ∈ Ω^{2} (J^{1}M) such that X_{1}⌋ω = 0 and ω ∧ dU = 0. Here X_{1} is the Reeb vector field – a contact vector field with generating function 1.

Let h be a nonvanishing smooth function on J^{1}M. It is clear that the forms ω and hω generate the same equation.

Let ω and ω̃ be effective differential 2-forms. Two Monge-Ampère equations E_{ω} and Eω̃ are (local) contact equivalent at a point a ∈ J^{1}M if there exists a contact diffeomorphism φ : J^{1}M → J^{1}M, φ (a) = a and some function h_{φ} ∈ C^{∞} (J^{1}M), h_{φ} (a) ≠ 0, such that

φ^{∗} (ω)_{ε} = h_{φ}ω̃.

Here φ^{∗} (ω)_{ε} is the effective part of φ^{∗} (ω).

Any effective differential form ω generates the non-holonomic field of endomorphisms

A_{ω} : J^{1}M ∋ a ⟼ A_{ω_{a}} ∈ End (C(a)) .  (6)

by the formula

X_{a} ⌋ ω_{a} = A_{ω,a}X_{a}⌋ Ω_{a}

for any tangent vector X_{a} ∈ C(a).

The operator A_{ω} is symmetric with respect to Ω, i.e.,

Ω (A_{ω}X, Y) = Ω (X, A_{ω}Y)

for any vector field X, Y ∈ D(C).

A function Pf (ω) ∈ C^{∞} (J^{1}M) is called a Pfaffian of the form ω if

Pf (ω) Ω ∧ Ω = ω ∧ ω.

Note that

Pf (hω) = h^{2} Pf (ω)

for a function h ∈ C^{∞} (J^{1}M).

For an effective 2-form ω the square of A_{ω} is scalar and

A_{ω}^{2} + Pf (ω) = 0.  (7)

We say that a Monge-Ampère equation E_{ω} (a form ω, an operator Δ_{ω}) are hyperbolic, elliptic or parabolic at a point a ∈ J^{1}M if Pf (ω) is negative, positive or zero at this point respectively. Hyperbolic and elliptic equations are called non-degenerate.

If Pf (ω) (a) ≠ 0, we can normalize the form ω in some neighborhood of the point a:

ω ↦ \frac{1}{\sqrt{|Pf (ω)|}}ω.

If |Pf(ω)| = 1, then the form ω is called normed. The Pfaffian of a normed form is equal to −1 for a hyperbolic form and +1 for an elliptic one.

The operator A_{ω} corresponding to the normed form ω is denoted by A. It is clear that for hyperbolic and elliptic equations we have A^{2} = 1 and A^{2} = −1 respectively.

So, for a non-degenerated operators we obtain a Monge-Ampère structure (C, A) on J^{1}M which generates an almost product structure P = (C_{+,}l, C_{−}) (see Example 4).

Note that non-degenerate Monge-Ampère equations, in contrast Monge-Ampère operators, generate AP-structures (C_{+}, l, C_{−}) up to the change C_{+} and C_{−}. Indeed, effective 2-forms ω_{1} = ω and ω_{2} = −ω generate the same equation, but C_{−}^{1} = C_{+}^{2} and C_{+}^{1} = C_{−}^{2}. Here C_{±}^{i} are eigensubspaces of the operators A_{ω_{i}} (i = 1, 2).

On the other hand, any pair of arbitrary real distributions C_{1,0} and C_{0,1} on J^{1}M such that

(1) dim C_{1,0} = dim C_{0,1} = 2;

(2) at each point a ∈ J^{1}M C(a) = C_{1,0}(a) ⊕ C_{0,1}(a);

(3) at each point a ∈ J^{1}M the subspaces C_{1,0}(a) and C_{0,1}(a) are skew-orthogonal with respect to the symplectic structure Ω_{a};

determines the operator A up to the signum. Therefore a hyperbolic Monge-Ampère equation can be regarded as such unordered pair {C_{1,0}, C_{0,1}}.

Analogously, an elliptic Monge-Ampère equation can be regarded as such unordered pair {C_{1,0}, C_{0,1}} of complex conjugate distributions on J^{1}M that

(1) dim_{ℂ} C_{1,0} = dim_{ℂ} C_{0,1} = 2;

(2) at each point a ∈ J^{1}M C(a)^{ℂ} = C_{1,0}(a) ⊕ C_{0,1}(a);

(3) at each point a ∈ J^{1}M the subspaces C_{1,0}(a) and C_{0,1}(a) are skew-orthogonal with respect to the complexification Ω_{a}^{ℂ} of the symplectic structure Ω_{a}.

3.2. Coordinate representations. We have the following representations of main objects in the canonical local coordinates (q,u, p) = (q_{1}, q_{2}, u, p_{1}, p_{2}) on the manifold J^{1}M:

• The Cartan form

U = du − p_{1}dq_{1} − p_{2}dq_{2};

• The Cartan distribution C is generated by the vector fields

\frac{d}{dq_{1}} = \frac{∂}{∂q_{1}} + p_{1} \frac{∂}{∂u}, \frac{d}{dq_{2}} = \frac{∂}{∂q_{2}} + p_{2} \frac{∂}{∂u}, \frac{∂}{∂p_{1}} , \frac{∂}{∂p_{2}} .  (8)

• The Reeb vector field X_{1} = ∂/∂u;

• An effective 2-form

ω = Edq_{1} ∧ dq_{2} + B (dq_{1} ∧ dp_{1} − dq_{2} ∧ dp_{2}) +  (9)
+ Cdq_{1} ∧ dp_{2} − Adq_{2} ∧ dp_{1} + Ddp_{1} ∧ dp_{2},

where A, B, C, D, E are some smooth functions on J^{1}M;

• The Monge-Ampère equation E_{ω}

Av_{xx} + 2Bv_{xy} + Cv_{yy} + D (v_{xx}v_{yy} − v_{xy}^{2}) + E = 0;  (10)

• The Pfaffian

Pf (ω) = AC − DE − B^{2}.

3.3. Tensor invariants of non-degenerate Monge-Ampère equations. In Example 8 we have four contact tensor invariants of a Monge-Ampère equation: d_{−1,1,1}, d_{1,1,−1}, d_{0,−1,2}, and d_{2,−1,0}.

In order to unify hyperbolic and elliptic types, we consider a complex tangent bundle T^{ℂ}(J^{1}M). Let us describe there structures:

\begin{matrix}d_{1,1,−1} = U^{ℂ} ∧ α_{1} ⊗ Q_{1} + U^{ℂ} ∧ α_{2} ⊗ Q_{2}, \\ d_{−1,1,1} = U^{ℂ} ∧ β_{1} ⊗ P_{1} + U^{ℂ} ∧ β_{2} ⊗ P_{2}, \\ d_{2,−1,0} = α_{3} ∧ α_{4} ⊗ Z, \\ d_{0,−1,2} = β_{3} ∧ β_{4} ⊗ Z. \end{matrix}  (11)

Here α_{i} ∈ Ω^{1}(C_{1,0}), β_{i} ∈ Ω^{1}(C_{0,1}), P_{j} ∈ D (C_{1,0}) , Q_{j} ∈ D (C_{0,1}) , Z ∈ D(l) (i = 1, … 3; j = 1, 2) .

These tensors can be regarded as maps

d_{1,1,−1} : C_{1,0}^{(1)} × C_{1,0}^{(1)} → C_{0,1},
d_{−1,1,1} : C_{0,1}^{(1)} × C_{0,1}^{(1)} → C_{1,0},
d_{2,−1,0} : C_{1,0} × C_{1,0} → l,
d_{0,−1,2} : C_{0,1} × C_{0,1} → l.

Let us explain a geometrical meaning of the tensors. Due to Theorem 1 the distribution C_{1,0}^{(1)} is completely integrable if and only if d_{1,1,−1} = 0 and the distribution C_{0,1}^{(1)} is completely integrable if and only if d_{−1,1,1} = 0.

Therefore, due to [15] we see that a non-degenerate Monge-Ampère equation is locally contact equivalent to the equation v_{xx} + εv_{yy} = 0 with ε = ±1 if and only if d_{1,1,−1} = d_{−1,1,1} = 0.

Example 9 (Non-linear Wave Equation). Consider the following nonlinear wave equation:

v_{xy} = f (x, y, v, v_{x}, v_{y}) .

Vector fields

P_{1} = \frac{d}{dq_{1}} + f \frac{∂}{∂p_{2}} ,
P_{2} = \frac{∂}{∂p_{1}}

constitute a basis for the distribution C_{1,0} and

Q_{1} = \frac{d}{dq_{2}} + f \frac{∂}{∂p_{1}} ,
Q_{2} = \frac{∂}{∂p_{2}}

for the distribution C_{0,1}.

The distribution l is generated by the following vector field

Z = \frac{∂}{∂u} + f_{p_{2}} \frac{∂}{∂p_{1}} + f_{p_{1}} \frac{∂}{∂p_{2}} .

The vector fields P_{1}, P_{2}, Z, Q_{1}, Q_{2} form a basis in vector fields on J^{1}M. The dual basis is

α_{1} = dq_{1},
α_{2} = dp_{1} + p_{1}f_{p_{2}}dq_{1} + (p_{2}f_{p_{2}} − f) dq_{2} − f_{p_{2}}du,
U = du − p_{1}dq_{1} − p_{2}dq_{2},
β_{1} = dq_{2},
β_{2} = dp_{2} + (p_{1}f_{p_{1}} − f) dq_{1} + p_{2}f_{p_{1}}dq_{2} − f_{p_{1}}du.

For this case we have the following representation of the four constructed tensor invariants:

d_{−1,1,1} = (ff_{p_{2}p_{2}}dq_{1} ∧ du − f_{p_{2}p_{2}}dp_{2} ∧ du − p_{1}f_{p_{2}p_{2}}dq_{1} ∧ dp_{2} − p_{2}f_{p_{2}p_{2}}dq_{2} ∧ dp_{2} + (f_{u} − p_{2}f_{p_{2}u} + f_{p_{1}}f_{p_{2}} − ff_{p_{1}p_{2}} − f_{q_{2}p_{2}}) dq_{2} ∧ du

+ (p_{1}f_{u} − p_{1}p_{2}f_{p_{2}u} − p_{2}ff_{p_{2}p_{2}} + p_{1}f_{p_{1}}f_{p_{2}} − p_{1}ff_{p_{1}p_{2}} − p_{1}f_{q_{2}p_{2}}) dq_{1} ∧ dq_{2})

⊗ \frac{∂}{∂p_{1}} ,

d_{1,1,−1} = (ff_{p_{1}p_{1}}dq_{2} ∧ du − f_{p_{1}p_{1}}dp_{1} ∧ du − p_{1}f_{p_{1}p_{1}}dq_{1} ∧ dp_{1} − p_{2}f_{p_{1}p_{1}}dq_{2} ∧ dp_{1} + (f_{u} + f_{p_{1}}f_{p_{2}} − p_{1}f_{p_{1}u} − ff_{p_{1}p_{2}} − f_{q_{1}p_{1}}) dq_{1} ∧ du

+(−p_{2}f_{u} − p_{2}f_{p_{1}}f_{p_{2}} + p_{1}p_{2}f_{p_{1}u} + p_{2}ff_{p_{1}p_{2}} + p_{1}ff_{p_{1}p_{1}} + p_{2}f_{q_{1}p_{1}}) dq_{1} ∧ dq_{2})

⊗ \frac{∂}{∂p_{2}} ,

d_{2,−1,0} = (dq_{1} ∧ dp_{1} − f_{p_{2}}dq_{1} ∧ du + (p_{2}f_{p_{2}} − f) dq_{1} ∧ dq_{2})

⊗ (\frac{∂}{∂u} + f_{p_{2}} \frac{∂}{∂p_{1}} + f_{p_{1}} \frac{∂}{∂p_{2}} ) ,

d_{0,−1,2} = (dq_{2} ∧ dp_{2} − f_{p_{1}}dq_{2} ∧ du − (p_{1}f_{p_{1}} − f) dq_{1} ∧ dq_{2})

⊗ (\frac{∂}{∂u} + f_{p_{2}} \frac{∂}{∂p_{1}} + f_{p_{1}} \frac{∂}{∂p_{2}} ) .

3.4. Contact linearization of Monge-Ampère equations. In this section we use the constructed tensors to solve the problem of contact linearization of non-degenerate Monge-Ampère equations. This problem is the following.

Find a class of Monge-Ampère equations that are locally contact equivalent to nonhomogeneous linear equations

v_{xx} + εv_{yy} = r (x, y) v_{x} + s (x, y) v_{y} + c (x, y) v + d (x, y) .  (12)

We assume that all possible derivatives of the distributions C_{±} are distributions also.

Note that for equation (12) the distributions C_{±}^{(2)} are completely integrable and dim C_{±}^{(2)} ≤ 4. This means that the tensors d_{−1,1,1} and d_{1,1,−1} have the forms

d_{1,1,−1} = U^{ℂ} ∧ α ⊗ Q, d_{−1,1,1} = U^{ℂ} ∧ β ⊗ P

for some α ∈ Ω^{1}(C_{1,0}), β ∈ Ω^{1}(C_{0,1}), P ∈ D (C_{1,0}), Q ∈ D (C_{0,1}). Let us define two complex differential 2-forms ξ_{1}, ξ_{2} ∈ Ω^{1,0,1}:

ξ_{1} =^{def} P ⌋d_{2,−1,0} (U^{ℂ} ∧ β) ,

ξ_{2} =^{def} Q⌋d_{0,−1,2} (U^{ℂ} ∧ α) .

Since d_{0,−1,2} and d_{2,−1,0} are tensors, we see that the forms ξ_{1} and ξ_{2} are contact invariant of the Monge-Ampère equation. Using representations (11) for d_{2,−1,0} and d_{0,−1,2}, we get the following forms of ξ_{1} and ξ_{2}:

ξ_{1} = P ⌋ (α_{3} ∧ α_{4} ∧ β) = (α_{3} (P) α_{4} − α_{4} (P) α_{3}) ∧ β,
ξ_{2} = Q⌋ (β_{3} ∧ β_{4} ∧ α) = (β_{3} (Q) β_{4} − β_{4} (Q) β_{3}) ∧ α.

Note that for elliptic equations the forms ξ_{1} and ξ_{2} are complex conjugate: ξ_{1} = \overline{ξ_{2}}.

Define differential 1-forms

ν_{1} =^{def} Q⌋ξ_{1},

ν_{2} =^{def} P ⌋ξ_{2}.

The vector fields Q and P are defined up to the multiplication by nonvanishing smooth functions and therefore the forms ν_{1} and ν_{2} are so. Therefore the unordered pair {ν_{1}, ν_{2}} is a relative contact invariant of a Monge-Ampère equation.

Theorem 2 (Linearization of MAE). Assume that in some neighborhood of a point a ∈ J^{1}M derivatives C_{±}^{(k)} (k = 1, 2) of the distributions C_{±} are distributions also and the distributions C_{±}^{(2)} are completely integrable and dim C_{±}^{(2)} = 4.

A Monge-Ampère equation E_{ω} is locally equivalent to a Monge-Ampère equation (12) if and only if ν_{1} = ν_{2} = 0 and the differential two-forms ξ_{1} and ξ_{2} are closed.

Proof. The proof of the necessity it trivial: it is not hard to check that for equation (12) conditions 1–3 hold. Let us prove the sufficiency.

From the first condition it follows that the equation E_{ω} is locally equivalent to a Monge-Ampère equation

v_{xx} + εv_{yy} = f(x, y, v, v_{x}, v_{y})  (13)

for some function f ∈ C^{∞} (J^{1}M) (see [15]). For this equation we have:

ν_{1} = h_{1} ((2\sqrt{ε}f − ιp_{1}f_{p_{2}} − \sqrt{ε}f_{p_{1}}) dq_{1} − p_{2} (ιf_{p_{2}} + \sqrt{ε}f_{p_{1}}) dq_{2} +

(ιf_{p_{2}} + \sqrt{ε}f_{p_{1}}) du − 2\sqrt{ε}dp_{1} − 2ιεdp_{2}) ,

ν_{2} = h_{2} (εp_{1} (\sqrt{ε}f_{p_{1}} − ιf_{p_{2}}) dq_{1} + ε (2ιf + \sqrt{ε}p_{2}f_{p_{1}} − ιp_{2}f_{p_{2}}) dq_{2} +

ε (ιf_{p_{2}} − \sqrt{ε}f_{p_{1}}) du + 2\sqrt{ε}dp_{1} − 2ιdp_{2}) ,

where h_{1} = εf_{p_{1}p_{1}} + 2ιf_{p_{1}p_{2}} − f_{p_{2}p_{2}} and h_{2} = εf_{p_{1}p_{1}} − 2ιf_{p_{1}p_{2}} − f_{p_{2}p_{2}}. We see that ν_{1} = ν_{2} = 0 if and only if h_{1} = h_{2} = 0. This means that

\{ \begin{matrix}f_{p_{1}p_{2}} = 0, \\ εf_{p_{1}p_{1}} − f_{p_{2}p_{2}}. \end{matrix}

Therefore the function f is linear with respect to p_{1} and p_{2}:

f (q, u, p) = r (q, u) p_{1} + s (q, u) p_{2} + g (q, u)

for some functions r, s, g ∈ C^{∞} (J^{0}M).

From this place the hyperbolic and elliptic cases we consider separately. Hyperbolic case. We see that the equation E_{ω} is contact equivalent to

the equation

v_{xx} − v_{yy} = r (x, y, v) v_{x} + s (x, y, v) v_{y} + g (x, y, v) .

The corresponding effective differential two-form is

ω = dq_{1} ∧ dp_{2} + dq_{2} ∧ dp_{1} + (rp_{1} + sp_{2} + g)dq_{1} ∧ dq_{2}.

170 ALEXEI KUSHNER The contact transformation

φ : q_{1} → \frac{q_{1} − q_{2}}{\sqrt{2}} , q_{2} → \frac{q_{1} + q_{2}}{\sqrt{2}} , u → u, p_{1} → \frac{p_{1} − p_{2}}{\sqrt{2}} , p_{2} → \frac{p_{1} + p_{2}}{\sqrt{2}}

takes it to the form

ω_{1} =^{def} φ^{∗} (ω) = dq_{1} ∧ dp_{1} − dq_{2} ∧ dp_{2} + (Rp_{1} + Sp_{2} + G)dq_{1} ∧ dq_{2}, (14)

where

R (q_{1,}q_{2}, u) = \frac{1}{\sqrt{2}} (s (\frac{q_{1} − q_{2}}{\sqrt{2}} , \frac{q_{1} + q_{2}}{\sqrt{2}} , u) + r (\frac{q_{1} − q_{2}}{\sqrt{2}} , \frac{q_{1} + q_{2}}{\sqrt{2}} , u)) ,
S (q_{1,}q_{2}, u) = \frac{1}{\sqrt{2}} (s (\frac{q_{1} − q_{2}}{\sqrt{2}} , \frac{q_{1} + q_{2}}{\sqrt{2}} , u) − r (\frac{q_{1} − q_{2}}{\sqrt{2}} , \frac{q_{1} + q_{2}}{\sqrt{2}} , u)) ,
G (q_{1,}q_{2}, u) = g (\frac{q_{1} − q_{2}}{\sqrt{2}} , \frac{q_{1} + q_{2}}{\sqrt{2}} , u) .

Then we get the following coordinate representation of the constructed tensors:

d_{1,1,−1} = (Rs + G_{u} + p_{2}S_{u} − R_{q_{1}}) (dq_{1} ∧ du − p_{2}dq_{1} ∧ dq_{2}) ⊗ \frac{∂}{∂p_{2}} ,
d_{−1,1,1} = (RS + G_{u} + p_{1}R_{u} − S_{q_{2}}) (dq_{2} ∧ du + p_{1}dq_{1} ∧ dq_{2}) ⊗ \frac{∂}{∂p_{1}} ,
d_{2,−1,0} = (dq_{1} ∧ dp_{1} − Sdq_{1} ∧ du − (G + p_{1}R) dq_{1} ∧ dq_{2}) ⊗

(\frac{∂}{∂u} + S \frac{∂}{∂p_{1}} + R\frac{∂}{∂p_{2}} ) ,

d_{0,−1,2} = (dq_{2} ∧ dp_{2} − Rdq_{2} ∧ du + (G + p_{2}S) dq_{1} ∧ dq_{2}) ⊗

(\frac{∂}{∂u} + S \frac{∂}{∂p_{1}} + R\frac{∂}{∂p_{2}} ) .

We can write the invariant 2-forms ξ_{1} and ξ_{2}:

ξ_{1} = (RS + G_{u} + p_{1}R_{u} − S_{q_{2}}) dq_{1} ∧ dq_{2},
ξ_{2} = − (RS + G_{u} + p_{2}S_{u} − R_{q_{1}}) dq_{1} ∧ dq_{2}.

Then

dξ_{1} = R_{u}dq_{1} ∧ dq_{2} ∧ dp_{1}

+ (SR_{u} + RS_{u} + G_{uu} + p_{1}R_{uu} − S_{uq_{2}}) dq_{1} ∧ dq_{2} ∧ du,

dξ_{2} = −S_{u}dq_{1} ∧ dq_{2} ∧ dp_{2} − (SR_{u} + RS_{u} + G_{uu} + p_{2}S_{uu} − R_{uq_{1}}) dq_{1} ∧ dq_{2} ∧ du.

We see that R_{u} = S_{u} = G_{uu} = 0 if and only if dξ_{1} = dξ_{2} = 0. In this case

G (q_{1}, q_{2}, u) = c(q_{1}, q_{2})u + z(q_{1}, q_{2})

and we get:

ω_{1} = 2 (R (q) p_{1} + S (q) p_{2} + c(q)u + z(q)) dq_{1} ∧dq_{2} +dq_{1} ∧dp_{1} −dq_{2} ∧dp_{2}.

Elliptic case. In this case the equation E_{ω} is contact equivalent to the equation

v_{xx} + v_{yy} = r (x, y, v) v_{x} + s (x, y, v) v_{y} + g (x, y, v) .

The invariant 2-form ξ_{1} is

ξ_{1} = (\frac{1}{4} (\frac{ds}{dq_{1}} − \frac{dr}{dq_{2}} )

+\frac{ι}{8} (r^{2} + s^{2} + 4g_{u} + 2(p_{1}r_{u} − r_{q_{1}} + p_{2}s_{u} − s_{q_{2}}))) dq_{1} ∧ dq_{2}

and ξ_{2} = \overline{ξ_{1}}. Then

dξ_{1} = \frac{1}{4} (−r_{u} + ιs_{u}) dq_{1} ∧ dq_{2} ∧ dp_{2} + \frac{1}{4} (ιr_{u} + s_{u}) dq_{1} ∧ dq_{2} ∧ dp_{1}+
\frac{1}{4} ((p_{1}s_{uu} − p_{2}r_{uu} − r_{uq_{2}} + s_{uq_{1}})+

ι (rr_{u} + ss_{u} + 2g_{uu} + p_{1}r_{uu} + p_{2}s_{uu} − s_{q_{2}u} − r_{q_{1}u})) dq_{1} ∧ dq_{2} ∧ du.

We see that r_{u} = s_{u} = g_{uu} = 0 if and only if dξ_{1} = 0. In this case

g (q_{1}, q_{2}, u) = c(q_{1}, q_{2})u + z(q_{1}, q_{2})

and we get the following effective 2-form:

ω = − (r (q) p_{1} + s (q) p_{2} + c(q)u + z(q)) dq_{1} ∧ dq_{2} + dq_{1} ∧ dp_{2} − dq_{2} ∧ dp_{1}.

□

Example 10 (The Hunter-Saxton equation). Let us consider the Hunter-Saxton equation

v_{tx} = vv_{xx} + κu_{x}^{2},

where κ is a constant. This equation is hyperbolic and it has applications in the theory of a director field of a liquid crystal [1] and in geometry of Einstein-Weil spaces. For this equation the corresponding effective differential 2-form is

ω = 2udq_{2} ∧ dp_{1} + dq_{1}∧dp_{1} − dq_{2} ∧ dp_{2} − 2 κp_{1}^{2} dq_{1}∧dq_{2}

and the corresponding operator

A_{ω} = ∥ \begin{matrix}1 & 2u & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & −2κp_{1}^{2} & 1 & 0 \\ 2κp_{1}^{2} & 0 & 2u & −1 \end{matrix} ∥ .

in the free basis

\frac{d}{dq_{1}} , \frac{d}{dq_{2}} , \frac{∂}{∂p_{1}} , \frac{∂}{∂p_{2}}

of the module D(C). Let us choice the following free basis of the module D(J^{1}M):

P_{1} = \frac{∂}{∂q_{1}} + p_{1} \frac{∂}{∂u} + κp_{1}^{2} \frac{∂}{∂p_{2}} ,
P_{2} = \frac{∂}{∂p_{1}} + u\frac{∂}{∂p_{2}} ,
Q_{1} = \frac{∂}{∂q_{2}} + κp_{1}^{2} \frac{∂}{∂p_{1}} − u\frac{∂}{∂q_{1}} + (p_{2} − up_{1}) \frac{∂}{∂u}
Q_{2} = \frac{∂}{∂p_{2}} ,
Z = \frac{∂}{∂u} + (2 κ − 1) p_{1} \frac{∂}{∂p_{2}} ,

and the following dual free basis of the module Ω^{1} (J^{1}M)

α_{1} = dq_{1} + udq_{2},
α_{2} = dp_{1} − κp_{1}^{2}dq_{2},
β_{1} = dq_{2},
β_{2} = dp_{2}+ (1 − 2κ) p_{1}du+ (κ−1) p_{1}^{2}dq_{1}+ (2κ − 1) p_{1}p_{2}dq_{2} − udp_{1} ,
U = du − p_{1}dq_{1} − p_{2}dq_{2}.

Here P_{1}, P_{2} ∈ D (C_{1,0}), Q_{1}, Q_{2} ∈ D (C_{0,1}), Z ∈ D (l), α_{1}, α_{2} ∈ Ω^{1,0,0} (J^{1}M), β_{1}, β_{2} ∈ Ω^{0,0,1} (J^{1}M).We get the coordinate representation of the tensors:

d_{−1,1,1} = − (p_{1}dq_{1} ∧ dq_{2} + dq_{2} ∧ du) ⊗ (\frac{∂}{∂q_{1}} + p_{1} \frac{∂}{∂u} + κp_{1}^{2} \frac{∂}{∂p_{2}} ) ,

d_{1,1,−1} = 2( κ − 1) (κp_{1}^{3}dq_{1} ∧ dq_{2} + κp_{1}^{2}dq_{2} ∧ du

− dp_{1} ∧ du − p_{1}dq_{1} ∧ dp_{1} − p_{2}dq_{2} ∧ dp_{1}) ⊗ \frac{∂}{∂p_{2}} ,

d_{2,−1,0} = (dq_{1} ∧ dp_{1} − κp_{1}^{2}dq_{1} ∧ dq_{2} + udq_{2} ∧ dp_{1})

⊗ (\frac{∂}{∂u} + (2 κ − 1) p_{1} \frac{∂}{∂p_{2}} )

d_{0,−1,2} = (dq_{2} ∧ dp_{2} + (1 − 2κ) p_{1}dq_{2} ∧ du

+ (1 − κ) p_{1}^{2}dq_{1} ∧ dq_{2} − udq_{2} ∧ dp_{1})

⊗ (\frac{∂}{∂u} + (2 κ − 1) p_{1} \frac{∂}{∂p_{2}} )

and the differential 2-forms ξ_{1}and ξ_{2}:

ξ_{1} = −dq_{2} ∧ dp_{1},
ξ_{2} = 2 (1 − κ) dq_{2} ∧ dp_{1}.

Due to the theorem, the Hunter-Saxton equation is linearized. The corresponding linear equation is the Euler-Poisson equation [14]

v_{tx} = \frac{1}{κ (t + x)}v_{t} + \frac{2 (1 − κ)}{κ (t + x)}v_{x} − \frac{2 (1 − κ)}{(κ (t + x))^{2}} u.

3.5. The equivalence problem.

3.5.1. Relative invariants. We consider a non-degenerate Monge-Ampère equation E =^{def} E_{ω}. Let P = (C_{1,0}, l, C_{0,1}) be the corresponding almostproduct structure. Let Z be a real vector field which is generates the complex distribution l and that is normed by the condition U (Z) = 1.

Define a function k by the formula:

k =^{def} ⟨⟨Z, d_{1,1,−1}⟩ , ⟨Z, d_{−1,1,1}⟩⟩ ,

where ⟨, ⟩ is a contraction.

Let Ẽ =^{def} Eω̃ be an another non-degenerate Monge-Ampère equation with the AP-structure P̃= (C̃_{1,0}, l̃, C̃_{0,1}, ). If φ : J^{1}M → J^{1}M, φ^{∗} (U) = λU is a contact transformation such that φ^{∗} (ω)_{ε} = hω̃ for some nonvanishing function h, then (up to permutation of 1st and 3rd members)

φ_{∗} \binom{}{P} = P and

φ_{∗}^{−1} (Z) = \frac{1}{λ}Z̃,
φ^{∗} (k) = \frac{1}{λ^{2}} k̃

Here Z̃ is a vector field that generates the distribution l̃, U \binom{}{Z} = 1. This means that Z and k are relative invariants of a Monge-Ampère equation.

3.5.2. Non-holonomic de Rham complex. Let us introduce submodules \underline{Ω}^{s} ⊂ Ω^{s} (J^{1}M) of vanishing on Z differential s-forms:

\underline{Ω}^{s} =^{def} \{α ∈ Ω^{s} (J^{1}M) | Z⌋ α = 0\} .

Elements of the submodules \underline{Ω}^{s} we call l-horizontal forms. The set of all l-horizontal forms form the algebra \underline{Ω}^{∗} with respect to the operation of exterior multiplication.

Define a projection Π : Ω^{s} (J^{1}M) → \underline{Ω}^{s} and an operator ∂ : Ω^{s} (J^{1}M) → \underline{Ω}^{s+1} by the following formulas:

Π (α) =^{def} α − U ∧ (Z⌋ α) ,

∂ =^{def} Π ∘ d.

The kernel of the operator Π is

ker Π = {U ∧ α| α ∈ Ω^{∗} (J^{1}M)} .

Lemma 2. Operators Π and ∂ are natural with respect to contact diffeomorphisms, i.e.

φ^{∗} ∘ Π = Π̃ ∘ φ^{∗}  (15)

and

φ^{∗} ∘ ∂ = ˜∂ ∘ φ^{∗}.  (16)

Proof. For an s-form α ∈ Ω^{s} (J^{1}M) we have:

φ^{∗} (Π (α)) = φ^{∗} (α) − φ^{∗} (U) ∧ φ^{∗} (Z⌋ α)

= φ^{∗} (α) − U ∧ (φ_{∗}^{−1} (Z)⌋ φ^{∗} (α))

= φ^{∗} (α) − λU ∧ (\frac{1}{λ}Z̃⌋ φ^{∗} (α))

= Π̃ (φ^{∗} (α)) .

Let us prove the second formula. For an arbitrary differential l-horizontal form α we have:

φ^{∗} (∂α) = φ^{∗} ∘ Π ∘ d (α) = Π̃ ∘ φ^{∗} ∘ d (α) =
= Π̃ ∘ d ∘ φ^{∗} (α) = ˜∂(φ^{∗} (α)) .
□

The restriction of ∂ to the algebra \underline{Ω}^{∗} we denote by δ:

δ : \underline{Ω}^{s} → \underline{Ω}^{s+1},

δ =^{def} ∂|\underline{Ω}^{∗}.

The sequence

0 →^{δ} \underline{Ω}^{0} →^{δ} \underline{Ω}^{1} →^{δ} \underline{Ω}^{2} →^{δ} \underline{Ω}^{3} →^{δ} \underline{Ω}^{4} →^{δ} 0 ,  (17)

where \underline{Ω}^{0} =^{def} C^{∞}(J^{1}M), is a complex, i.e. δ^{2} = 0. This complex we will call the non-holonomic de Rham complex.

Note that

δ (α ∧ β) = δα ∧ β + (−1)^{deg α} α ∧ δβ

for any differential forms α, β ∈ \underline{Ω}^{s}.

A differential l-horizontal 2-form ω is called l-effective if ω ∧ ∂U = 0. Note that

∂ (gU) = g∂U

for any function g ∈ C^{∞} (J^{1}M) and therefore this definition is correct.

Let ω be an l-effective differential 2-form and let φ : J^{1}M → J^{1}M be a contact diffeomorphism such that φ^{∗} (ω) = ω̂. Then the form ω̂ is l̂-effective also.

3.5.3. e-structures. For any differential l-horizontal 2-form α we construct an l-effective part

α_{l} =^{def} α − s_{α}∂U ∧ ∂U,

where the function s_{α} is defined by the formula

α ∧ ∂U = s_{α}∂U ∧ ∂U.

The effective 2-form ω and the l-effective part ω_{l} are generating the same Monge-Ampère equation.

Now we can formulate the condition of contact equivalence of Monge-Ampère equations E_{ω} and Eω̃ in terms of l-effective forms: φ^{∗} (ω_{l}) = hω̃l̃.

From this place we suppose that ω and ω̃ are l-effective and l̃-effective differential 2-forms respectively and φ^{∗}(U) = λU, φ^{∗} (ω) = hω̃.

Define a function F and an operator A : D(C) → D(C) ω by the following formulas:

F ∂U ∧ ∂U = ω ∧ ω,  (18)
AX⌋∂U = X⌋ω,

where the vector field X ∈ D(C).

Then

φ^{∗}(F) = \frac{h^{2}}{λ^{2}} F̃.

Indeed,

h^{2}ω̃ ∧ ω̃ = φ^{∗} (F ∂U ∧ ∂U) = φ^{∗} (F) φ^{∗} (∂U) ∧ φ^{∗} (∂U)
= φ^{∗} (F) ˜∂φ^{∗} (U) ∧ ˜∂φ^{∗} (U) = λ^{2}φ^{∗} (F) ˜∂U ∧ ˜∂U.

On the other hand ω̃ ∧ ω̃ = F̃˜∂U ∧ ˜∂U. Then, λ^{2}φ^{∗} (F) = h^{2}F̃. The square of the operator A is scalar and A^{2} + F = 0. The differential 2-form ∂U is non-degenerate on the module of vector

fields from the Cartan distribution. This means that if X⌋∂U = 0 for X ∈ D(C), then X = 0.

For a function H ∈ C^{∞}(J^{1}M) the formula

X_{H}⌋∂U = −∂H  (19)

uniquely defines a vector field X_{H} ∈ D (C).

Note that

φ_{∗}^{−1} (X_{H}) = \frac{1}{λ}X̃_{φ^{∗}(H)}.

We need two technical lemmas.

Lemma 3. We have:

Ã = \frac{λ}{h}φ_{∗}^{−1} ∘ A ∘ φ_{∗}.

Proof. Applying φ to the both parts of (18)_{2} we have:

λφ_{∗}^{−1} (AX)⌋˜∂U = hφ_{∗}^{−1} (X)⌋ω̃.

Moreover,

φ_{∗}^{−1} (X)⌋ω̃ = Ãφ_{∗}^{−1} (X)⌋˜∂U.

Since ∂U is non-degenerate, we get: λφ_{∗}^{−1} ∘ A = hÃ ∘ φ_{∗}^{−1}. Therefore Ã = \frac{λ}{h}φ_{∗}^{−1} ∘ A ∘ φ_{∗}. □

Lemma 4. For any function h ∈ C^{∞}(J^{1}M) we have:

∂h ∧ ω = \frac{1}{2}AX_{h}⌋ (∂U ∧ ∂U) .

Proof. The form ω is l-effective, i.e., ω ∧ ∂U = 0. Then

0 = X_{h}⌋ (ω ∧ ∂U) = (AX_{h}⌋ ∂U) ∧ ∂U − ∂h ∧ ω.
□

Now we can construct an e-structure for the equation E. Define a vector field W which lies in the Cartan distribution. This vector fields is uniquely determined by the following relations:

W ⌋ (∂U ∧ ∂U) = 2∂ω,
U (W) = 0.

Suppose that k ≠ 0. Let us introduce the function

F_{0} =^{def} \frac{F}{k} .

and the vector field

V =^{def} \frac{1}{k}AW

Since Lemma 3, and the facts that

φ_{∗}^{−1} (W) = \frac{1}{λ^{2}} (hW̃ + ÃX̃_{h}) ,
φ^{∗} (F_{0}) = h^{2}F̃_{0},

we obtain:

φ_{∗}^{−1} (V) = \frac{1}{φ^{∗} (k)} (φ_{∗}^{−1} ∘ A ∘ φ_{∗}) ∘ φ_{∗}^{−1} (W) =

= \frac{h^{2}}{λ} Ṽ − \frac{h}{λ}F̃_{0}X̃_{h}

For the vector fields X_{F_{0}} we have:

φ_{∗}^{−1} (X_{F_{0}}) = \frac{h^{2}}{λ} X̃F̃_{0} + \frac{2h}{λ}F̃_{0}X̃_{h}.

Define vector fields Z ∈ D (l) and Y by the formulas

U (Z) = \frac{1}{\sqrt{|k|}}

and

Y =^{def} \frac{\sqrt{|k|}}{F_{0}} (X_{F_{0}} + 2V) .

The vector fields Z and Y are invariant (up to multiplication by −1) of E:

φ_{∗}^{−1} (Y) = Ỹ.

The vector field Y splits into the sum

Y = Y_{1,0} + Y_{0,1},

where Y_{1,0} ∈ D (C_{1,0}) and Y_{0,1} ∈ D (C_{0,1}).

Applying tensors d_{−1,1,1} and d_{1,1,−1} to Z, Y_{0,1}, Y_{1,0} we get two invariant vector fields from the distributions C_{1,0} and C_{0,1} respectively:

X_{1,0} =^{def} d_{−1,1,1} (Z, Y_{0,1}) ,

X_{0,1} =^{def} d_{1,1,−1} (Z, Y_{1,0}) ,

For the case of general Monge-Ampère equation the vector fields Z, X_{1,0}, Y_{1,0}, X_{0,1}, Y_{0,1} form an e-structure on J^{1}M. Denote the constructed e-structure by

e_{E} = (Z, {X_{1,0}, X_{0,1}} , {Y_{1,0}, Y_{0,1}}) .

The e-structure is real for a hyperbolic equation and complex for elliptic one. In the last case we can construct a real e-structure using and operation of complex conjugate.

Theorem 3. Two non-degenerate Monge-Ampère equations E and Ẽ are contact equivalent if their constructed e-structures e_{E} and eẼ are equivalent.

So, the problem of contact equivalence of hyperbolic Monge-Ampère equations is a problem of equivalence of e-structures.

Example 11 (Non-linear wave equation). Construct an e-structure for a non-linear wave equation

v_{xy} = f (x, y, v, v_{x}, v_{y}) .

For this equation

Z = \frac{∂}{∂u} + f_{p_{2}} \frac{∂}{∂p_{1}} + f_{p_{1}} \frac{∂}{∂p_{2}}

and

Π (ω) = (p_{1}f_{p_{1}} + p_{2}f_{p_{2}} − 2f) dq_{1} ∧ dq_{2} + dq_{1} ∧ dp_{1}
− dq_{2} ∧ dp_{2} − f_{p_{2}}dq_{1} ∧ du + f_{p_{1}}dq_{2} ∧ du,
∂U = (p_{2}f_{p_{2}} − p_{1}f_{p_{1}}) dq_{1} ∧ dq_{2} + dq_{1} ∧ dp_{1}
+ dq_{2} ∧ dp_{2} − f_{p_{2}}dq_{1} ∧ du − f_{p_{1}}dq_{2} ∧ du.

Below the form Π (ω) is denoted by ω. We see that ω ∧ ∂U = 0, k = f_{p_{1}p_{1}}f_{p_{2}p_{2}} and F = −1. Suppose that the function k is non-vanishing. Then

F_{0} = −\frac{1}{f_{p_{1}p_{1}}f_{p_{2}p_{2}}} .

In the free basis P_{1}, P_{2}, Q_{1}, Q_{2} the operator A has a diagonal form:

A = ∥ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} ∥ .

Moreover

W = f_{p_{2}} \frac{∂}{∂p_{1}} − f_{p_{1}} \frac{∂}{∂p_{2}} ,
V = \frac{1}{f_{p_{1}p_{1}}f_{p_{2}p_{2}}} (f_{p_{2}} \frac{∂}{∂p_{1}} + f_{p_{1}} \frac{∂}{∂p_{2}} ) ,
X_{F_{0}} = k^{−2} (f_{p_{1}p_{1}}f_{p_{2}p_{2}p_{2}} + f_{p_{2}p_{2}}f_{p_{1}p_{1}p_{2}}) \frac{d}{dq_{2}}

+ k^{−2} (f_{p_{1}p_{1}}f_{p_{1}p_{2}p_{2}} + f_{p_{2}p_{2}}f_{p_{1}p_{1}p_{1}}) \frac{d}{dq_{1}}

− k^{−2}(p_{2} (f_{p_{1}p_{1}}f_{p_{2}p_{2}u} + f_{p_{2}p_{2}}f_{p_{1}p_{1}u}) + f_{p_{1}p_{1}}f_{q_{2}p_{2}p_{2}} + f_{p_{2}p_{2}}f_{q_{2}p_{1}p_{1}})\frac{∂}{∂p_{2}}

− k^{−2}(p_{1} (f_{p_{1}p_{1}}f_{p_{2}p_{2}u} + f_{p_{2}p_{2}}f_{p_{1}p_{1}u}) + f_{p_{1}p_{1}}f_{q_{1}p_{2}p_{2}} + f_{p_{2}p_{2}}f_{q_{1}p_{1}p_{1}})\frac{∂}{∂p_{1}} .

We obtain the following e-structure:

Y_{1,0} = −k^{−1/2}(f_{p_{1}}_{p_{2}}_{p_{2}}f_{p_{1}}_{p_{1}} + f_{p_{2}}_{p_{2}}f_{p_{1}}_{p_{1}}_{p_{1}}) (\frac{d}{dq_{1}} + f \frac{∂}{∂p_{2}} )

+ k^{−1/2}(−2f_{p_{2}}f_{p_{2}p_{2}}f_{p_{1}p_{1}} + p_{1}f_{p_{2}p_{2}u}f_{p_{1}p_{1}} + ff_{p_{2}p_{2}p_{2}}f_{p_{1}p_{1}}

+ p_{1}f_{p_{2}p_{2}}f_{p_{1}p_{1}u} + ff_{p_{2}p_{2}}f_{p_{1}p_{1}p_{2}} + f_{p_{1}p_{1}}f_{q_{1}p_{2}p_{2}} + f_{p_{2}p_{2}}f_{q_{1}p_{1}p_{1}})\frac{∂}{∂p_{1}} ,

Y_{0,1} = −k^{−1/2}(f_{p_{2}p_{2}p_{2}}f_{p_{1}p_{1}} + f_{p_{2}p_{2}}f_{p_{1}p_{1}p_{2}}) (\frac{d}{dq_{2}} + f \frac{∂}{∂p_{1}} )

k^{−1/2}(p_{2}f_{p_{2}p_{2}u}f_{p_{1}p_{1}} + f_{p_{1}p_{1}}(ff_{p_{1}p_{2}p_{2}} + f_{q_{2}p_{2}p_{2}})+

f_{p_{2}p_{2}}(−2f_{p_{1}}f_{p_{1}p_{1}} + p_{2}f_{p_{1}p_{1}u} + ff_{p_{1}p_{1}p_{1}} + f_{q_{2}p_{1}p_{1}}))\frac{∂}{∂p_{2}} ,

Z=k^{−1/2} (\frac{∂}{∂u} + f_{p_{2}} \frac{∂}{∂p_{1}} + f_{p_{1}} \frac{∂}{∂p_{2}} ) ,

180 ALEXEI KUSHNER

X_{1,0} = \frac{1}{f_{p_{1}p_{1}}} (p_{2}f_{p_{2}p_{2}u}f_{p_{1}p_{1}} + p_{2}f_{p_{1}}(f_{p_{2}p_{2}p_{2}}f_{p_{1}p_{1}} + f_{p_{2}p_{2}}f_{p_{1}p_{1}p_{2}})

− \frac{1}{f_{p_{2}p_{2}}} (f_{p_{2}p_{2}p_{2}}f_{p_{1}p_{1}} + f_{p_{2}p_{2}}f_{p_{1}p_{1}p_{2}})(−f_{u} + p_{2}f_{p_{2}u} − f_{p_{2}}f_{p_{1}}

+ p_{2}f_{p_{2}p_{2}}f_{p_{1}} + ff_{p_{1}p_{2}} + f_{q_{2}p_{2}}) + f_{p_{1}p_{1}}(ff_{p_{1}p_{2}p_{2}} + f_{q_{2}p_{2}p_{2}})

+ f_{p_{2}p_{2}}(−2f_{p_{1}}f_{p_{1}p_{1}} + p_{2}f_{p_{1}p_{1}u} + ff_{p_{1}p_{1}p_{1}} + f_{q_{2}p_{1}p_{1}}))\frac{∂}{∂p_{1}}

X_{0,1} = \frac{1}{f_{p_{2}p_{2}}} (−2f_{p_{2}}f_{p_{2}p_{2}}f_{p_{1}p_{1}} + p_{1}f_{p_{2}p_{2}u}f_{p_{1}p_{1}} + ff_{p_{2}p_{2}p_{2}}f_{p_{1}p_{1}}

+p_{1}f_{p_{2}p_{2}}f_{p_{1}p_{1}u} + ff_{p_{2}p_{2}}f_{p_{1}p_{1}p_{2}} + p_{1}f_{p_{2}}(f_{p_{1}p_{2}p_{2}}f_{p_{1}p_{1}} + f_{p_{2}p_{2}}f_{p_{1}p_{1}p_{1}})

+ f_{p_{1}p_{1}}f_{q_{1}p_{2}p_{2}} − \frac{1}{f_{p_{1}p_{1}}} (f_{p_{1}p_{2}p_{2}}f_{p_{1}p_{1}} + f_{p_{2}p_{2}}f_{p_{1}p_{1}p_{1}}(−f_{u} + p_{1}f_{p_{1}u}

+ ff_{p_{1}p_{2}} + f_{p_{2}}(−f_{p_{1}} + p_{1}f_{p_{1}p_{1}}) + f_{q_{1}p_{1}}) + f_{p_{2}p_{2}}f_{q_{1}p_{1}p_{1}})\frac{∂}{∂p_{2}} .

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Astrakhan State University E-mail address: kushnera@mail.ru

Received August 4, 2006