Lobachevskii Journal of Mathematics http://ljm.ksu.ru ISSN 1818-9962 Vol. 23, 2006, 151–181

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 $a \in N$ .

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 $P = (P_{1}, \dots, P_{r})$ be an ordered set of real or complex distributions on $N$ , i.e.

P_{i} : N ∋ a \mapsto P_{i} (a) \subset T_{a} N,

P_{i} : N ∋ a \mapsto P_{i} (a) \subset T_{a} N^{ℂ} = T_{a} N \otimes ℂ,

$i = 1, \dots, r$ .

The set $P$ is called an almost product structure ( $A P$ -structure) on $N$ if at each point $a \in N$ 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 $P_{1} (a), \dots, P_{r} (a)$ , i.e.

\oplus_{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}) \overset{def}{=} \{X \in D (P) ∣ X_{a} \in P (a) \forall a \in M\} .

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

Ω^{s} (P_{i}) \overset{def}{=} \{α \in Ω^{s} (N)| X⌋ α = 0 \forall X \in D (P_{j}), j = 1, \dots, r; j \neq 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 $a \in N$ the linear operator $A_{a} : T_{a} N \to T_{a} N$ is semi-simple.

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

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

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

T_{a} N = M_{a} \oplus L_{a},

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

For hyperbolic $f$ -structure the tangent space $T_{a} N$ at each point $a \in N$ 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} \oplus L_{a}^{+} \oplus 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} \oplus L_{a}^{+} \oplus 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:

$η (ξ) = 1$ ,
$η \circ Φ = 0$ ,
$Φ (ξ) = 0$ ,
$Φ^{2} = - ɛ + η \otimes ξ$ ,

where $ɛ = \pm 1$ (see [2]). Similar to the previous example the almost contact structure is called hyperbolic or elliptic if $ɛ = - 1$ or $ɛ = 1$ 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 $Φ_{a} ∣_{ker η_{a}}$ , $a \in N .$

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 $C : a \mapsto C (a) \subset T_{a} N$ and let $J$ be a ”non-holonomic” ﬁeld of endomorphisms2 $J$ , $J_{a} : C (a) \to C (a)$ , $J^{2} = ɛ$ , where $ɛ = \pm 1$ .

Suppose that the distribution $C$ is generated by the differential $1$ -form $U$ on $N$ : $C (a) = ker U_{a}$ for each $a \in N$ 3. Let $Ω_{a}$ is a restriction of $d U$ to $C (a)$ : $Ω_{a} = d U ∣_{C (a)}$ . 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)

$\forall X, Y \in C (a)$ , $\forall a \in N$ .

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

A Monge-Ampère structure generates an almost-product structure $P = (C_{+,} l, C_{-})$ on $N$ , where the $2$ -dimensional distributions4 $C_{\pm} : a \mapsto C_{\pm} (a)$ are generated by the eigensubspaces $C_{\pm} (a)$ of $J$ and $1$ -dimensional distribution $l$ is generated by the intersection of the ﬁrst derivatives5 $C_{+}^{(1)}$ , $C_{-}^{(1)}$ of the distributions $C_{+}$ and $C_{-}$ : $l (a) = C_{+}^{(1)} (a) \cap C_{-}^{(1)} (a)$ .

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_{\pm}, Y_{\pm}]) = - d U (X_{\pm}, Y_{\pm}) \neq 0

for any $X_{\pm}, Y_{\pm}$ $\in D (C_{\pm})$ . Similarly, $U^{ℂ} ([X_{\pm}, Y_{\pm}]) \neq 0$ 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) \oplus l (a) \oplus 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: $C_{+} (a) = \bar{C_{-} (a)}$ . Then the subspaces $C_{+}^{(1)} (a)$ and $C_{-}^{(1)} (a)$ are complex conjugated also, and its intersection is complex conjugate to itself: $l (a) = \bar{l (a)}$ . Therefore this complex line is generated by a real vector $Z$ : $l (a) = ℂ Z_{a}$ , $Z \in T_{a} (N)$ (see [10]).

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

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 $a \in N$ the vector space $P (a) \subset T_{a} N$ 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 $P_{3} (a) \overset{def}{=} P_{1}^{(1)} (a) \cap P_{2}^{(2)} (a)$ be an intersection of the ﬁrst derivatives of the distributions $P_{1}$ and $P_{2}$ at a point $a \in N$ . Suppose that $P_{3} : a \mapsto P_{3} (a)$ 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 $A = \oplus_{k} A^{k}$ be a differential $r$ -graded algebra over a ﬁeld of characteristic $0$ with a differential $d$ , i.e.

d (a \cdot b) = d a \cdot b + {(- 1)}^{a} a \cdot d b .

Here $k$ is a multi-index, $k = (k_{1}, \dots, k_{r})$ , $k_{i} \in \{0, 1, \dots, n_{i}\}$ , $n_{i} \in ℕ$ are some numbers, $|k| = k_{1} + \dots + k_{r}$ , and ${(- 1)}^{a}$ = ${(- 1)}^{deg a}$ where $deg a \overset{def}{=} s$ if $a \in A_{s} \overset{def}{=} \oplus_{|k| = s} A^{k}$ . We assume also that $A$ is a super-commutative algebra, i.e.

a \cdot b = {(- 1)}^{a b} b \cdot a,

and the differential and multiplication are compatible with grad:

A^{k} \cdot A^{t} \subset A^{k + t}

d A_{s} \subset A_{s + 1}

The differential $d$ splits in the following direct sum:

d = \underset{|σ| = 1}{\oplus} d_{σ} = (\underset{i = 1}{\overset{r}{\oplus}} d_{i}) \oplus (\underset{|t| = 1}{\oplus} d_{t}),

where $σ = (σ_{1}, \dots, σ_{r})$ , $σ_{j} \in I_{j} = \{z \in ℤ| |z| \leq n_{j}\}$ , $d_{σ} : A^{k} \to A^{k + σ}$ $(|σ| = 1)$ , $d_{i} = d_{(0 \dots 1_{i} \dots 0)}$ 6 and $t$ has negative components.

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

d_{i} (a \cdot b) = d_{i} a \cdot b + {(- 1)}^{a} a \cdot d_{i} b

and

d_{t} (a \cdot b) = d_{t} a \cdot b + {(- 1)}^{a} a \cdot d_{t} b

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

d_{t} (a \cdot b) = a \cdot d_{t} b

for any $a_{0} \in A_{0}$ and for any $a, b \in A$ .

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

d (a \cdot b) = \sum_{|σ| = 1} d_{σ} (a \cdot b) = \sum_{i = 1}^{r} d_{i} (a \cdot b) + \sum_{|t| = 1} d_{t} (a \cdot b),

On the other hand,

\begin{array}{l} d (a \cdot b) & = d a \cdot b + {(- 1)}^{a} a \cdot d b \\ = (\sum_{i = 1}^{r} d_{i} a + \sum_{|t| = 1} d_{t} a) \cdot b + {(- 1)}^{a} a \cdot (\sum_{i = 1}^{r} d_{i} b + \sum_{|t| = 1} d_{t} b) . \end{array}

Therefore $d_{i} a \cdot b \notin A^{k + t}$ for each $i = 1, \dots, r$ and one gets that $d_{t} (a \cdot b) = a \cdot d_{t} b$ . □

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

A^{s} (N) = \underset{p + q = s}{\oplus} A^{p, q} (N),

$s = 0, 1, 2, 4$ and

d = d_{1, 0} \oplus d_{0, 1} \oplus d_{2, - 1} \oplus 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).

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) = \underset{|k| = s}{\oplus} Λ^{k} (T_{a}^{*} N),

where $k$ is a multi-index, $k = (k_{1}, \dots, k_{r})$ , $k_{i} \in \{0, 1, \dots, dim P_{i}\}$ and

Λ^{k} (T_{a}^{*} N) = \otimes_{i = 1}^{r} Λ^{k_{i}} (P_{i} (a)) .

Here

\begin{matrix} Λ^{k_{i}} (P_{i} (a)) \overset{def}{=} \{α \in Λ^{k_{i}} (T_{a}^{*} N)| X⌋ α = 0 \\ \forall X \in P_{j} (a), j = 1, \dots, r; j \neq i \} . \end{matrix}

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

Ω^{s} (N) = \underset{|k| = s}{\oplus} Ω^{k},

(1)

where

Ω^{k} \overset{def}{=} \otimes_{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 = \underset{|σ| = 1}{\oplus} d_{σ},

where

σ_{j} \in I_{j} \overset{def}{=} \{z \in ℤ| |z| \leq dim P_{j}\}

and

d_{σ} : Ω^{k} \to Ω^{k + σ} .

From Lemma 1 it follows that if one of the component $t_{i}$ of a multi-index $t$ 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 $θ \otimes X$ , where $θ$ is a differential $s$ -form and $X$ is a $(s - 1)$ -vector ﬁeld on $N$ .

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

\begin{array}{l} (θ \otimes X) (α) & = (X⌋ α) θ, \\ (θ \otimes X) (Y) & = (Y⌋ θ) X \end{array}

respectively. Therefore a tensor $θ \otimes X$ for $θ \in Ω^{s} (P_{i})$ and $X \in D^{s - 1} (P_{j})$ 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 $A P$ - and $A C$ -structures on $ℝ^{4}$ ). Let us consider a classical $A P$ -structure (or a classical $A C$ -structure) $J$ on $ℝ^{4}$ . The tangent space $T_{a} ℝ^{4}$ (or the complexiﬁcation ${(T_{a} ℝ^{4})}^{ℂ}$ for $A C$ -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 $A C$ -structure), $i = 1, 2$ . The module $Ω^{s} (ℝ^{4})$ (or its complexiﬁcation for $A C$ -structure) falls into the direct sum of $Ω^{p, q}$ , where $p + q = s$ , and

Ω^{p, q} \overset{def}{=} Ω^{p} (P_{1}) \otimes Ω^{q} (P_{2}) .

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

d = d_{1, 0} \oplus d_{0, 1} \oplus d_{2, - 1} \oplus d_{- 1, 2} .

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

Note that for $A C$ -structure the tensors $d_{- 1, 2}$ and $d_{2, - 1}$ are complex conjugated: $\bar{d_{- 1, 2}} = d_{2, - 1}$ . 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 $A P$ -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:

\begin{array}{l} Ω^{0} (ℝ^{5}) & = C^{\infty} (ℝ^{5}), \\ Ω^{1} (ℝ^{5}) & = Ω^{1, 0, 0} \oplus Ω^{0, 1, 0} \oplus Ω^{0, 0, 1}, \\ Ω^{2} (ℝ^{5}) & = Ω^{2, 0, 0} \oplus Ω^{1, 1, 0} \oplus Ω^{1, 0, 1} \oplus Ω^{0, 1, 1} \oplus Ω^{0, 0, 2}, \\ Ω^{3} (ℝ^{5}) & = Ω^{2, 1, 0} \oplus Ω^{2, 0, 1} \oplus Ω^{1, 1, 1} \oplus Ω^{1, 0, 2} \oplus Ω^{0, 1, 2}, \\ Ω^{4} (ℝ^{5}) & = Ω^{2, 1, 1} \oplus Ω^{2, 0, 2} \oplus Ω^{1, 1, 2}, \\ Ω^{5} (ℝ^{5}) & = Ω^{2, 1, 2}, \end{array}

and the decomposition of exterior differential:

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 $q_{j, k}^{s} \overset{def}{=} d_{1_{j} + 1_{k} - 1_{s}} : Ω^{1_{s}} \to Ω^{1_{j} + 1_{k}}$ ( $s \neq k, j$ ) can be regarded as a map

q_{j, k}^{s} : D (P_{j}) \times D (P_{k}) \to D (P_{s}) .

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

q_{j, k}^{s} (X, Y) \overset{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} \overset{def}{=} \sum_{s, j, k (s \neq j, k)} q_{j, k}^{s} .

It is not hard to see that

Q_{P} (X, Y) = - \sum_{s, j, k (s \neq j, k)} P_{s} [P_{j} X, P_{k} Y] .

2.3. Subdistributions. Let

P_{I} \overset{def}{=} \underset{i \in I}{\oplus} P_{i},

(2)

where $I \subset \{1, 2, \dots, r\}$ . 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 multi-indices. Let $Ann (P)$ be an annihilator of a distribution $P$ :

Ann (P) \overset{def}{=} \{α \in Ω^{1} (N)| α (X) = 0 \forall X \in D (P)\} .

Let us introduce multi-indices of length $r$ : $1 \overset{def}{=} (1, \dots, 1)$ , $1_{i} \overset{def}{=} (0, \dots, 0, 1_{i}, 0, \dots, 0)$ 7, $k \overset{def}{=} \sum_{i \in I} 1_{i}$ , and $\bar{k}$ , where $\bar{k} + k = 1$ , and put $(a, b) \overset{def}{=} a_{1} b_{1} + \dots a_{r} b_{r}$ for $a = (a_{1}, \dots, a_{r})$ , $b = (b_{1}, \dots, b_{r})$ .

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

Ann (P_{k}) = \underset{(1_{i}, \bar{k}) = 1}{\oplus} 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 $(t, \bar{k}) = - 1$ .

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

\bar{k} = (\underset{l}{\underset{︸}{1, \dots, 1}}, \underset{r - l}{\underset{︸}{0 \dots 0}}) .

Then

Ann (P_{k}) = 〈α_{1}, \dots, α_{p_{1}}, α_{p_{1} + 1}, \dots, α_{p_{1} + p_{2}}, \dots, α_{p_{1} + \dots + p_{l}}〉,

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

Let $α_{j} \in Ω^{1_{j}} \subset Ann (P_{k})$ and $t = (t_{1}, \dots, t_{r})$ , where $t_{j} = - 1$ . If $t_{s} = 0$ for $s \in \{1, \dots, \hat{j}, \dots, l\}$ , then

α_{1} \land \dots \land α_{p_{1} + \dots + p_{l}} \land d_{t} α_{j} = 0 \Leftrightarrow d_{t} = 0 .

If $t_{s} \neq 0$ for $s \in \{1, \dots, \hat{j}, \dots, l\}$ , then $α_{1} \land \dots \land α_{p_{1} + \dots + p_{l}} \land d_{t} α_{j} = 0$ . So, $α_{1} \land \dots \land α_{p_{1} + \dots + p_{l}} \land d α_{j} = 0 \Leftrightarrow d_{t} = 0$ for all $t$ such that $(t, \bar{k}) = - 1$ . □

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 \in J^{1} M \to C (a) \subset T_{a} (J^{1} M)

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

At each point $a \in J^{1} M$ the $2$ -form

Ω_{a} \overset{def}{=} d U ∣_{C (a)} \in Λ^{2} (C^{*} (a))

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

Ω : J^{1} M ∋ a \mapsto Ω_{a} \in Λ^{2} (C^{*} (a))

on $J^{1} M$ .

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

Δ_{ω} (v) = ω ∣_{j^{1} (v) (M)} .

(3)

Here $v \in C^{\infty} (M)$ is a smooth function and $j_{1} (v) (M) \subset J^{1} M$ 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} \overset{def}{=} \{Δ_{ω} (v) = 0\} \subset J^{2} M

is called a Monge-Ampère equation.

But constructed map ”differential $2$ -forms” $\to$ ”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 $2$ -forms.

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^{*} = \underset{s \geq 0}{\oplus} I^{s},

$I^{s} \subset$ $Ω^{s} (J^{1} M)$ . The ideal $I^{*}$ is generated by forms

U \land α + d U \land 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 $s \geq 3 .$

Elements of the quotient module $Ω^{s} (J^{1} M) ∕ I^{s}$ we call effective $s$ -forms $(s \leq 2)$ :

Ω_{ɛ}^{s} (J^{1} M) ≅ Ω^{s} (J^{1} M) ∕ I^{s} .

(5)

For each element of $Ω_{ɛ}^{2} (J^{1} M)$ one can choice a unique representative $ω \in Ω^{2} (J^{1} M)$ such that $X_{1}⌋ ω = 0$ and $ω \land d U = 0$ . 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 $h ω$ generate the same equation.

Let $ω$ and $\tilde{ω}$ be effective differential $2$ -forms. Two Monge-Ampère equations $E_{ω}$ and $E_{\tilde{ω}}$ are (local) contact equivalent at a point $a \in J^{1} M$ if there exists a contact diffeomorphism $ϕ : J^{1} M \to J^{1} M$ , $ϕ (a) = a$ and some function $h_{ϕ} \in C^{\infty} (J^{1} M)$ , $h_{ϕ} (a) \neq 0$ , such that

ϕ^{*} {(ω)}_{ɛ} = h_{ϕ} \tilde{ω} .

Here $ϕ^{*} {(ω)}_{ɛ}$ 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 \mapsto A_{ω_{a}} \in End (C (a)) .

(6)

by the formula

X_{a}⌋ ω_{a} = A_{ω, a} X_{a} ⌋ Ω_{a}

for any tangent vector $X_{a} \in C (a)$ .

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 $X, Y \in D (C)$ .

A function $Pf (ω) \in C^{\infty} (J^{1} M)$ is called a Pfaffian of the form $ω$ if

Pf (ω) Ω \land Ω = ω \land ω .

Note that

Pf (h ω) = h^{2} Pf (ω)

for a function $h \in C^{\infty} (J^{1} M)$ .

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 $a \in J^{1} M$ if $Pf (ω)$ is negative, positive or zero at this point respectively. Hyperbolic and elliptic equations are called non-degenerate.

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

ω \mapsto \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 $A P$ -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 $C_{\pm}^{i}$ 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

$dim C_{1, 0} = dim C_{0, 1} = 2$ ;
at each point $a \in J^{1} M$ $C (a) = C_{1, 0} (a) \oplus C_{0, 1} (a)$ ;
at each point $a \in J^{1} M$ 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

${dim}_{ℂ} C_{1, 0} = {dim}_{ℂ} C_{0, 1} = 2;$
at each point $a \in J^{1} M$ $C {(a)}^{ℂ} = C_{1, 0} (a) \oplus C_{0, 1} (a)$ ;
at each point $a \in J^{1} M$ 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 $U = d u - p_{1} d q_{1} - p_{2} d q_{2};$
The Cartan distribution $C$ is generated by the vector ﬁelds
$\frac{d}{d q_{1}} = \frac{\partial}{\partial q_{1}} + p_{1} \frac{\partial}{\partial u}, \frac{d}{d q_{2}} = \frac{\partial}{\partial q_{2}} + p_{2} \frac{\partial}{\partial u}, \frac{\partial}{\partial p_{1}}, \frac{\partial}{\partial p_{2}} .$ (8)
The Reeb vector ﬁeld $X_{1} = \partial ∕ \partial u$ ;
An effective $2$ -form $\begin{align} ω & = E d q_{1} \land d q_{2} + B (d q_{1} \land d p_{1} - d q_{2} \land d p_{2}) + & (9) \\ + C d q_{1} \land d p_{2} - A d q_{2} \land d p_{1} + D d p_{1} \land d p_{2}, \end{align}$
where $A, B, C, D, E$ are some smooth functions on $J^{1} M$ ;
The Monge-Ampère equation $E_{ω}$
$A v_{x x} + 2 B v_{x y} + C v_{y y} + D (v_{x x} v_{y y} - v_{x y}^{2}) + E = 0;$ (10)
The Pfaffian $Pf (ω) = A C - D E - B^{2} .$

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} \land α_{1} \otimes Q_{1} + U^{ℂ} \land α_{2} \otimes Q_{2}, \\ d_{- 1, 1, 1} = U^{ℂ} \land β_{1} \otimes P_{1} + U^{ℂ} \land β_{2} \otimes P_{2}, \\ d_{2, - 1, 0} = α_{3} \land α_{4} \otimes Z, \\ d_{0, - 1, 2} = β_{3} \land β_{4} \otimes Z . \end{matrix}

(11)

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

These tensors can be regarded as maps

\begin{array}{l} d_{1, 1, - 1} & : C_{1, 0}^{(1)} \times C_{1, 0}^{(1)} \to C_{0, 1}, \\ d_{- 1, 1, 1} & : C_{0, 1}^{(1)} \times C_{0, 1}^{(1)} \to C_{1, 0}, \\ d_{2, - 1, 0} & : C_{1, 0} \times C_{1, 0} \to l, \\ d_{0, - 1, 2} & : C_{0, 1} \times C_{0, 1} \to l . \end{array}

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 $v_{x x} + ɛ v_{y y} = 0$ with $ɛ = \pm 1$ if and only if $d_{1, 1, - 1} =$ $d_{- 1, 1, 1} = 0$ .

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

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

Vector ﬁelds

\begin{array}{l} P_{1} & = \frac{d}{d q_{1}} + f \frac{\partial}{\partial p_{2}}, \\ P_{2} & = \frac{\partial}{\partial p_{1}} \end{array}

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

\begin{array}{l} Q_{1} & = \frac{d}{d q_{2}} + f \frac{\partial}{\partial p_{1}}, \\ Q_{2} & = \frac{\partial}{\partial p_{2}} \end{array}

for the distribution $C_{0, 1}$ .

The distribution $l$ is generated by the following vector ﬁeld

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

\begin{array}{l} α_{1} & = d q_{1}, \\ α_{2} & = d p_{1} + p_{1} f_{p_{2}} d q_{1} + (p_{2} f_{p_{2}} - f) d q_{2} - f_{p_{2}} d u, \\ U & = d u - p_{1} d q_{1} - p_{2} d q_{2}, \\ β_{1} & = d q_{2}, \\ β_{2} & = d p_{2} + (p_{1} f_{p_{1}} - f) d q_{1} + p_{2} f_{p_{1}} d q_{2} - f_{p_{1}} d u . \end{array}

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

\begin{matrix} d_{- 1, 1, 1} = (f f_{p_{2} p_{2}} d q_{1} \land d u - f_{p_{2} p_{2}} d p_{2} \land d u - p_{1} f_{p_{2} p_{2}} d q_{1} \land d p_{2} \\ - p_{2} f_{p_{2} p_{2}} d q_{2} \land d p_{2} + (f_{u} - p_{2} f_{p_{2} u} + f_{p_{1}} f_{p_{2}} - f f_{p_{1} p_{2}} - f_{q_{2} p_{2}}) d q_{2} \land d u \\ + (p_{1} f_{u} - p_{1} p_{2} f_{p_{2} u} - p_{2} f f_{p_{2} p_{2}} + p_{1} f_{p_{1}} f_{p_{2}} - p_{1} f f_{p_{1} p_{2}} - p_{1} f_{q_{2} p_{2}}) d q_{1} \land d q_{2}) \\ \otimes \frac{\partial}{\partial p_{1}}, \end{matrix}

\begin{matrix} d_{1, 1, - 1} = (f f_{p_{1} p_{1}} d q_{2} \land d u - f_{p_{1} p_{1}} d p_{1} \land d u - p_{1} f_{p_{1} p_{1}} d q_{1} \land d p_{1} \\ - p_{2} f_{p_{1} p_{1}} d q_{2} \land d p_{1} + (f_{u} + f_{p_{1}} f_{p_{2}} - p_{1} f_{p_{1} u} - f f_{p_{1} p_{2}} - f_{q_{1} p_{1}}) d q_{1} \land d u \\ + (- p_{2} f_{u} - p_{2} f_{p_{1}} f_{p_{2}} + p_{1} p_{2} f_{p_{1} u} + p_{2} f f_{p_{1} p_{2}} + p_{1} f f_{p_{1} p_{1}} + p_{2} f_{q_{1} p_{1}}) d q_{1} \land d q_{2}) \\ \otimes \frac{\partial}{\partial p_{2}}, \end{matrix}

\begin{matrix} d_{2, - 1, 0} = (d q_{1} \land d p_{1} - f_{p_{2}} d q_{1} \land d u + (p_{2} f_{p_{2}} - f) d q_{1} \land d q_{2}) \\ \otimes (\frac{\partial}{\partial u} + f_{p_{2}} \frac{\partial}{\partial p_{1}} + f_{p_{1}} \frac{\partial}{\partial p_{2}}), \end{matrix}

\begin{matrix} d_{0, - 1, 2} = (d q_{2} \land d p_{2} - f_{p_{1}} d q_{2} \land d u - (p_{1} f_{p_{1}} - f) d q_{1} \land d q_{2}) \\ \otimes (\frac{\partial}{\partial u} + f_{p_{2}} \frac{\partial}{\partial p_{1}} + f_{p_{1}} \frac{\partial}{\partial p_{2}}) . \end{matrix}

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_{x x} + ɛ v_{y y} = 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 $C_{\pm}$ are distributions also.

Note that for equation (12) the distributions $C_{\pm}^{(2)}$ are completely integrable and $dim C_{\pm}^{(2)} \leq 4$ . This means that the tensors $d_{- 1, 1, 1}$ and $d_{1, 1, - 1}$ have the forms

d_{1, 1, - 1} = U^{ℂ} \land α \otimes Q, d_{- 1, 1, 1} = U^{ℂ} \land β \otimes P

for some $α \in Ω^{1} (C_{1, 0})$ , $β \in Ω^{1} (C_{0, 1})$ , $P \in D (C_{1, 0})$ , $Q \in D (C_{0, 1})$ .

Let us deﬁne two complex differential $2$ -forms $ξ_{1}, ξ_{2} \in Ω^{1, 0, 1}$ :

\begin{array}{l} ξ_{1} \overset{def}{=} P ⌋ d_{2, - 1, 0} (U^{ℂ} \land β), \\ ξ_{2} \overset{def}{=} Q ⌋ d_{0, - 1, 2} (U^{ℂ} \land α) . \end{array}

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 $ξ_{2}$ :

\begin{array}{l} ξ_{1} & = P ⌋ (α_{3} \land α_{4} \land β) = (α_{3} (P) α_{4} - α_{4} (P) α_{3}) \land β, \\ ξ_{2} & = Q ⌋ (β_{3} \land β_{4} \land α) = (β_{3} (Q) β_{4} - β_{4} (Q) β_{3}) \land α . \end{array}

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

Deﬁne differential $1$ -forms

\begin{array}{l} ν_{1} \overset{def}{=} Q ⌋ ξ_{1}, \\ ν_{2} \overset{def}{=} P ⌋ ξ_{2} . \end{array}

The vector ﬁelds $Q$ and $P$ are deﬁned up to the multiplication by non-vanishing 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 $a \in J^{1} M$ derivatives $C_{\pm}^{(k)}$ ( $k = 1, 2$ ) of the distributions $C_{\pm}$ are distributions also and the distributions $C_{\pm}^{(2)}$ are completely integrable and $dim C_{\pm}^{(2)} = 4$ .

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_{x x} + ɛ v_{y y} = f (x, y, v, v_{x}, v_{y})

(13)

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

\begin{array}{l} ν_{1} = h_{1} ((2 \sqrt{ɛ} f - ι p_{1} f_{p_{2}} - \sqrt{ɛ} f_{p_{1}}) d q_{1} - p_{2} (ι f_{p_{2}} + \sqrt{ɛ} f_{p_{1}}) d q_{2} + \\ (ι f_{p_{2}} + \sqrt{ɛ} f_{p_{1}}) d u - 2 \sqrt{ɛ} d p_{1} - 2 ι ɛ d p_{2}), \\ ν_{2} = h_{2} (ɛ p_{1} (\sqrt{ɛ} f_{p_{1}} - ι f_{p_{2}}) d q_{1} + ɛ (2 ι f + \sqrt{ɛ} p_{2} f_{p_{1}} - ι p_{2} f_{p_{2}}) d q_{2} + \\ ɛ (ι f_{p_{2}} - \sqrt{ɛ} f_{p_{1}}) d u + 2 \sqrt{ɛ} d p_{1} - 2 ι d p_{2}), \end{array}

where $h_{1} = ɛ f_{p_{1} p_{1}} + 2 ι f_{p_{1} p_{2}} - f_{p_{2} p_{2}}$ and $h_{2} = ɛ f_{p_{1} p_{1}} - 2 ι f_{p_{1} p_{2}} - f_{p_{2} p_{2}}$ . 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, \\ ɛ f_{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 $r, s, g \in C^{\infty} (J^{0} M)$ .

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_{x x} - v_{y y} = r (x, y, v) v_{x} + s (x, y, v) v_{y} + g (x, y, v) .

The corresponding effective differential two-form is

ω = d q_{1} \land d p_{2} + d q_{2} \land d p_{1} + (r p_{1} + s p_{2} + g) d q_{1} \land d q_{2} .

The contact transformation

ϕ : q_{1} \to \frac{q_{1} - q_{2}}{\sqrt{2}}, q_{2} \to \frac{q_{1} + q_{2}}{\sqrt{2}}, u \to u, p_{1} \to \frac{p_{1} - p_{2}}{\sqrt{2}}, p_{2} \to \frac{p_{1} + p_{2}}{\sqrt{2}}

takes it to the form

ω_{1} \overset{def}{=} ϕ^{*} (ω) = d q_{1} \land d p_{1} - d q_{2} \land d p_{2} + (R p_{1} + S p_{2} + G) d q_{1} \land d q_{2},

(14)

where

\begin{array}{l} 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) . \end{array}

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

\begin{array}{l} d_{1, 1, - 1} & = (R s + G_{u} + p_{2} S_{u} - R_{q_{1}}) (d q_{1} \land d u - p_{2} d q_{1} \land d q_{2}) \otimes \frac{\partial}{\partial p_{2}}, \\ d_{- 1, 1, 1} & = (R S + G_{u} + p_{1} R_{u} - S_{q_{2}}) (d q_{2} \land d u + p_{1} d q_{1} \land d q_{2}) \otimes \frac{\partial}{\partial p_{1}}, \\ d_{2, - 1, 0} & = (d q_{1} \land d p_{1} - S d q_{1} \land d u - (G + p_{1} R) d q_{1} \land d q_{2}) \otimes \\ (\frac{\partial}{\partial u} + S \frac{\partial}{\partial p_{1}} + R \frac{\partial}{\partial p_{2}}), \\ d_{0, - 1, 2} & = (d q_{2} \land d p_{2} - R d q_{2} \land d u + (G + p_{2} S) d q_{1} \land d q_{2}) \otimes \\ (\frac{\partial}{\partial u} + S \frac{\partial}{\partial p_{1}} + R \frac{\partial}{\partial p_{2}}) . \end{array}

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

\begin{array}{l} ξ_{1} & = (R S + G_{u} + p_{1} R_{u} - S_{q_{2}}) d q_{1} \land d q_{2}, \\ ξ_{2} & = - (R S + G_{u} + p_{2} S_{u} - R_{q_{1}}) d q_{1} \land d q_{2} . \end{array}

Then

\begin{matrix} d ξ_{1} = R_{u} d q_{1} \land d q_{2} \land d p_{1} \\ + (S R_{u} + R S_{u} + G_{u u} + p_{1} R_{u u} - S_{u q_{2}}) d q_{1} \land d q_{2} \land d u, \end{matrix}

\begin{matrix} d ξ_{2} = - S_{u} d q_{1} \land d q_{2} \land d p_{2} \\ - (S R_{u} + R S_{u} + G_{u u} + p_{2} S_{u u} - R_{u q_{1}}) d q_{1} \land d q_{2} \land d u . \end{matrix}

We see that $R_{u} = S_{u} = G_{u u} = 0$ if and only if $d ξ_{1} = d ξ_{2} = 0$ . 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)) d q_{1} \land d q_{2} + d q_{1} \land d p_{1} - d q_{2} \land d p_{2} .

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

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

The invariant $2$ -form $ξ_{1}$ is

\begin{matrix} ξ_{1} = (\frac{1}{4} (\frac{d s}{d q_{1}} - \frac{d r}{d q_{2}}) \\ + \frac{ι}{8} (r^{2} + s^{2} + 4 g_{u} + 2 (p_{1} r_{u} - r_{q_{1}} + p_{2} s_{u} - s_{q_{2}}))) d q_{1} \land d q_{2} \end{matrix}

and $ξ_{2} = \bar{ξ_{1}}$ . Then

\begin{array}{l} d ξ_{1} & = \frac{1}{4} (- r_{u} + ι s_{u}) d q_{1} \land d q_{2} \land d p_{2} + \frac{1}{4} (ι r_{u} + s_{u}) d q_{1} \land d q_{2} \land d p_{1} + \\ \frac{1}{4} ((p_{1} s_{u u} - p_{2} r_{u u} - r_{u q_{2}} + s_{u q_{1}}) + \\ ι (r r_{u} + s s_{u} + 2 g_{u u} + p_{1} r_{u u} + p_{2} s_{u u} - s_{q_{2} u} - r_{q_{1} u})) d q_{1} \land d q_{2} \land d u . \end{array}

We see that $r_{u} = s_{u} = g_{u u} = 0$ if and only if $d ξ_{1} = 0$ . 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)) d q_{1} \land d q_{2} + d q_{1} \land d p_{2} - d q_{2} \land d p_{1} .

□

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

v_{t x} = v v_{x x} + κ u_{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 = 2 u d q_{2} \land d p_{1} + d q_{1} \land d p_{1} - d q_{2} \land d p_{2} - 2 κ p_{1}^{2} d q_{1} \land d q_{2}

and the corresponding operator

A_{ω} = ↑\begin{matrix} 1 & 2 u & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & - 2 κ p_{1}^{2} & 1 & 0 \\ 2 κ p_{1}^{2} & 0 & 2 u & - 1 \end{matrix}↑ .

in the free basis

\frac{d}{d q_{1}}, \frac{d}{d q_{2}}, \frac{\partial}{\partial p_{1}}, \frac{\partial}{\partial p_{2}}

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

\begin{array}{l} P_{1} & = \frac{\partial}{\partial q_{1}} + p_{1} \frac{\partial}{\partial u} + κ p_{1}^{2} \frac{\partial}{\partial p_{2}}, \\ P_{2} & = \frac{\partial}{\partial p_{1}} + u \frac{\partial}{\partial p_{2}}, \\ Q_{1} & = \frac{\partial}{\partial q_{2}} + κ p_{1}^{2} \frac{\partial}{\partial p_{1}} - u \frac{\partial}{\partial q_{1}} + (p_{2} - u p_{1}) \frac{\partial}{\partial u} \\ Q_{2} & = \frac{\partial}{\partial p_{2}}, \\ Z & = \frac{\partial}{\partial u} + (2 κ - 1) p_{1} \frac{\partial}{\partial p_{2}}, \end{array}

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

\begin{array}{l} α_{1} & = d q_{1} + u d q_{2}, \\ α_{2} & = d p_{1} - κ p_{1}^{2} d q_{2}, \\ β_{1} & = d q_{2}, \\ β_{2} & = d p_{2} + (1 - 2 κ) p_{1} d u + (κ - 1) p_{1}^{2} d q_{1} + (2 κ - 1) p_{1} p_{2} d q_{2} - u d p_{1}, \\ U & = d u - p_{1} d q_{1} - p_{2} d q_{2} . \end{array}

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

d_{- 1, 1, 1} = - (p_{1} d q_{1} \land d q_{2} + d q_{2} \land d u) \otimes (\frac{\partial}{\partial q_{1}} + p_{1} \frac{\partial}{\partial u} + κ p_{1}^{2} \frac{\partial}{\partial p_{2}}),

\begin{matrix} d_{1, 1, - 1} = 2 (κ - 1) (κ p_{1}^{3} d q_{1} \land d q_{2} + κ p_{1}^{2} d q_{2} \land d u \\ - d p_{1} \land d u - p_{1} d q_{1} \land d p_{1} - p_{2} d q_{2} \land d p_{1}) \otimes \frac{\partial}{\partial p_{2}}, \end{matrix}

\begin{matrix} d_{2, - 1, 0} = (d q_{1} \land d p_{1} - κ p_{1}^{2} d q_{1} \land d q_{2} + u d q_{2} \land d p_{1}) \\ \otimes (\frac{\partial}{\partial u} + (2 κ - 1) p_{1} \frac{\partial}{\partial p_{2}}) \end{matrix}

\begin{matrix} d_{0, - 1, 2} = (d q_{2} \land d p_{2} + (1 - 2 κ) p_{1} d q_{2} \land d u \\ + (1 - κ) p_{1}^{2} d q_{1} \land d q_{2} - u d q_{2} \land d p_{1}) \\ \otimes (\frac{\partial}{\partial u} + (2 κ - 1) p_{1} \frac{\partial}{\partial p_{2}}) \end{matrix}

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

\begin{array}{l} ξ_{1} & = - d q_{2} \land d p_{1}, \\ ξ_{2} & = 2 (1 - κ) d q_{2} \land d p_{1} . \end{array}

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

v_{t x} = \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 $E \overset{def}{=} E_{ω}$ . Let $P = (C_{1, 0}, l, C_{0, 1})$ be the corresponding almost-product 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 \overset{def}{=} 〈〈Z, d_{1, 1, - 1}〉, 〈Z, d_{- 1, 1, 1}〉〉,

where $〈,〉$ is a contraction.

Let $\tilde{E} \overset{def}{=} E_{\tilde{ω}}$ be an another non-degenerate Monge-Ampère equation with the $A P$ -structure $\tilde{P} = ({\tilde{C}}_{1, 0}, \tilde{l}, {\tilde{C}}_{0, 1},)$ . If $ϕ : J^{1} M \to J^{1} M,$ $ϕ^{*} (U) = λ U$ is a contact transformation such that $ϕ^{*} {(ω)}_{ɛ} = h \tilde{ω}$ for some non-vanishing function $h$ , then (up to permutation of 1st and 3rd members) $ϕ_{*} (\tilde{P}) = P$ and

\begin{array}{l} ϕ_{*}^{- 1} (Z) & = \frac{1}{λ} \tilde{Z}, \\ ϕ^{*} (k) & = \frac{1}{λ^{2}} \tilde{k} \end{array}

Here $\tilde{Z}$ is a vector ﬁeld that generates the distribution $\tilde{l}$ , $U (\tilde{Z}) = 1$ . 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 ${\underline{Ω}}^{s} \subset Ω^{s} (J^{1} M)$ of vanishing on $Z$ differential $s$ -forms:

{\underline{Ω}}^{s} \overset{def}{=} \{α \in 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 $Π : Ω^{s} (J^{1} M) \to {\underline{Ω}}^{s}$ and an operator $\partial : Ω^{s} (J^{1} M) \to {\underline{Ω}}^{s + 1}$ by the following formulas:

\begin{array}{l} Π (α) \overset{def}{=} α - U \land (Z ⌋ α), \\ \partial \overset{def}{=} Π \circ d . \end{array}

The kernel of the operator $Π$ is

ker Π = \{U \land α| α \in Ω^{*} (J^{1} M)\} .

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

ϕ^{*} \circ Π = \tilde{Π} \circ ϕ^{*}

(15)

and

ϕ^{*} \circ \partial = \tilde{\partial} \circ ϕ^{*} .

(16)

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

\begin{array}{l} ϕ^{*} (Π (α)) & = ϕ^{*} (α) - ϕ^{*} (U) \land ϕ^{*} (Z ⌋ α) \\ = ϕ^{*} (α) - U \land (ϕ_{*}^{- 1} (Z) ⌋ ϕ^{*} (α)) \\ = ϕ^{*} (α) - λ U \land (\frac{1}{λ} \tilde{Z} ⌋ ϕ^{*} (α)) \\ = \tilde{Π} (ϕ^{*} (α)) . \end{array}

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

\begin{array}{l} ϕ^{*} (\partial α) & = ϕ^{*} \circ Π \circ d (α) = \tilde{Π} \circ ϕ^{*} \circ d (α) = \\ = \tilde{Π} \circ d \circ ϕ^{*} (α) = \tilde{\partial} (ϕ^{*} (α)) . \end{array}

□

The restriction of $\partial$ to the algebra ${\underline{Ω}}^{*}$ we denote by $δ$ :

\begin{array}{l} δ & : {\underline{Ω}}^{s} \to {\underline{Ω}}^{s + 1}, \\ δ \overset{def}{=} \partial ∣_{{\underline{Ω}}^{*}} . \end{array}

The sequence

0 \overset{δ}{\to} {\underline{Ω}}^{0} \overset{δ}{\to} {\underline{Ω}}^{1} \overset{δ}{\to} {\underline{Ω}}^{2} \overset{δ}{\to} {\underline{Ω}}^{3} \overset{δ}{\to} {\underline{Ω}}^{4} \overset{δ}{\to} 0,

(17)

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

Note that

δ (α \land β) = δ α \land β + {(- 1)}^{deg α} α \land δ β

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

A differential $l$ -horizontal $2$ -form $ω$ is called $l$ -effective if $ω \land \partial U = 0$ . Note that

\partial (g U) = g \partial U

for any function $g \in C^{\infty} (J^{1} M)$ and therefore this deﬁnition is correct.

Let $ω$ be an $l$ -effective differential $2$ -form and let $ϕ : J^{1} M \to J^{1} M$ be a contact diffeomorphism such that $ϕ^{*} (ω) = \hat{ω}$ . Then the form $\hat{ω}$ is $\hat{l}$ -effective also.

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

α_{l} \overset{def}{=} α - s_{α} \partial U \land \partial U,

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

α \land \partial U = s_{α} \partial U \land \partial 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 $E_{\tilde{ω}}$ in terms of $l$ -effective forms: $ϕ^{*} (ω_{l}) = h {\tilde{ω}}_{\tilde{l}}$ .

From this place we suppose that $ω$ and $\tilde{ω}$ are $l$ -effective and $\tilde{l}$ -effective differential $2$ -forms respectively and $ϕ^{*} (U) = λ U,$ $ϕ^{*} (ω) = h \tilde{ω}$ .

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

\begin{align} F \partial U \land \partial U & = ω \land ω, & (18) \\ A X ⌋ \partial U & = X ⌋ ω, \end{align}

where the vector ﬁeld $X \in D (C)$ .

Then

ϕ^{*} (F) = \frac{h^{2}}{λ^{2}} \tilde{F} .

Indeed,

\begin{array}{l} h^{2} \tilde{ω} \land \tilde{ω} & = ϕ^{*} (F \partial U \land \partial U) = ϕ^{*} (F) ϕ^{*} (\partial U) \land ϕ^{*} (\partial U) \\ = ϕ^{*} (F) \tilde{\partial} ϕ^{*} (U) \land \tilde{\partial} ϕ^{*} (U) = λ^{2} ϕ^{*} (F) \tilde{\partial} U \land \tilde{\partial} U . \end{array}

On the other hand $\tilde{ω} \land \tilde{ω} = \tilde{F} \tilde{\partial} U \land \tilde{\partial} U$ . Then, $λ^{2} ϕ^{*} (F) =$ $h^{2} \tilde{F}$ .

The square of the operator $A$ is scalar and $A^{2} + F = 0$ .

The differential $2$ -form $\partial U$ is non-degenerate on the module of vector ﬁelds from the Cartan distribution. This means that if $X ⌋ \partial U = 0$ for $X \in D (C)$ , then $X = 0$ .

For a function $H \in C^{\infty} (J^{1} M)$ the formula

X_{H} ⌋ \partial U = - \partial H

(19)

uniquely deﬁnes a vector ﬁeld $X_{H} \in D (C)$ .

Note that

ϕ_{*}^{- 1} (X_{H}) = \frac{1}{λ} {\tilde{X}}_{ϕ^{*} (H)} .

We need two technical lemmas.

Lemma 3. We have:

\tilde{A} = \frac{λ}{h} ϕ_{*}^{- 1} \circ A \circ ϕ_{*} .

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

λ ϕ_{*}^{- 1} (A X) ⌋ \tilde{\partial} U = h ϕ_{*}^{- 1} (X) ⌋ \tilde{ω} .

Moreover,

ϕ_{*}^{- 1} (X) ⌋ \tilde{ω} = \tilde{A} ϕ_{*}^{- 1} (X) ⌋ \tilde{\partial} U .

Since $\partial U$ is non-degenerate, we get: $λ ϕ_{*}^{- 1} \circ A = h \tilde{A} \circ ϕ_{*}^{- 1}$ . Therefore $\tilde{A} = \frac{λ}{h} ϕ_{*}^{- 1} \circ A \circ ϕ_{*}$ . □

Lemma 4. For any function $h \in C^{\infty} (J^{1} M)$ we have:

\partial h \land ω = \frac{1}{2} A X_{h} ⌋ (\partial U \land \partial U) .

Proof. The form $ω$ is $l$ -effective, i.e., $ω \land \partial U = 0 .$ Then

0 = X_{h} ⌋ (ω \land \partial U) = (A X_{h} ⌋ \partial U) \land \partial U - \partial h \land 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:

\begin{array}{l} W ⌋ (\partial U \land \partial U) & = 2 \partial ω, \\ U (W) & = 0 . \end{array}

Suppose that $k \neq 0$ . Let us introduce the function

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

and the vector ﬁeld

V \overset{def}{=} \frac{1}{k} A W

Since Lemma 3, and the facts that

\begin{array}{l} ϕ_{*}^{- 1} (W) & = \frac{1}{λ^{2}} (h \tilde{W} + \tilde{A} {\tilde{X}}_{h}), \\ ϕ^{*} (F_{0}) & = h^{2} {\tilde{F}}_{0}, \end{array}

we obtain:

\begin{array}{l} ϕ_{*}^{- 1} (V) & = \frac{1}{ϕ^{*} (k)} (ϕ_{*}^{- 1} \circ A \circ ϕ_{*}) \circ ϕ_{*}^{- 1} (W) = \\ = \frac{h^{2}}{λ} \tilde{V} - \frac{h}{λ} {\tilde{F}}_{0} {\tilde{X}}_{h} \end{array}

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

ϕ_{*}^{- 1} (X_{F_{0}}) = \frac{h^{2}}{λ} {\tilde{X}}_{{\tilde{F}}_{0}} + \frac{2 h}{λ} {\tilde{F}}_{0} {\tilde{X}}_{h} .

Deﬁne vector ﬁelds $Z \in D (l)$ and $Y$ by the formulas

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

and

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

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

ϕ_{*}^{- 1} (Y) = \tilde{Y} .

The vector ﬁeld $Y$ splits into the sum

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

where $Y_{1, 0} \in D (C_{1, 0})$ and $Y_{0, 1} \in D (C_{0, 1})$ .

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:

\begin{array}{l} X_{1, 0} \overset{def}{=} d_{- 1, 1, 1} (Z, Y_{0, 1}), \\ X_{0, 1} \overset{def}{=} d_{1, 1, - 1} (Z, Y_{1, 0}), \end{array}

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 $\tilde{E}$ are contact equivalent if their constructed $e$ -structures $e_{E}$ and $e_{\tilde{E}}$ 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_{x y} = f (x, y, v, v_{x}, v_{y}) .

For this equation

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

and

\begin{array}{l} Π (ω) & = (p_{1} f_{p_{1}} + p_{2} f_{p_{2}} - 2 f) d q_{1} \land d q_{2} + d q_{1} \land d p_{1} \\ - d q_{2} \land d p_{2} - f_{p_{2}} d q_{1} \land d u + f_{p_{1}} d q_{2} \land d u, \\ \partial U & = (p_{2} f_{p_{2}} - p_{1} f_{p_{1}}) d q_{1} \land d q_{2} + d q_{1} \land d p_{1} \\ + d q_{2} \land d p_{2} - f_{p_{2}} d q_{1} \land d u - f_{p_{1}} d q_{2} \land d u . \end{array}

Below the form $Π (ω)$ is denoted by $ω$ . We see that $ω \land \partial U = 0$ , $k = f_{p_{1} p_{1}} f_{p_{2} p_{2}}$ 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

\begin{array}{l} W & = f_{p_{2}} \frac{\partial}{\partial p_{1}} - f_{p_{1}} \frac{\partial}{\partial p_{2}}, \\ V & = \frac{1}{f_{p_{1} p_{1}} f_{p_{2} p_{2}}} (f_{p_{2}} \frac{\partial}{\partial p_{1}} + f_{p_{1}} \frac{\partial}{\partial 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}{d q_{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}{d q_{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{\partial}{\partial 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{\partial}{\partial p_{1}} . \end{array}

We obtain the following $e$ -structure:

\begin{array}{l} 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}{d q_{1}} + f \frac{\partial}{\partial p_{2}}) \\ + k^{- 1 ∕ 2} (- 2 f_{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}} + f f_{p_{2} p_{2} p_{2}} f_{p_{1} p_{1}} \\ + p_{1} f_{p_{2} p_{2}} f_{p_{1} p_{1} u} + f f_{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{\partial}{\partial 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}{d q_{2}} + f \frac{\partial}{\partial p_{1}}) \\ k^{- 1 ∕ 2} (p_{2} f_{p_{2} p_{2} u} f_{p_{1} p_{1}} + f_{p_{1} p_{1}} (f f_{p_{1} p_{2} p_{2}} + f_{q_{2} p_{2} p_{2}}) + \\ f_{p_{2} p_{2}} (- 2 f_{p_{1}} f_{p_{1} p_{1}} + p_{2} f_{p_{1} p_{1} u} + f f_{p_{1} p_{1} p_{1}} + f_{q_{2} p_{1} p_{1}})) \frac{\partial}{\partial p_{2}}, \\ Z & = k^{- 1 ∕ 2} (\frac{\partial}{\partial u} + f_{p_{2}} \frac{\partial}{\partial p_{1}} + f_{p_{1}} \frac{\partial}{\partial p_{2}}), \end{array}

\begin{array}{l} 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}} + f f_{p_{1} p_{2}} + f_{q_{2} p_{2}}) + f_{p_{1} p_{1}} (f f_{p_{1} p_{2} p_{2}} + f_{q_{2} p_{2} p_{2}}) \\ + f_{p_{2} p_{2}} (- 2 f_{p_{1}} f_{p_{1} p_{1}} + p_{2} f_{p_{1} p_{1} u} + f f_{p_{1} p_{1} p_{1}} + f_{q_{2} p_{1} p_{1}})) \frac{\partial}{\partial p_{1}} \\ X_{0, 1} & = \frac{1}{f_{p_{2} p_{2}}} (- 2 f_{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}} + f f_{p_{2} p_{2} p_{2}} f_{p_{1} p_{1}} \\ + p_{1} f_{p_{2} p_{2}} f_{p_{1} p_{1} u} + f f_{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} \\ + f f_{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{\partial}{\partial p_{2}} . \end{array}

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Astrakhan State University

E-mail address: kushnera@mail.ru

Received August 4, 2006