Lobachevskii Journal of Mathematics Vol. 4, 1999, 109–162 \begin{matrix}c \\ ◯ \end{matrix}O.P. Tchij

O.P. Tchij

CONTACT GEOMETRY OF HYPERBOLIC MONGE–AMPÈRE EQUATIONS

(submitted by B.N.Shapukov)

Abstract. This paper is devoted to the geometry of hyperbolic Monge–Ampère equations with large symmetry algebras. We classify all hyperbolic Monge–Ampére equations whose Lie algebra of contact symmetries is transitive. We also give the explicit construction for Cartan connections associated with generic hyperbolic Monge–Ampére equations and find those equations which correspond to connections with vanishing curvature tensor.

1. Hyperbolic Monge–Ampère equations with transitive symmetry algebra

1.1. Hyperbolic Monge–Ampr̀e equations. The classical Monge–Ampère equation is a second-order partial differential equation with two independent variables x, y and one unknown function u(x, y):

au_{xx} + 2bu_{xy} + cu_{yy} + f + e Hess u = 0,  (1)

where a, b, c, f, e are functions of x, y, u, u_{x}, u_{y} and Hess u = u_{xx}u_{yy} − u_{xy}^{2}. Geometrically, this equation can be viewed as a hypersurface E in the space J^{2}(ℝ^{3}) of second order jets of 2-dimensional manifolds in ℝ^{3}.

Let C be the contact structure on the manifold M = J^{1}(ℝ^{3}) defined by a contact form ω_{0}. A local diffeomorphism φ: M → M is said to be contact if φ_{∗}(C) = C, where φ_{∗} is the differential of φ. Two Monge–Ampère equations E_{1} and E_{2} are said to be locally equivalent at two points m_{1}, m_{2} ∈ M if there exists a contact local diffeomorphism φ such that φ(m_{1}) = m_{2} and φ^{(2)}(E_{1}) ⊂ E_{2}, where φ^{(2)} is the natural extension of φ to the space J^{2}(ℝ^{3}). A contact vector field X ∈ D(M) is called an infinitesimal symmetry of E if X_{p}^{(2)} ∈ T_{p}(E) for all p ∈ E. The set of all infinitesimal symmetries, denoted sym(E), forms a Lie algebra. We shall say that a Monge–Ampère equation E has a transitive symmetry algebra provided sym(E)_{m} = T_{m}M for all m ∈ M.

1991 Mathematical Subject Classification. 35A30, 58G35.

Key words and phrases. Contact transformations, characteristic distribution, Cartan connections, symmetries, homogeneous spaces.

1.1.1. Exterior Monge–Ampère systems. It can be assumed that, relative to the standard coordinate system (x, y, u, p, q, r, s, t) in J^{2}(ℝ^{3}), the contact form ω_{0} on M = J^{1}(ℝ^{3}) has the form

ω_{0} = du − pdx − qdy.

Let ϑ be an arbitrary differential 2-form on M. The differential system

Σ = {ω_{0}, dω_{0}, ϑ, dϑ}

generated by the forms ω_{0} and ϑ is called an exterior differential Monge– Ampère system (see [33]).

We can assume without loss of generality that the second generator ϑ of Σ has the form

ϑ = a · dp ∧ dy + b · (dx ∧ dp + dq ∧ dy) + c · dx ∧ dq + f · dx ∧ dy + e · dp ∧ dq. (2)

Indeed, any differential 2-form coincides with such a form ϑ modulo {ω_{0}, dω_{0}}. It follows that there is a one-to-one correspondence between Monge–Ampère equations and exterior Monge–Ampère systems.

If u = u(x, y) is a solution of Eq. (1), then the submanifold

L_{u} = {(x, y, u(x, y), u_{x}(x, y), u_{y}(x, y))} ⊂ M

satisfies the conditions ω_{0}|_{L_{u}} = ϑ|_{L_{u}} = 0, so that L_{u} is an integral manifold of Σ. Conversely, every 2-dimensional submanifold L in M such that ω_{0}|_{L} = ϑ|_{L} = 0 will be called a (multi-valued) generalized solution of the equation. If the projection of L onto ℝ^{3} has no singularities, we get an ordinary classical solution. In what follows, unless otherwise stated, by a solution of a Monge– Ampère equation, we shall mean a generalized solution.

Let φ be a contact transformation of M, and let Σ be an exterior Monge– Ampère system. Then φ^{∗}Σ is again an exterior Monge–Ampère system. Thus the family of exterior Monge–Ampère systems on M is invariant under the group of contact transformations of M. It follows that the class of Monge– Ampère equations is invariant under contact transformations and is therefore a natural object to be studied from the point of view of contact geometry.

1.1.2. Characteristic distributions and types of hyperbolic equations. The Pfaffian of Eq. (1) is computed by the formula Pf(E) = ac − b^{2} − ef and coincides with the Pfaffian of the corresponding differential 2-form ϑ (ϑ ∧ ϑ = Pf(ϑ)dω_{0} ∧ dω_{0}). In the definition domain of a hyperbolic equation, we always have Pf(ϑ) < 0. In the following we shall restrict our discussion to hyperbolic Monge–Ampère equations.

For an exterior Monge–Ampère system Σ, we define its characteristic system Char(Σ) as follows: Char(Σ) = ∪_{m∈M} Char(Σ)_{m}, where

Char(Σ)_{m} = {v ∈ C_{m} | v⌟ϑ_{m} = 0 mod (v⌟(dω_{0})_{m}, (ω_{0})_{m})}

for all m ∈ M.

Proposition 1. For every point m ∈ M, there exist two 2-dimensional subspaces E_{m} and F_{m} such that Char(Σ)_{m} = E_{m} ∪ F_{m} and

(a) C_{m} = E_{m} ⊕ F_{m};

(b) (dω_{0})_{m}(E_{m}, F_{m}) = 0.

Proof. Suppose λ_{1} and λ_{2} are the roots of the characteristic equation λ^{2} + Pf(ϑ) = 0. Define two subspaces E_{m} and F_{m} as follows:

E_{m} = {v ∈ C_{m} | v⌟(ϑ − λ_{1}dω_{0}) = 0},
F_{m} = {v ∈ C_{m} | v⌟(ϑ − λ_{2}dω_{0}) = 0}.

Since the 2-forms (ϑ+λ_{i}dω_{0})_{|C} are decomposable, they can be written as κ_{1}∧κ_{2} and ν_{1} ∧ ν_{2} respectively, where κ_{1}, κ_{2}, ν_{1}, ν_{2} are linear functions on C_{m}. Then E_{m} (resp. F_{m}) is precisely the subspace on which the forms κ_{1}, κ_{2} (resp. ν_{1}, ν_{2}) vanish, and hence Char(Σ)_{m} = E_{m} ∪ F_{m}.

Since dω_{0} ∧ dω_{0}_{|C} ≠ 0, the functions κ_{1}, κ_{2}, ν_{1}, ν_{2} are linearly independent, and it follows that C_{m} = E_{m} ⊕ F_{m}.

Now since (λ_{1} − λ_{2})dω_{0}_{|C} = κ_{1} ∧ κ_{2} − ν_{1} ∧ ν_{2}, it can be seen that E_{m} and F_{m} are orthogonal relative to dω_{0}, which completes the proof of the proposition.

Because we are concerned with a local problem in the neighborhood of a point of general position, we can assume that E = ∪_{m∈M}E_{m} and F = ∪_{m∈M}F_{m} are distributions on M. The two distributions E, F so obtained are called the characteristic distributions of the hyperbolic Monge–Ampère equation in question.

Conversely, given a pair of distributions (E, F) satisfying conditions (a) and (b), there exists a unique exterior Monge–Ampère system Σ such that Char(Σ) = E ∪ F.

Indeed, to any exterior Monge–Ampère system, given by a form ϑ, one can assign a unique operator field J on the contact distribution C by letting

ϑ(X, Y)_{|C} = dω_{0}(JX, Y) = dω_{0}(X, JY),

where X, Y ∈ C. It can be shown by straightforward computation that J^{2} + Pf(ϑ) = 0 and tr J = 0. At each point m ∈ M, the subspaces E_{m} and F_{m} are eigen-spaces of J corresponding to the eigen-values λ_{1} and λ_{2} respectively. Therefore, J determines an almost-product structure at each point and can be uniquely recovered from the pair of characteristic distributions.

The following theorem allows to reduce the problem of equivalence of hyperbolic equations to the equivalence of pairs of characteristic distributions.

Theorem 1. Two hyperbolic Monge–Ampère equations E_{1} and E_{2} are equivalent by means of a contact diffeomorphism φ if and only if so are the corresponding pairs of characteristic distributions, i.e. φ_{∗}(E_{1}, F_{1}) = (E_{2}, F_{2}).

The proof of this theorem can be found in [14, 33].

Thus we see that the correspondence between equations and characteristic distributions is in agreement with the action of the group of contact transformations.

We shall now establish the form of this correspondence in terms of the coordinate system in M = J^{1}(ℝ^{3}) that we fixed above.

The vector fields

\{\frac{∂}{∂x} + p\frac{∂}{∂u}, \frac{∂}{∂p}, \frac{∂}{∂y} + q \frac{∂}{∂u}, \frac{∂}{∂q} \}

form a basis of the contact distribution at each point in M. In terms of this basis, the restrictions of the forms ϑ and dω_{0} to the contact distribution have the following matrices:

ϑ_{|C} = (\begin{matrix}0 & b & f & c \\ −b & 0 & a & e \\ −f & −a & 0 & −b \\ −c & −e & b & 0 \end{matrix}) , (dω_{0})_{|C} = (\begin{matrix}0 & 1 & 0 & 0 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \end{matrix}) .

The matrix represeentation of the operator field J is as follows:

J = (dω_{0})_{|C}^{(−1)} · ϑ_{|C} = (\begin{matrix}b & 0 & −a & −e \\ 0 & b & f & c \\ c & e & −b & 0 \\ −f & −a & 0 & −b \end{matrix}) .

The eigen-space of J corresponding to the eigen-value λ = ±\sqrt{b^{2} − ac + ef} is generated by the following vector fields:

\begin{matrix}e_{1} = e (\frac{∂}{∂x} + p\frac{∂}{∂u}) − c\frac{∂}{∂p} + (b − λ)\frac{∂}{∂q} , \\ e_{2} = (b + λ)\frac{∂}{∂p} + e (\frac{∂}{∂y} + q \frac{∂}{∂u}) − a\frac{∂}{∂q} , \\ e_{3} = a (\frac{∂}{∂x} + p\frac{∂}{∂u}) − f \frac{∂}{∂p} + (b − λ) (\frac{∂}{∂y} + q \frac{∂}{∂u}) , \\ e_{4} = (b + λ) (\frac{∂}{∂x} + p\frac{∂}{∂u}) + c (\frac{∂}{∂y} + q \frac{∂}{∂u}) − f \frac{∂}{∂q} , \end{matrix}  (3)

of which only two are linearly independent at every point of M.

Conversely, given a pair of characterstic distributions E = ⟨j_{1}, j_{2}⟩ and F = ⟨j_{3}, j_{4}⟩, let H = {j_{1}, j_{2}, j_{3}, j_{4}} be the 4 × 4 matrix composed of the coefficients of the vectors j_{1}, j_{2}, j_{3}, j_{4} in terms of our fixed basis in C. Since the operator J determines an almost-product structure in C_{m} for all m ∈ M, we can assume that, relative to the basis {j_{1}, j_{2}, j_{3}, j_{4}}, it has the form

B = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix}) .

Thus J can be derived from the equality:

J = H · B · H^{−1}.

Now the coefficients of an equation with given characteristic system can be determined from the formula:

ϑ_{|C} = (dω_{0})_{|C} · J = (dω_{0})_{|C} · H · B · H^{−1}.  (4)

Let R be an arbitrary distribution on M. We define the derived distributions R^{(k)} (k = 0, 1, 2, …) by induction as follows: R^{(0)} = R, R_{m}^{(1)} = ⟨X_{m}, Y_{m}, [X, Y]_{m}⟩ at each point m ∈ M, where X, Y are vector fields lying in R, and R^{(k+1)} = R^{(k)}^{(1)}.

Lemma 1. The characteristic distributions E and F of a hyperbolic equation are no completely integrable.

Proof. Let X, Y be vector fields lying in E. Assume that E is completely integrable, so that [X, Y] ∈ E. Since ω_{0}(X) = ω_{0}(Y) = 0, we have dω_{0}(X, Y) = −ω_{0}([X, Y]) = 0. Therefore dω_{0}(E, E) = 0, and E is a Lagrangean subspace (maximal isotropic subspace) with respect to dω_{0}. In view of dω_{0}(E, F) = 0, this means that dω_{0}(E, C) = 0, which contradicts the fact that dω_{0} must be nondegenerate on the contact distribution C.

Thus for the characteristic distributions E, F we have E ≠ E^{(1)}, F ≠ F^{(1)}, and dim E^{(1)} = dim F^{(1)} = 3.

Now let R be a not completely integrable 2-dimensional distribution on M. Consider the derived distributions R^{(k)} (k = 1, 2, 3). The only four possibilities for the non-decreasing triple (dim R^{(1)}, dim R^{(2)}, dim R^{(3)}) are

(1) — (3, 3, 3);
(2) — (3, 4, 4);
(3) — (3, 4, 5);
(4) — (3, 5, 5).

We shall say that a Monge–Ampère equation is of type H_{(i,j)} if the triple corresponding to E is (i), while the triple corresponding to F is (j), where i, j = 1, …, 4. To be definite, we shall assume that j ⩽ i.

Since φ_{∗}[X, Y] = [φ_{∗}X, φ_{∗}Y] for any pair X, Y of vector fields on M and for any diffeomorphism φ, we see that the notion of derived distribution is invariant under diffeomorphism. Therefore the type of the equation is preserved by contact transformations.

Example 1. By way of example, consider the wave equation u_{xy} = 0. According to formulas (3), the characteristic distributions are generated by the vector fields

E = ⟨\frac{∂}{∂x} + p\frac{∂}{∂u}, \frac{∂}{∂p}⟩ , F = ⟨\frac{∂}{∂y} + q \frac{∂}{∂u}, \frac{∂}{∂q} ⟩ .

The first derivations of these distributions are

E^{(1)} = ⟨\frac{∂}{∂x}, \frac{∂}{∂u}, \frac{∂}{∂p}⟩ , F^{(1)} = ⟨\frac{∂}{∂y} , \frac{∂}{∂u}, \frac{∂}{∂q} ⟩ .

It can be seen that E^{(1)} and F^{(1)} are completely integrable distributions, and hence E^{(1)} = E^{(2)} = E^{(3)}, F^{(1)} = F^{(2)} = F^{(3)}. Therefore the characteristic distributions both have type (1), so that our equation is of type H_{(1,1)}.

We shall now see how the classical results on the classification of hyperbolic equations up to the group of contact transformations can be formulated in terms of equation types.

The next theorem goes back Sophus Lie:

Theorem 2 ([31]). All Monge–Ampère equations of type H_{(1,1)} are equivalent to each other.

Thus every equation of type H_{(1,1)} is locally equivalent to the wave equation u_{xy} = 0.

In Goursat’s textbook [26] there is a proof of the following fact (see also [15, 36]):

Theorem 3. Every Monge–Ampère equation of type H_{(2,i)} (i = 1, 2) is locally equivalent to an equation of the form

u_{xy} + f(x, y, u, u_{x}, u_{y}) = 0.  (5)

In [31], Lie determines canonical forms for equations of the form (5) with nontrivial finite or infinite group of contact transformations.

Therefore, to obtain a complete classification of hyperbolic equations, one has to find normal forms of the equations of type H_{(3,i)} (i = 1, 2, 3) and H_{(4,i)} (i = 1, 2, 3, 4) with transitive algebra of contact symmetries. The rest of the present paper is devoted to this task.

These two classes of equations are extremely important because classical analytical methods, such as the method of intermediate integral or Monge’s method, are not applicable to them.

1.1.3. Method of intermediate integral. Let (E, F) be the characteristic system for some hyperbolic Monge–Ampère equation. Let I be an integral of one of the distributions E or F. We shall now show how the knowledge of I can be used to construct a solution of the initial equation that will depend on a function and a constant. We remark that in order that such an integral I exist, it is necessary and sufficient that the distribution be of type (1) or (2), because only in these two cases the last member of the derived series of distributions does not coincides with all of the tangent space.

Lemma 2. A 2-dimensional submanifold in L ⊂ M is an integral manifold of an exterior Monge–Ampère system (i.e., a many-valued solution of an equation) if and only if

dim(L_{m} ∩ E_{m}) = dim(L_{m} ∩ F_{m}) = 1

for every point m ∈ L.

Proof. Let L be an integral manifold of an exterior Monge–Ampère system. Then ϑ_{|L} = ω_{0}_{|L} = 0, where ω_{0} is a contact form on M and ϑ ∈ Λ^{2}(M) is the differential form that determines the initial equation. Thus L_{m} ⊂ C_{m}, and since dω_{0}_{|L} = 0, we see that at each point m ∈ L, the subspace L_{m} is isotropic with respect to the restriction of dω_{0} to the contact distribution. On the other hand, the condition ϑ_{|L} = 0 implies that L_{m} is also an isotropic subspace with respect to the form ϑ. In a certain basis of the tangent space at the point m, the restrictions of the matrices A = (dω_{0})_{m} and B = ϑ_{m} to the contact distribution have the form

A|_{C_{m}} = (\begin{matrix}J & 0 \\ 0 & J \end{matrix}) ,  B|_{C_{m}} = (\begin{matrix}J & 0 \\ 0 & −J \end{matrix}) ,  J = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) ,

assuming that E is generated by the vectors ⟨(1, 0, 0, 0), (0, 1, 0, 0)⟩, and F is generated by the vectors ⟨(0, 0, 1, 0), (0, 0, 0, 1)⟩. Therefore, if L_{m} is a Lagrangean subspace relative to the forms A and B, then it has the form L_{m} = (L_{m} ∩ E_{m}) ⊕ (L_{m} ∩ F_{m}).

Conversely, if L is a submanifold in M such that L_{m} = (L_{m}∩E_{m})⊕(L_{m}∩F_{m}), then by virtue of dω_{0}(E, F) = 0, L is an integral manifold of the exterior Monge–Ampère system.

Thus we have reformulated the notion of solution of a Monge–Ampère equation in terms of characteristic distributions.

Proposition 2. Let I be an integral of one of the distributions E or F. Let L be a 2-dimensional Legendrean submanifold (i.e. ω_{0}_{|L} = 0) such that I(L) = const, so that L lies on some level surface of I. Then L is an integral manifold of the exterior Monge–Ampère system corresponding to E and F.

Proof. Let K be the completely integrable distribution given by the integral I, i.e. K = {dI = 0}. By the preceding lemma, in order to prove our assertion, it will suffice to show that dim(L_{m} ∩ E_{m}) = dim(L_{m} ∩ F_{m}) = 1 for all m ∈ L. To be definite, let I be an integral of the distribution E.

Since L is a Legendrean submanifold, we have L_{m} ⊂ C_{m}, and since L lies on the level surface I, we have L_{m} ⊂ K_{m}. The contact distribution is not completely integrable and dim C_{m} = 4. Therefore dim K_{m} ∩ C_{m} = dim K_{m} + dim C_{m} − dim M = 3. On the other hand E_{m} ⊂ K_{m} ∩ C_{m} and L_{m} ⊂ K_{m} ∩ C_{m}. The subspace L_{m} is isotropic relative to dω_{0}, and hence L_{m} and E_{m} differ from each other. It follows that dim L_{m}∩E_{m} = dim L_{m}+dim E_{m}−dim(K_{m}∩C_{m}) = 1. Finally, from C_{m} = E_{m} ⊕ F_{m}, we get dim L_{m} ∩ F_{m} = 1.

Example 2. Consider the following equation of gas dynamics:

u_{xx}u_{yy} − u_{xy}^{2} + y^{−4} = 0,

(see [12], Ch. I, §12). Then ϑ = dp ∧ dq + y^{−4}dx ∧ dy and Pf(ϑ) = −y^{−4}. The distribution E is generated by the vector fields:

X_{1} = \frac{∂}{∂x} + p\frac{∂}{∂u} − y^{−2} \frac{∂}{∂q} , X_{2} = \frac{∂}{∂y} + q \frac{∂}{∂u} + y^{−2} \frac{∂}{∂p},

while the distribution F is generated by

X_{3} = \frac{∂}{∂x} + p\frac{∂}{∂u} + y^{−2} \frac{∂}{∂q} , X_{4} = \frac{∂}{∂y} + q \frac{∂}{∂u} − y^{−2} \frac{∂}{∂p}.

The equation is of type H_{(1,1)}, so that there exist two functionally independent integrals of E, namely I_{1} = p + y^{−1} and I_{2} = u − xp − yq − \frac{x}{y}, and two for F, namely I_{3} = p − y^{−1} and I_{4} = u − xp − yq + \frac{x}{y}. To solve the initial equation, one has to solve the following partial differential equations of the first order: u_{x} = ±\frac{1}{y} + c_{1} and xu_{x} + yu_{y} = u ∓ \frac{x}{y} − c_{2}.

As a result, we get two classes of solutions for the initial equation: u = ±\frac{x}{y} + c_{1}x + f(y) and u = yg (\frac{x}{y}) ± \frac{x}{y} + c_{2}, where f, g are arbitrary smooth functions. Consequently, the general solution of the equation has the form:

u = ±\frac{x}{y} + f(y) + yg (\frac{x}{y}) .

We have thus shown that the classical method for the integration of Monge– Ampère equations admits a natural interpretation in terms of contact geometry.

1.1.4. Geometric interpretation of Monge–Ampère equations. In this section, we assign to each hyperbolic Monge–Ampère equation a certain geometric structure, namely a sub-bundle in the bundle of linear frames or a G-structure, and show that the problem of equivalence for equations can be reduced to a purely geometric problem. A detailed presentation of the theory of Gstructures can be found in [13].

Definition 1. By a linear frame p at a point m ∈ M we mean an ordered basis {X_{1}, X_{2}, X_{3}, X_{4}, X_{5}} of the tangent space T_{m}M.

Let L(M) be the set of all linear frames p at all points of M, and let π be the map from L(M) to M taking a linear frame p at m to the point m. As a result we obtain a vector bundle L(M), which is called the bundle of linear frames over M.

Let V = ℝ^{5} be 5-dimensional real vector space with standard basis

{u_{1} = (1, 0, 0, 0, 0), …, u_{5} = (0, 0, 0, 0, 1)}.

Then the group GL(V) can be viewed as the structure group of the bundle L(M). Let G be a subgroup of GL(V), and 𝔤 the corresponding Lie algebra.

Definition 2. A G-structure on M is a reduction of the bundle of linear frames L(M) to G.

Let P and P′ be two G-structures on M. Given a diffeomorphism φ: M → M, the tangent map φ_{∗}: L(M) → L(M) is an isomorphism of the bundle of frames. A diffeomorphism φ: M → M is called an isomorphism of the G-structures P and P′ if φ_{∗}(P) = P′.

Suppose π: P → M is the canonical projection. Any linear frame p = (X_{1}, …, X_{5}) ∈ P such that π(p) = m can be seen as a linear isomorphism p: V → T_{m}(M) such that p(u_{i}) = X_{i} for i = 1, …, 5. There is a canonical V -valued linear differential form ω defined on the bundle of frames: ω(Y) = p^{−1}(π_{∗}(Y)) for Y ∈ T_{p}(P). Every G-structure P is a submanifold in L(M). The restriction of ω to P is a V -valued form on P, which will be denoted by the same letter ω. Relative to the standard basis of V, we can write ω = ω^{1}u_{1} + ⋯ + ω^{5}u_{5}.

Locally, we shall identify every G-structure P with a direct product U × G, where U ⊂ M. This allows to view the canonical left-invariant 𝔤-valued form θ on the Lie group G as a form on P. Then the forms (ω, θ) generate the cotangent bundle at each point in P.

Let (E, F) be a characteristic Monge–Ampère system. Since dim E^{(1)} = dim F^{(1)} = 3, the intersection E^{(1)} ∩F^{(1)} is a 1-dimensional distribution that is invariantly connected with the initial equation. In the bundle of linear frames, we choose a sub-bundle P_{7} such that p(⟨u_{2}, u_{3}⟩) = E_{m}, p(⟨u_{4}, u_{5}⟩) = F_{m}, p(⟨u_{1}⟩) = (E^{(1)} ∩ F^{(1)})_{m} for π(p) = m and such that the restriction of ω to P_{7} has the following structure equations:

dω^{1} = −ω^{2} ∧ ω^{3} − ω^{4} ∧ ω^{5} mod ⟨ω^{1}, θ⟩,

\begin{matrix}dω^{2} = 0 \\ dω^{3} = 0 \end{matrix} mod ⟨ω^{1}, ω^{2}, ω^{3}, θ⟩,

\begin{matrix}dω^{4} = 0 \\ dω^{5} = 0 \end{matrix} mod ⟨ω^{1}, ω^{4}, ω^{5}, θ⟩.

Straightforward computation shows that the group G_{7} = SL(2, ℝ)×SL(2, ℝ)× ℝ is the structure group of the bundle P_{7} and that its Lie algebra 𝔤_{7} has the form:

𝔤_{7} = \{(\begin{matrix}2λ & 0 & 0 & 0 & 0 \\ 0 & a_{1} + λ & b_{1} & 0 & 0 \\ 0 & c_{1} & −a_{1} + λ & 0 & 0 \\ 0 & 0 & 0 & a_{2} + λ & b_{2} \\ 0 & 0 & 0 & c_{2} & −x_{2} + λ \end{matrix})| a_{1}, b_{1}, c_{1}, a_{2}, b_{2}, c_{2}, λ ∈ ℝ\} .

Thus, to each hyperbolic Monge–Ampère equation we assign a G-structure P_{7} on M with structure group G_{7}.

Example 3. For the wave equation u_{xy} = 0, consider the following ordered set of vector fields:

V_{1} = \frac{∂}{∂u}, V_{2} = \frac{∂}{∂x} + p\frac{∂}{∂u}, V_{3} = \frac{∂}{∂p}, V_{4} = \frac{∂}{∂y} + q \frac{∂}{∂u}, V_{5} = \frac{∂}{∂q} .

Then E = ⟨V_{2}, V_{3}⟩, F = ⟨V_{4}, V_{5}⟩, E^{(1)} ∩ F^{(1)} = ⟨V_{1}⟩, and [V_{2}, V_{3}] = V_{1}, [V_{4}, V_{5}] = V_{1}, [V_{i}, V_{j}] = 0 for i = 2, 3, j = 4, 5. Consequently, the vector fields (V_{1}, V_{2}, V_{3}, V_{4}, V_{5}) determine a section s: M → P_{7} of the G-structure P_{7}.

Proposition 3. For a diffeomorphism φ: M → M to be an isomorphism of two characteristic systems (E, F) and (E′, F′), it is necessary and sufficient that φ be an isomorphism of the G-structures P_{7} and P′_{7}.

Proof. Suppose that φ_{∗}(E) = E′ and φ_{∗}(F) = F′, so that the two characteristic systems are equivalent by means of φ. We show that in this case, φ is an isomorphism of the corresponding G-structures P_{7} and P′_{7}. Let s: M → P_{7} be an arbitrary section of P_{7}, given by a set of vector fields (V_{1}, V_{2}, V_{3}, V_{4}, V_{5}), where E = ⟨V_{2}, V_{3}⟩ and F = ⟨V_{4}, V_{5}⟩. Then, by the definition of G-structure, these vector fields satisfy the following conditions:

[V_{2}, V_{3}] = V_{1} mod E,
[V_{4}, V_{5}] = V_{1} mod F,
[V_{i}, V_{j}] = 0 mod C for i = 2, 3, j = 4, 5.

We apply φ to both parts of these equalities:

[φ_{∗}(V_{2}), φ_{∗}(V_{3})] = φ_{∗}(V_{1}) mod φ_{∗}(E),
[φ_{∗}(V_{4}), φ_{∗}(V_{5})] = φ_{∗}(V_{1}) mod φ_{∗}(F),
[φ_{∗}(V_{i}), φ_{∗}(V_{j})] = 0 mod φ_{∗}(C) for i = 2, 3, j = 4, 5.

Since C = E′ ⊕ F′, we have φ_{∗}(C) = C and

[φ_{∗}(V_{2}), φ_{∗}(V_{3})] = φ_{∗}(V_{1}) mod E′,
[φ_{∗}(V_{4}), φ_{∗}(V_{5})] = φ_{∗}(V_{1}) mod F′,
[φ_{∗}(V_{i}), φ_{∗}(V_{j})] = 0 mod C for i = 2, 3, j = 4, 5.

It follows that the vector fields (φ_{∗}(V_{1}), φ_{∗}(V_{2}), φ_{∗}(V_{3}), φ_{∗}(V_{4}), φ_{∗}(V_{5})) define a section s′ of P′_{7}. Thus φ_{∗}(P_{7}) = P′_{7}, and so φ is an isomorphism of the Gstructures P_{7} and P′_{7}.

Conversely, let φ be an isomorphism of P_{7} and P′_{7}. Let us show that φ_{∗}(E) = E′. The subspace W = ⟨u_{2}, u_{3}⟩ is invariant under the action of the group G_{7} in V, i.e. g(W) = W for all g ∈ G_{7}. By the definition of G-structure, for any linear frame p ∈ P_{7} such that π(p) = m, we have E_{m} = p(W) for all m ∈ M. Similarly, for every frame p′ ∈ P′_{7} we get E′_{m} = p′(W). If φ is an isomorphism of the G-structures P_{7} and P′_{7}, then φ_{∗}(p) ∈ P′_{7}. It follows that

φ_{∗}(E_{m}) = φ_{∗}(p(W)) = (φ_{∗}(p))(W) = E′_{φ(m)},

as was to be proved.

Thus, the problem of classifying hyperbolic Monge–Ampère equations is equivalent to classifying the corresponding G-structures up to isomorphism.

2. Equations of type H_{(4,i)}

2.1. Equations of type H_{(4,i)} and associated Cartan connections. This section is devoted to the study of Monge–Ampère equations of type H_{(4,i)} from the geometric point of view. Equations of this type are equations of general position, with which we invariantly associate Cartan connections. Differentialgeometric tools are used to solve the problem of local equivalence of equations up to the group of contact transformations. Normal forms are obtained for equations with contact transformation groups of maximal dimension.

2.1.1. Systems of type H_{(4,i)} and G-structures. Let M = J^{1}(ℝ^{3}), and let C be a contact structure on M, given by a local differential 1-form ω_{0}. With every hyperbolic Monge–Ampère equation we have associated a pair of characteristic distributions (E, F). Denote by L(M) the bundle of linear frames over M, and by ω its canonical form. Let V = ℝ^{5} be 5-dimensional real vector space with standard basis {u_{1}, …, u_{5}}.

Suppose that E is of type (4), so that our characteristic system is of type H_{(4,i)}. Then locally, it is possible to choose a sub-bundle P in the bundle of linear frames such that p(⟨u_{2}, u_{3}⟩) = E_{m}, p(⟨u_{4}, u_{5}⟩) = F_{m} for π(p) = m and such that the restriction of the canonical form ω to P has the following structure equations:

dω^{1} = ε · ω^{2} ∧ ω^{3} − ω^{4} ∧ ω^{5} mod ⟨ω^{1}, θ⟩,

\begin{matrix}dω^{2} = 0 \\ dω^{3} = 0 \end{matrix} mod ⟨ω^{1}, ω^{2}, ω^{3}, θ⟩,

\begin{matrix}dω^{4} = −3ω^{1} ∧ ω^{2} \\ dω^{5} = −3ω^{1} ∧ ω^{3} \end{matrix} mod ⟨ω^{4}, ω^{5}, θ⟩,

where ε = ±1. Depending on whether ε is equal to +1 or −1, we shall say that the equation has type H_{(4,i)}^{+} or H_{(4,i)}^{−}.

Straightforward computation shows that the group G = SL(2, ℝ) is the structure group of P and that its Lie algebra 𝔤 has the form

𝔤 = \{(\begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & x_{1} & x_{2} & 0 & 0 \\ 0 & x_{3} & −x_{1} & 0 & 0 \\ 0 & 0 & 0 & x_{1} & x_{2} \\ 0 & 0 & 0 & x_{3} & −x_{1} \end{matrix})| x_{1}, x_{2}, x_{3} ∈ ℝ\} .

Choose a basis {e_{1}, e_{2}, e_{3}} in 𝔤 in such a way that 𝔤 = x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3}. The canonical 𝔤-valued form θ can be written in the form θ = θ^{1}e_{1} + θ^{2}e_{2} + θ^{3}e_{3}. Then the structure equations assume the form

dω^{1} = ε · ω^{2} ∧ ω^{3} − ω^{4} ∧ ω^{5} mod ⟨ω^{1}⟩,

\begin{matrix}dω^{2} = −θ^{1} ∧ ω^{2} − θ^{2} ∧ ω^{3} \\ dω^{3} = θ^{1} ∧ ω^{3} − θ^{3} ∧ ω^{2} \end{matrix} mod ⟨ω^{1}, ω^{2}, ω^{3}⟩,

\begin{matrix}dω^{4} = −θ^{1} ∧ ω^{4} − θ^{2} ∧ ω^{5} − 3ω^{1} ∧ ω^{2} \\ dω^{5} = θ^{1} ∧ ω^{5} − θ^{3} ∧ ω^{4} − 3ω^{1} ∧ ω^{3} \end{matrix} mod ⟨ω^{4}, ω^{5}⟩

(6)

([13], Ch. VII, §2).

Thus, to each system of type H_{(4,i)} we assign a G-structure P on M with structure group SL(2, ℝ). Similar to Proposition 3, we can prove the following:

Proposition 4. A necessary and sufficient condition for a diffeomorphism φ: M → M to be an isomorphism of two characteristic systems (E, F) and (E′, F′) is that φ be an isomorphism of the G-structures P and P′.

Consequently, we can classify all systems of type H_{(4,i)} by classifying the corresponding G-structures up to isomorphism.

2.1.2. Cartan connections. We shall now give basic definitions of the theory of Cartan connections, which we shall use later on. For a detailed presentation of this theory, see [3].

Let \overline{G} be a Lie group, G a Lie subgroup of \overline{G}, and (𝔤̄, 𝔤) the corresponding pair of Lie algebras. Let P be the principal fiber bundle with structure group G. Denote by X^{∗} the fundamental vector field corresponding to an element X ∈ 𝔤.

Definition 3. A Cartan connection of type (\overline{G}, G) in the bundle P is a differential 1-form ω̄ on P with values in 𝔤̄ such that

1) ω̄(X^{∗}) = X for all X ∈ 𝔤;

2) R_{g}^{∗}(ω̄) = (Ad g^{−1})ω̄ for all g ∈ G;

3) ω̄_{p}: T_{p}P → 𝔤̄ is a linear isomorphism for all p ∈ P.

Two Cartan connections ω̄ and ω̄′ of type (\overline{G}, G) on principal fiber bundles P and P′ are said to be isomorphic if there exists an isomorphism of bundles F: P → P′ such that F^{∗}(ω̄′) = ω̄. The 2-form Ω = dω̄ + [ω̄, ω̄]/2 is called the curvature of the Cartan connection ω̄.

Definition 4. The function c: P → L(𝔤̄/𝔤 ∧ 𝔤̄/𝔤, 𝔤̄) such that

c(p): (X + 𝔤, Y + 𝔤) ↦ Ω(\overline{X}, \overline{Y}),

where \overline{X} is the vector field on P defined by ω̄(\overline{X}) = X for X ∈ 𝔤̄, is called the structure function of the Cartan connection ω̄.

The pair (\overline{G}, G) is called the model of the Cartan connection.

We fix two pairs of Lie groups (\overline{G}, G), which we shall later use as models for Cartan connections.

Let \overline{G}_{1} = SL(3, ℝ), and let SL(2, ℝ) be the following subgroup of SL(3, ℝ):

\{(\begin{matrix}A & 0 \\ 0 & 1 \end{matrix})| A ∈ SL(2, ℝ)\} .

Let {e_{1}, e_{2}, e_{3}, u_{1}, u_{2}, u_{3}, u_{4}, u_{5}} be a basis for 𝔰𝔩(3, ℝ) such that

𝔰𝔩(3, ℝ) = {x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3} + x_{4}u_{1} + x_{5}u_{2} + x_{6}u_{3} + x_{7}u_{4} + x_{8}u_{5}} =

= \{(\begin{matrix}x_{1} + x_{4} & x_{2} & x_{5} + x_{7} \\ x_{3} & −x_{1} + x_{4} & x_{6} + x_{8} \\ −x_{6} + x_{8} & x_{5} − x_{7} & −2x_{4} \end{matrix})| x_{1}, …, x_{8} ∈ ℝ\} .

Let \overline{G}_{2} = SU(1, 2), where the invariant Hermitean quadratic form is |z_{1}|^{2} − |z_{2}|^{2} − |z_{3}|^{2}, and let SU(1, 1) be the subgroup in SU(1, 2) of the form

\{(\begin{matrix}A & 0 \\ 0 & 1 \end{matrix})| A ∈ SU(1, 1)\} .

Let {e_{1}, e_{2}, e_{3}, u_{1}, u_{2}, u_{3}, u_{4}, u_{5}} be a basis for 𝔰𝔲(1, 2) such that

𝔰𝔲(1, 2) = {x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3} + x_{4}u_{1} + x_{5}u_{2} + x_{6}u_{3} + x_{7}u_{4} + x_{8}u_{5}} =

= \{ (\begin{matrix}i(x_{2} − \frac{1}{4} x_{3} + x_{4}) & x_{1} − i(x_{2} + \frac{1}{4} x_{3}) & x_{5} + \frac{1}{2} x_{8} + i(x_{7} − \frac{1}{2} x_{6}) \\ x_{1} + i(x_{2} + \frac{1}{4} x_{3}) & i(−x_{2} + \frac{1}{4} x_{3} + x_{4}) & x_{5} − \frac{1}{2} x_{8} + i(x_{7} + \frac{1}{2} x_{6}) \\ x_{5} + \frac{1}{2} x_{8} + i(−x_{7} + \frac{1}{2} x_{6}) & −x_{5} + \frac{1}{2} x_{8} + i(x_{7} + \frac{1}{2} x_{6}) & −2 i x_{4} \end{matrix})| x_{1}, …, x_{8} ∈ ℝ\}.

Since the groups SL(2, ℝ) and SU(1, 1) are isomorphic, we shall always identify them. Relative to the bases just fixed, the structure equations for the Cartan connection of type (SL(3, ℝ), SL(2, ℝ)) with ε = 1 and of type (SU(1, 2), SU(1, 1)) with ε = −1 assume the form:

dω^{1} = ε · ω^{2} ∧ ω^{3} − ω^{4} ∧ ω^{5} + Ω^{1},
dω^{2} = −θ^{1} ∧ ω^{2} − θ^{2} ∧ ω^{3} − 3ε · ω^{1} ∧ ω^{4} + Ω^{2},
dω^{3} = θ^{1} ∧ ω^{3} − θ^{3} ∧ ω^{2} − 3ε · ω^{1} ∧ ω^{5} + Ω^{3},
dω^{4} = −θ^{1} ∧ ω^{4} − θ^{2} ∧ ω^{5} − 3ω^{1} ∧ ω^{2} + Ω^{4},
dω^{5} = θ^{1} ∧ ω^{5} − θ^{3} ∧ ω^{4} − 3ω^{1} ∧ ω^{3} + Ω^{5},
dθ^{1} = −θ^{2} ∧ θ^{3} − ω^{2} ∧ ω^{5} − ω^{3} ∧ ω^{4} + Θ^{1},
dθ^{2} = −2θ^{1} ∧ θ^{2} + 2ω^{2} ∧ ω^{4} + Θ^{2},
dθ^{3} = 2θ^{1} ∧ θ^{3} − 2ω^{3} ∧ ω^{5} + Θ^{3},

where Ω^{i} = ∑ \frac{1}{2}c_{jk}^{i}ω^{j} ∧ ω^{k}, Θ^{i} = ∑ \frac{1}{2}c_{ijk}ω^{j} ∧ ω^{k}, and c_{jk}^{i}, c_{ijk} are the structure functions of the Cartan connection.

Now let P be the principal SL(2, ℝ)-bundle over the 5-dimensional manifold M, and let ω be a given V -valued form ω. Let us consider the question whether there exists an 𝔰𝔩(2, ℝ)-valued form θ such that (ω, θ) is a Cartan connection of type (SL(3, ℝ), SL(2, ℝ)) (resp. (SU(1, 2), SU(1, 1)) in P. It follows from the definition of Cartan connection that such a form exists if and only if the form ω satisfies the following conditions:

(a) ω(X) = 0 iff the vector X is vertical.

(b) R_{g}^{∗}ω = (Ad g^{−1})ω for all g ∈ SL(2, ℝ), where Ad g^{−1} is the transformation of V (= 𝔤̄/𝔤) induced by Ad g^{−1} : 𝔤̄ → 𝔤̄.

We define a W ⊂ L(𝔤̄/𝔤 ∧ 𝔤̄/𝔤, 𝔤̄) by means of the following system of equations:

\begin{matrix}c_{13}^{2} = c_{12}^{3} = 0, c_{12}^{2} = c_{13}^{3}; \\ c_{23}^{2} = c_{23}^{3} = c_{45}^{4} = c_{45}^{5} = 0; \\ c_{24}^{3} = c_{24}^{5} = c_{35}^{2} = c_{35}^{4} = 0; \\ c_{25}^{2} + c_{34}^{2} = c_{35}^{3}, c_{25}^{4} + c_{34}^{4} = c_{35}^{5}; \\ c_{34}^{3} + c_{25}^{3} = c_{24}^{2}, c_{34}^{5} + c_{25}^{5} = c_{24}^{4}, \end{matrix}  (7)

where Ω^{i} = ∑ \frac{1}{2}c_{jk}^{i}ω^{j} ∧ ω^{k}. The subspace W is invariant under the natural action of SL(2, ℝ) in the space L(𝔤̄/𝔤 ∧ 𝔤̄/𝔤, 𝔤̄).

Theorem 4. Suppose ω = (ω^{i}) is a V -valued form on P satisfying conditions (a) and (b). There exists a unique Cartan connection (ω^{i}, θ^{j}) of type (SL(3, ℝ), SL(2, ℝ)) (resp. (SU(1, 2), SU(1, 1))) whose structure function takes values in W.

Proof. Uniqueness. Let (ω, θ̄) be another Cartan connection with given ω. In view of (a) we can write θ̄^{i} − θ^{i} = ∑ a_{ij}ω^{j}, where a_{ij} are some functions on P. We denote the curvature of (ω, θ̄) by (Ω̄, Θ̄) and write Ω̄^{i} in the form Ω̄^{i} = ∑ \frac{1}{2}c̄_{kl}^{i}ω^{k} ∧ ω^{l}.

From the structure equations of the Cartan connection we immediately get

(θ̄^{1} − θ^{1}) ∧ ω^{2} + (θ̄^{2} − θ^{2}) ∧ ω^{3} = (Ω̄^{2} − Ω^{2}),
−(θ̄^{1} − θ^{1}) ∧ ω^{3} + (θ̄^{3} − θ^{3}) ∧ ω^{2} = (Ω̄^{3} − Ω^{3}),
(θ̄^{1} − θ^{1}) ∧ ω^{4} + (θ̄^{2} − θ^{2}) ∧ ω^{5} = (Ω̄^{4} − Ω^{4}),
−(θ̄^{1} − θ^{1}) ∧ ω^{5} + (θ̄^{3} − θ^{3}) ∧ ω^{4} = (Ω̄^{5} − Ω^{5}).

Choosing necessary coefficients, we fet a sistem of linear equations for a_{ij}:

(a_{22} − a_{13}) = (c̄_{23}^{2} − c_{23}^{2}), a_{14} + (c̄_{24}^{2} − c_{24}^{2}) = 0, a_{15} + (c̄_{25}^{2} − c_{25}^{2}) = 0;
a_{24} + (c̄_{34}^{2} − c_{34}^{2}) = 0, a_{25} + (c̄_{35}^{2} − c_{35}^{2}) = 0, a_{12} + a_{33} + (c̄_{23}^{3} − c_{23}^{3}) = 0;
a_{14} = (c̄_{34}^{3} − c_{34}^{3}), a_{15} = (c̄_{35}^{3} − c_{35}^{3}), a_{34} + (c̄_{24}^{3} − c_{24}^{3}) = 0;
a_{35} + (c̄_{25}^{3} − c_{25}^{3}) = 0, a_{11} = (c̄_{12}^{2} − c_{12}^{2}), a_{12} = (c̄_{24}^{4} − c_{24}^{4});
a_{13} = (c̄_{34}^{4} − c_{34}^{4}), (a_{24} − a_{15}) = (c̄_{45}^{4} − c_{45}^{4}), a_{21} = (c̄_{13}^{2} − c_{13}^{2});
a_{22} = (c̄_{25}^{4} − c_{25}^{4}), a_{23} = (c̄_{35}^{4} − c_{35}^{4}), a_{11} + (c̄_{13}^{3} − c_{13}^{3}) = 0;
a_{12} + (c̄_{25}^{5} − c_{25}^{5}) = 0, a_{13} + (c̄_{35}^{5} − c_{35}^{5}) = 0, a_{14} + a_{35} + (c̄_{45}^{5} − c_{45}^{5}) = 0;
a_{31} = (c̄_{12}^{3} − c_{12}^{3}), a_{32} = (c̄_{24}^{5} − c_{24}^{5}), a_{33} = (c̄_{34}^{5} − c_{34}^{5}).

Since equations (7) must be satisfied, we get a_{ij} = 0, so that θ̄ = θ, which proves the eniqueness of our Cartan connection.

Existence. Assume that wer are given a Cartan connection (ω, θ). Let us find a system of functions a_{ij} for which (ω, θ̄) satisfies conditions (7). It will suffice to put

a_{11} = \frac{1}{2}(c_{13}^{3} − c_{12}^{2}), a_{21} = −c_{13}^{2}, a_{31} = −c_{12}^{3};

a_{12} = \frac{1}{3}(c_{34}^{5} + c_{25}^{5} + c_{23}^{3} − c_{24}^{4}), a_{22} = −\frac{1}{3}(c_{25}^{4} + c_{34}^{4} − c_{35}^{5}) − \frac{2}{3}c_{23}^{2}, a_{32} = −c_{24}^{5}; a_{13} = \frac{1}{3}((c_{35}^{5} + c_{23}^{2}) − (c_{25}^{4} + c_{34}^{4})), a_{23} = −c_{35}^{4}, a_{33} = −\frac{1}{3}(c_{34}^{5} + c_{25}^{5} − c_{24}^{4}) + \frac{2}{3}c_{23}^{3}; a_{14} = \frac{1}{3}((c_{24}^{2} + c_{45}^{5}) − (c_{34}^{3} + c_{25}^{3})), a_{24} = −\frac{1}{3}(c_{35}^{3} − c_{25}^{2} − c_{34}^{2}) − \frac{2}{3}c_{45}^{4}, a_{34} = c_{24}^{3}; a_{15} = \frac{1}{3}((c_{25}^{2} + c_{34}^{2} + c_{45}^{4}) − c_{35}^{3}), a_{25} = c_{35}^{2}, a_{35} = −\frac{1}{3}(c_{24}^{2} − c_{34}^{3} − c_{25}^{3}) + \frac{2}{3}c_{45}^{5}.

Since the subspace W is invariant under the action of SL(2, ℝ) in L(𝔤̄/𝔤 ∧ 𝔤̄/𝔤, 𝔤̄), we have R_{g}^{∗}θ̄ = (Ad g^{−1})θ̄ for all g ∈ G. Thus the 𝔤̄-valued form (ω, θ̄) is the Cartan connection that we have been looking for.

To complete the proof, we need to show that there exists at least one Cartan connection with given ω. Since P admits an ordinary connection, let θ be its connection form. Since θ(X^{∗}) = X for any X ∈ 𝔤, R_{g}^{∗}θ = (Ad g^{−1})θ for any g ∈ G, it follows from conditions (a) and (b) that the 𝔤̄-valued form (ω, θ) is a Cartan connection.

Corollary. With any Monge–Ampère equation of type H_{(4,i)}^{ε}, it is possible to associate a Cartan connection of type (SL(3, ℝ), SL(2, ℝ)) if ε = 1 and of type (SU(1, 2), SU(1, 1)) if ε = −1.

Proof. If P is the G-structure corresponding to the system H_{(4,i)} and if ω is its canonical form, then conditions (a) and (b) are certainly satisfied.

Let (E, F) be a characteristic system of type H_{(4,i)}, and let P be the corresponding G-structure with structure group SL(2, ℝ). The Cartan connection from Theorem 4 is said to be associated with the system H.

Proposition 5. Let (ω, θ) and (ω′, θ′) be the Cartan connections associated with two systems (E, F) and (E′, F′). A map F: P → P′ is an isomorphism of these Cartan connections if and only if F = φ_{∗}|P, where φ is an isomophism of the systems H and H′.

Proof. Suppose F: P → P′ takes (ω′, θ′) to (ω, θ). Then F takes ω′ to ω, and by the properties of the canonical form on the bundle of lienar frames, there exists a diffeomorphism φ: M → M such that φ_{∗}(P) = P′ ([3], Ch. 2, §1, Lemma 1). Thus φ is an isomorphism of the G-structures P and P′.

Conversely, let φ be an isomorphism of the two characteristic systems. Then φ is an isomorhism of the G-structures P and P′, and hence it takes the canonical form ω′ to the canonical form ω. By Theorem 4, the forms ω and ω′ can be uniquely extended to Cartan connections in P and P′ respectively. Thus F = φ_{∗}|P is an isomorphism of Cartan connections.

We see that the problem of local classification of equations reduces to that of classifying Cartan connections of type (SL(3, ℝ), SL(2, ℝ)) and (SU(1, 2), SU(1, 1)) up to isomorphism. A Cartan connection is a special case of an absolute parallelism structure, for which the equivalence problem has already been solved ([13, 22]).

Contact symmetries of an equation of type H_{(4,i)} can be extended, in a unique way, to symmetries of the associated Cartan connection on the 8-dimensional manifold P. The dimension of the symmetry group of an absolute parallelism structure does not exceed the dimension of the manifold on which it is defined (see [13]). Consequently, the group of contact symmetries of an equation is a Lie group of dimension ⩽ 8.

Example 4. As an illustration, we construct the Cartan connection for the equation

Hess u + u^{−4} = 0, u > 0.  (8)

This is an equation of type H_{(4,4)}, and the vector fields

V_{1} = 2u\frac{∂}{∂u} + 2p\frac{∂}{∂p} + 2q \frac{∂}{∂q} ,

V_{2} = u^{3/2} (\frac{∂}{∂x} + p\frac{∂}{∂u} − u^{−2} \frac{∂}{∂q} ) ,

V_{3} = u^{3/2} (\frac{∂}{∂y} + q \frac{∂}{∂u} + u^{−2} \frac{∂}{∂p}) ,

V_{4} = u^{3/2} (\frac{∂}{∂x} + p\frac{∂}{∂u} + u^{−2} \frac{∂}{∂q} ) ,

V_{5} = u^{3/2} (\frac{∂}{∂y} + q \frac{∂}{∂u} − u^{−2} \frac{∂}{∂p})

on M determine a local section s: M → P of the G-structure with the structure group SL(2, ℝ). Having chosen a section, we can view a Cartan connection ω̄ on the principal fiber bundle P as a form on M with standard transition formulas for passing to another section. In our coordinate system, the canonical V -valued form ω is defined by the dual forms {ω^{1}, …, ω^{5}} such that ω^{i}(V_{j}) = δ_{ij}. Then the form θ such that

θ^{1} = \frac{1}{2u}(−pdx + qdy),

θ^{2} = −\frac{q}{u}dx,

θ^{3} = −\frac{p}{u}dy

makes, together with ω, the Cartan connection ω̄ = (ω, θ) of type (SL(3, ℝ), SL(2, ℝ)) mentioned in Theorem 4. The structure function of this Cartan connection is identically zero. It follows that the local group of contact symmetries of equation (8) is isomorphic to SL(3, ℝ), and every equation for which the Cartan connection of type (SL(3, ℝ), SL(2, ℝ)) has zero structure function is locally equivalent to equation (8).

2.1.3. Equations with maximal symmetry group. On the homogeneous spaces SL(3, ℝ)/SL(2, ℝ) and SU(1, 2)/SU(1, 1), there exist invariant characteristic Monge–Ampère systems of type H_{(4,4)}. Let E_{SL} and E_{SU} denote the equations corresponding to these systems. Each of these equations has a maximal 8dimensional group of contact symmetries. The equation E_{SL} has the normal form (8), while the normal form of E_{SU} is

(1 + 2xy) Hess u − 2xu_{y}u_{xx} − 2(2u − xu_{x} − yu_{y})u_{xy} − 2yu_{x}u_{yy} + 1 + 2u_{x}u_{y} = 0.

Theorem 5. Given an equation of type H_{(4,i)}^{ε}, the following conditions are equivalent:

1) The group of contact symmetries of the equation is 8-dimensional.

2) The structure function of the associated Cartan connection is identically equal to zero.

3) The equation is locally equivalent to the equation E_{SL} if ε = 1 or to the equation E_{SU} if ε = −1.

Proof. The symmetry group of a Cartan connection ω̄ on P has dimension 8 if and only if the structure function c of ω̄ is constant. Since c(pg) = g^{−1}.c(p) for all p ∈ P and g ∈ G, the constant value c ∈ W of the structure function is invariant under the action of the group G on W. All invariant elements of the SL(2, ℝ)-module W form an 8-dimensional submodule W_{0}, so that c ∈ W_{0}. On the other hand, c satisfies the structure equations (6) and dΩ = [dω̄, ω̄]. The only element c that meets all these requirements is the zero element, which proves the equivalence of conditions 1) and 2).

The Cartan connections corresponding to the equations E_{SL} and E_{SU} are precisely the canonical Maurer–Cartan forms on the Lie groups SL(3, ℝ) and SU(1, 2). The curvature of the Cartan connection for such forms is identically equal to zero, and the equivalence of conditions 2) and 3) now follows immediately.

2.2. Equations whose transformation groups are not simply transitive. The object of this section is to classify Monge–Ampère equations of type H_{(4,i)} whose algebras of contact symmetries are not simply transitive, that is equations with dim sym(E) ⩾ 6. We shall classify them in terms of characteristic distributions and associated Cartan connections. We restrict ourselves to local considerations in a neighborhood of some point of general position. This allows to state the problem in the language of abstract Lie algebras and modules over Lie algebras.

From now on, we shall assume that the functions that appear in our equations are complex analytical functions. For this special case, the results of the preceding section may be rephrased as follows: with every equation of type H_{(4,i)} (i = 1, 2, 3, 4) one can associate, in a natural way, a subbundle P in the bundle of linear frames over M with structure group SL(2, ℂ) and a Cartan connection Θ in P whose model is the homogeneous space SL(3, ℂ)/SL(2, ℂ) ([33]). Since dim SL(3, ℂ) ⩽ 8, the dimension of sym(E) for equations of type H_{(4,i)} is at most 8. If it is equal to 8, then, by Theorem 5, the equation is contactly equivalent to an equation of type H_{(4,4)} on the homogeneous space SL(3, ℂ)/SL(2, ℂ). The case sym(E) = 5 will be discussed later on (see 2.3). It now remains to find normal forms of equations for which the dimension of sym(E) is equal to 6 or 7. As we shall see shortly, only the case dim sym(E) = 6 is possible, and the set of equivalence classes of equations may depend on some parameters.

2.2.1. Algebraic model of invariant Monge–Ampère equations. Let 𝔤̄ be a finite dimensional transitive Lie algebra of contact vector fields on M, and let 𝔤 = 𝔤̄_{o} be a subalgebra of 𝔤̄ consisting of all the vector fields that vanish at some point o. The subalgebra 𝔤 is effective in 𝔤̄, i.e. it contains no nonzero ideals of 𝔤̄.

Definition 5. The 𝔤-module 𝔤̄/𝔤 defined by

x.(y + 𝔤) = [x, y] + 𝔤, x ∈ 𝔤, y ∈ 𝔤̄

is called the isotropic module of the pair (𝔤̄, 𝔤).

Definition 6. The representation ρ: 𝔤 → 𝔤𝔩(𝔤̄/𝔤) with

ρ(x)(y + 𝔤) = [x, y] + 𝔤, x ∈ 𝔤, y ∈ 𝔤̄,

corresponding to the isotropic module is called the isotropic representation of the pair (𝔤̄, 𝔤).

Let E and F be the characteristic distributions of some Monge–Ampère equation. The tangent space T_{o}M can be identified with 𝔤̄/𝔤 in the canonical way (see [11]). The subspaces C_{o}, E_{o}, and F_{o} are then identified with subspaces in 𝔤̄/𝔤, which we shall also denote C, E, and F.

There is a one-to-one correspondence between invariant (under 𝔤̄) distributions on M and submodules of the isotropic module of the pair (𝔤̄, 𝔤). Thus the subspace C is the direct sum of the submodules E and F.

Definition 7. Let D be a submodule of the isotropic module of the pair (𝔤̄, 𝔤), and let V denote the quotient space 𝔤̄/𝔤. The alternating bilinear map

\begin{aligned}Ω_{D} : & D × D & ⟶ V/D, \\ & (x + 𝔤, y + 𝔤) & ⟼ ([x, y] + 𝔤) + D, x, y ∈ 𝔤̄ \end{aligned}

is called the curvature form of D.

For the contact form ω_{0}, the restriction of the 2-form dω_{0} to the contact distribution C is nondegenerate, and hence the curvature form Ω_{C} of the subset C ⊂ V is also nondegenerate. The condition dω_{0}(E, F) = 0 can then be superseded by the equivalent requirement that Ω_{C}(E, F) = 0. Thus there exists a one-to-one correspondence between Monge–Ampère equations with transitive symmetry algebras and 4-tuples (𝔤̄, 𝔤, C, E), where 𝔤 is an effective subalgebra in 𝔤̄ of codimension 5, C is a 4-dimensional submodule of the isomorphic module whose curvature form Ω_{C} is nondegenerate, and E is a 2-dimensional submodule such that E is not a Lagrangean subspace relative to the alternating form Ω_{C}. By virtue of Ω_{C}(E, F) = 0, the submodule F can be uniquely recovered from E.

Definition 8. Two 4-tuples (𝔤̄_{1}, 𝔤, C_{1}, E_{1}) and (𝔤̄_{2}, 𝔤̄, C_{2}, E_{2}) are said to be equivalent if there exists an isomorphism α: 𝔤̄_{1} → 𝔤̄_{2}, such that α(𝔤_{1}) = 𝔤_{2} and such that the induced mapping

\begin{aligned}β : & 𝔤̄_{1}/𝔤_{1} & ⟶ 𝔤̄_{2}/𝔤_{2}, \\ & x + 𝔤_{1} & ⟼ α(x) + 𝔤_{2}, \end{aligned}

satisfies the conditions β(C_{1}) = C_{2} and β(E_{1}) = E_{2}.

Locally, a 4-tuple (𝔤̄, 𝔤, C, E) uniquely determines a Monge–Ampère equation with transitive symmetry algebra.

Proposition 6. Let E_{1} and E_{2} be two Monge–Ampère equations with transitive symmetry Lie algebras 𝔤̄_{1} and 𝔤̄_{2}, respectively. Let o_{1}, o_{2} ∈ M, and let (𝔤̄_{1}, 𝔤_{1}, C_{1}, E_{1}) and (𝔤̄_{2}, 𝔤_{2}, C_{2}, E_{2}) be the 4-tuples corresponding to the equations E_{1} and E_{2}. In order that there exist a contact diffeomorphism φ such that φ(o_{1}) = o_{2}, φ_{∗}(𝔤̄_{1}) = 𝔤̄_{2}, and φ_{∗}(E_{1}) = E_{2}, it is necessary and sufficient that the 4-tuples (𝔤̄_{1}, 𝔤_{1}, C_{1}, E_{1}) and (𝔤̄_{2}, 𝔤_{2}, C_{2}, E_{2}) be equivalent.

Proof. Locally, every homogeneous space M is uniquely determined by the corresponding pair of Lie algebras (𝔤̄, 𝔤). Moreover, for any Lie algebra isomorphism α: 𝔤̄_{1} → 𝔤̄_{2} such that α(𝔤_{1}) = 𝔤_{2}, there exists a unique local diffeomorphism φ of M for which φ_{∗}(𝔤̄_{1}) = 𝔤̄_{2} ([11], Ch. I, §4). The pairs (C_{1}, E_{1}) and (C_{2}, E_{2}) uniquely define the contact distribution and the characteristic distributions in some neighborhood of the points o_{1} and o_{2}. Hence φ is a contact diffeomorphism and a symmetry of the characteristic distributions if and only if the induced map β: 𝔤̄_{1}/𝔤_{1} → 𝔤̄_{2}/𝔤_{2} takes the pair (C_{1}, E_{1}) to the pair (C_{2}, E_{2}).

We shall now find out what conditions should be imposed on a 4-tuple (𝔤̄, 𝔤, C, E) in order that the distribution E on M be of type (4).

Definition 9. Suppose E is a submodule of the isotropic module of the pair (𝔤̄, 𝔤). The first derived submodule E^{(1)} is the subspace in 𝔤̄/𝔤 defined by

E^{(1)} = E + {[x, y] + 𝔤 | x + 𝔤, y + 𝔤 ∈ E}.

It is easily seen that E^{(1)} is well defined. We now define the (i+1)-th derived submodule by induction:

E^{(i+1)} = E^{(i)} + {[x, y] + 𝔤 | x + 𝔤 ∈ E, y + 𝔤 ∈ E^{(i)}}.

We shall say that a submodule E has type (4) if dim E^{(1)} = 3 and dim E^{(2)} = 5.

The pair (𝔤̄, 𝔤) defines a homogeneous space with invariant Cartan connection Θ of type SL(3, ℂ)/SL(2, ℂ). Then Θ may be viewed as an invariant connection in the bundle P̄ that is obtained from P by extending the structure group SL(2, ℂ) to SL(3, ℂ) ([3], Ch. 4, §3). From the description of all invariant connections on a homogeneous space ([4], Ch. 2, §11), it follows that the Lie algebra 𝔤̄ can be regarded as a subspace in 𝔰𝔩(3, ℂ) such that 𝔰𝔩(3, ℂ) = 𝔰𝔩(2, ℂ) + 𝔤̄, 𝔤 = 𝔰𝔩(2, ℂ) ∩ 𝔤̄, and [x, y] = [x, y]_{𝔰𝔩(3,ℂ)}, where x ∈ 𝔤, y ∈ 𝔤̄ and [·, ·]_{𝔰𝔩(3,ℂ)} is the Lie bracket operation in 𝔰𝔩(3, ℂ).

The isotropic module of the pair (𝔰𝔩(3, ℂ), 𝔰𝔩(2, ℂ)) is faithful and has the form:

\{(\begin{matrix}0 & 0 & 0 \\ 0 & A & 0 \\ 0 & 0 & A \end{matrix})| A ∈ 𝔰𝔩(2, ℂ)\} .

Hence the isotropic module of (𝔤̄, 𝔤) is also faithful, and it will suffice to restrict ourselves to the case dim 𝔤 = 1, 2. Up to the conjugation in 𝔰𝔩(2, ℂ), the subalgebra 𝔤 can have one of the following three forms:

\{(\begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & x & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & x \\ 0 & 0 & 0 & 0 & 0 \end{matrix}) , (\begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & x & 0 & 0 & 0 \\ 0 & 0 & −x & 0 & 0 \\ 0 & 0 & 0 & x & 0 \\ 0 & 0 & 0 & 0 & −x \end{matrix}) , (\begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & x & y & 0 & 0 \\ 0 & 0 & −x & 0 & 0 \\ 0 & 0 & 0 & x & y \\ 0 & 0 & 0 & 0 & −x \end{matrix})\} x, y ∈ ℂ,

which we shall denote by 𝔤_{1}, 𝔤_{2}, 𝔤_{3} respectively.

We choose a basis {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}} for the quotient space V = 𝔤̄/𝔤 in such a way that ⟨u_{2}, u_{3}⟩ = E, ⟨u_{4}, u_{5}⟩ = F, and ⟨u_{1}⟩ = E^{(1)} ∩ F^{(1)}. We identify V with some complement of 𝔤 in 𝔤̄. The condition Ω_{C}(E, F) = 0 may be written as follows:

(A) [E, F] ⊂ 𝔤 + C.

Since the form Ω_{C} is nondegenerate and since E^{(1)} = ⟨u_{1}, u_{2}, u_{3}⟩, F^{(1)} = ⟨u_{1}, u_{4}, u_{5}⟩, we get

(B) [u_{2}, u_{3}] = v_{23} + u_{1}, [u_{4}, u_{5}] = v_{45} + u_{1}, where v_{23} ∈ 𝔤 + E, v_{45} ∈ 𝔤 + F Let [u_{1}, u_{2}] = v_{12} + a_{11}u_{4} + a_{21}u_{5} and [u_{1}, u_{3}] = v_{13} + a_{12}u_{4} + a_{22}u_{5}, where

v_{12}, v_{13} ∈ 𝔤 + E^{(1)} and A = (a_{ij}) (i, j = 1, 2). Then the condition that E be of type (4) can be replaced by the equivalent condition that

(C) rank A = 2.

For each of the isotropy subalgebras 𝔤_{i}(i = 1, 2, 3), we write out the Lie brackets in 𝔤̄ = 𝔤_{i} ⊕ V with undefined coefficients, taking into account the conditions (A), (B), (C), and substitute them into the Jacobi identities. As a result, we find that

(1) the case 𝔤_{3} is impossible;

(2) for 𝔤_{1} and 𝔤_{2}, the Lie algebra 𝔤̄ is nonsolvable.

Thus, the 4-tuples (𝔤̄, 𝔤, C, E) in question can be classified by using the following algorithm:

Step 1: Classify (up to isomorphism) all nonsolvable Lie algebras 𝔤̄ of di-

mension 6 over the complex numbers. Step 2: Classify (up to the group Aut(𝔤̄)) all 1-dimensional subalgebras 𝔤

in 𝔤̄ for which the isotropic representation is conjugate to either 𝔤_{1} or 𝔤_{2}. Step 3: Classify (up to the group Aut(𝔤̄, 𝔤)) all 4-dimensional submod-

ules C in 𝔤̄/𝔤 with nondegenerate curvature form Ω_{C}. Step 4: Classify (up to the group Aut(𝔤̄, 𝔤, C)) all 2-dimensional submod-

ules E in C that have type (4) and are not Lagrangean relative to the curvature form Ω_{C}.

Finally, from each 4-tuple (𝔤̄, 𝔤, C, E) obtained, the corresponding Monge– Ampère equation must be recovered.

2.2.2. The classification of equations. Denote by 𝔫_{3} the 3-dimensional Heisenberg algebra, and suppose that {e_{1}, e_{2}, e_{3}} is a basis for 𝔫_{3} such that [e_{1}, e_{2}] = e_{3} and [e_{1}, e_{3}] = [e_{2}, e_{3}] = 0.

Lemma 3. Let 𝔤̄ be a nonsolvable Lie algebra of dimension 6 over the field of complex numbers. Then 𝔤̄ is isomorphic to one and only one of the following Lie algebras:

1) 𝔤̄ = (𝔰𝔩(2, ℂ) ⋌ ℂ^{2}) × ℂ;

2) 𝔤̄ = 𝔤𝔩(2, ℂ) ⋌ ℂ^{2};

3) 𝔤̄ = 𝔰 ⋌ 𝔫_{3}, where 𝔰 ≅ 𝔰𝔩(2, ℂ) and

𝔰 = \{(\begin{matrix}A & 0 \\ 0 & 0 \end{matrix})| A ∈ 𝔰𝔩(2, ℂ)\};

4) 𝔤̄ = 𝔰𝔬(3, ℂ) ⋌ ℂ^{3};

5) 𝔤̄ = 𝔰𝔩(2, ℂ) × 𝔰𝔩(2, ℂ);

6) 𝔤̄ = 𝔰𝔩(2, ℂ) × ℂ^{3};

7) 𝔤̄ = 𝔰𝔩(2, ℂ) × 𝔫_{3};

8) 𝔤̄ = 𝔰𝔩(2, ℂ) × 𝔞, where

𝔞 = \{(\begin{matrix}x & x \\ 0 & x \end{matrix}) ⋌ ℂ^{2}| x ∈ ℂ\} ,

9) 𝔤̄ = 𝔰𝔩(2, ℂ) × 𝔞, where

𝔞 = \{(\begin{matrix}x & 0 \\ 0 & γx \end{matrix}) ⋌ ℂ^{2}| x ∈ ℂ\} ,

(the Lie algebras with parameters γ and 1/γ are isomorphic to each other).

Proof. Consider a Levy decomposition 𝔤̄ = 𝔰 ⋌ 𝔯, where 𝔯 is the radical of 𝔤̄, and 𝔰 is a Levy subalgebra. If 𝔯 = {0}, then 𝔤̄ is a semisimple Lie algebra of dimension 6, and hence 𝔤̄ ≅ 𝔰𝔩(2, ℂ) × 𝔰𝔩(2, ℂ). Suppose now that 𝔯 ≠ {0}. Then dim 𝔯 = 3 and the subalgebra 𝔰 is isomorphic to 𝔰𝔩(2, ℂ).

If the Levy decomposition is a direct sum of Lie algebras, then 𝔤̄ = 𝔰𝔩(2, ℂ)× 𝔞, where 𝔞 is a 3-dimensional Lie algebra, viewed up to isomorphism. In this way we obtain the subalgebras 6) through 9). Otherwise, the radical 𝔯 is a solvable 3-dimensional Lie algebra with nonsolvable derivation algebra. Every Lie algebra of this kind is isomorphic to ⟨id_{ℂ^{2}}⟩ ⋌ ℂ^{2}, 𝔫_{3}, and ℂ^{3}. In the first two cases get the subalgebras 2) and 3), respectively. If the radical 𝔯 is ℂ^{3}, then 𝔤𝔩(3, ℂ) is the algebra of derivations of 𝔯, and up to conjugation, there exist two subalgebras in 𝔤𝔩(3, ℂ) which are isomorphic to 𝔰𝔩(2, ℂ), namely:

\{(\begin{matrix}x & y & 0 \\ z & −x & 0 \\ 0 & 0 & 0 \end{matrix}) , (\begin{matrix}0 & x & y \\ −x & 0 & z \\ −y & −z & 0 \end{matrix})\} x, y, z ∈ ℂ.

They correspond to the Lie algebras 1) and 4).

We note that the only Lie algebras that actually serve as symmetry algebras for some Monge–Ampère equations of type H_{(4,i)} are those under the numbers 1) through 5).

Example 5. By way of an example, we shall now classify the Monge–Ampère equations corresponding to the Lie algebra 𝔤̄ = (𝔰𝔩(2, ℂ) ⋌ ℂ^{2}) × ℂ. We choose a basis {e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}} for 𝔤̄ as follows:

𝔤̄ = ⟨xe_{1} + ye_{2} + ze_{3} + pe_{4} + qe_{5} + re_{6}⟩ = \{(\begin{matrix}x & y & 0 \\ z & −x & 0 \\ 0 & 0 & 0 \end{matrix}) ⊕ (\begin{matrix}p \\ q \\ r \end{matrix})\} ,

x, y, z, p, q, r ∈ ℂ.

The group Aut(𝔤̄) is isomorphic to (GL(2, ℂ) ⋌ ℂ^{2}) × ℂ^{∗}. Up to the action of this group, every 1-dimensional subalgebra 𝔤 in 𝔤̄ for which the isotropic representation of the pair (𝔤̄, 𝔤) is conjugate to a subalgebra of the form 𝔤_{1} or 𝔤_{2}, is equivalent to the subalgebra 𝔤 = ⟨e_{2}⟩. The vectors u_{1} = e_{1} + 𝔤, u_{2} = e_{3} + 𝔤, u_{3} = e_{4} + 𝔤, u_{4} = e_{5} + 𝔤, u_{5} = e_{6} + 𝔤 form a basis of the quotient space V = 𝔤̄/𝔤. Up to the natural action of the group Aut(𝔤̄, 𝔤) in V, every 4-dimensional submodule C with nondegenerate curvature form has the form C = ⟨u_{1}, u_{2}, u_{3}, u_{4} + u_{5}⟩. Consider the action of the group Aut(𝔤̄, 𝔤, C) in C. Up to this action, there exists a unique submodule E of type (4) which is not Lagrangean relative to the curvature form Ω_{C}. Then E = ⟨u_{1} + u_{3}, u_{1} + u_{2} + u_{4} + u_{5}⟩, F = ⟨u_{1} − u_{3}, u_{2} − u_{3} − u_{4} − u_{5}⟩, and the submodule F is also of type (4).

Consider the 5-dimensional contact homogeneous space corresponding to the pair (𝔤̄, 𝔤). Suppose that g ∈ \overline{G} has the form

g = \{(\begin{matrix}a_{1} & a_{2} \\ a_{3} & a_{4} \end{matrix}) , k, m, n\} ,

where a_{1}a_{4} − a_{2}a_{3} = 1, k, m, n ∈ ℂ. Define the action of the group \overline{G} = (SL(2, ℂ) ⋌ ℂ^{2}) × ℂ on the contact manifold:

g.(x, y, u, p, q) = (a_{1}x + a_{2}y + k, a_{3}x + a_{4}y + m, u + n, a_{4}p − a_{3}q, −a_{2}p + a_{1}q).

The stationary subalgebra for the point o = (0, 0, 0, 0, 1) coincides with 𝔤. Using formula (4), we define, at the point o, the 2-form ϑ (see 1.1.1) that determines the Monge–Ampère equation:

ϑ_{o} = 2dp ∧ dy + dx ∧ dy + dp ∧ dq.

The corresponding invariant form ϑ on all of the manifold has the form

ϑ = (q + 1)^{2}dp ∧ dy − (q + 1)p(dx ∧ dp + dq ∧ dy) + p^{2}dx ∧ dq + dx ∧ dy + dp ∧ dq,

and hence the equation in question has the form

Hess u + 1 + (u_{y} + 1)^{2}u_{xx} − 2(u_{y} + 1)u_{x}u_{xy} + u_{x}^{2}u_{yy} = 0.

The coordinate transformation ū = u + y gives

Hess u + 1 + u_{y}^{2}u_{xx} − 2u_{x}u_{y}u_{xy} + u_{x}^{2}u_{yy} = 0,

and a further coordinate transformation, ū = exp(u), brings our equation to the following normal form:

Hess u + u^{2} = 0.  (9)

Thus, every Monge–Ampère equation of type H_{(4,i)} for which the Lie algebra (𝔰𝔩(2, ℂ) ⋌ ℂ^{2}) × ℂ is a transitive symmetry algebra, is locally equivalent to the equation (9). In terms of characteristic functions, the symmetry algebra has the form

sym(E) = ⟨p, q, xp − yq, yp, xq, u⟩.

For the remaining cases, we present only the results of the main steps of the classification.

2) 𝔤̄ = 𝔤𝔩(2, ℂ) ⋌ ℂ^{2},

𝔤̄ = ⟨xe_{1} + ye_{2} + ze_{3} + pe_{4} + qe_{5} + re_{6}⟩ = \{(\begin{matrix}x + p & y \\ z & −x + p \end{matrix}) ⊕ (\begin{matrix}q \\ r \end{matrix})\} ,

x, y, z, p, q, r ∈ ℂ.

The isotropy subalgebra is 𝔤 = ⟨e_{2}⟩. The contact submodule is

C = ⟨e_{1} + 𝔤, e_{3} + 𝔤, e_{5} + 𝔤, e_{4} + e_{6} + 𝔤⟩.

The characteristic system has the form

E = ⟨e_{1} + e_{5} + 𝔤, αe_{1} + e_{3} + e_{4} + e_{6} + 𝔤⟩,
F = ⟨e_{1} − e_{5} + 𝔤, e_{3} − αe_{5} − e_{4} − e_{6} + 𝔤⟩,

where α ≠ −1. The normal form of the equation is

Hess u + exp(4u) = 0, sym(E) = ⟨p, q, xp − yq, yp, xq, u + xp + yq⟩;

Hess u + u^{γ} = 0, γ ≠ −4, 0, 2, sym(E) = ⟨p, q, xp − yq, yp, xq, u + (\frac{γ}{2} − 1)xp⟩.

For γ = −4, the last equation has the following maximal 8-dimensional algebra of contact symmetries:

sym(E) = ⟨p, q, xp − yq, yp, xq, u − 3xp, xu − x^{2}p − xyq, yu − xyp − y^{2}q⟩.

3) 𝔤̄ = 𝔰 ⋌ 𝔫_{3}, where

𝔰 = ⟨xe_{1} + ye_{2} + ze_{3}⟩ = \{(\begin{matrix}x & y \\ z & −x \end{matrix})| x, y, z ∈ ℂ\}

and [e_{4}, e_{5}] = 2e_{6}.

The isotropy subalgebra is 𝔤 = ⟨e_{2}⟩. The contact submodule is

C = ⟨e_{1} + 𝔤, e_{3} + e_{6} + 𝔤, e_{4} + 𝔤, e_{5} + 𝔤⟩.

The characteristic system is

E = ⟨αe_{1} + e_{4} + 𝔤, e_{1} + α(e_{3} + e_{6}) + e_{5} + 𝔤⟩,
F = ⟨e_{1} − αe_{4} + 𝔤, e_{3} + e_{4} + e_{6} − αe_{5} + 𝔤⟩,

131 HYPERBOLIC MONGE–AMPÈRE EQUATIONS

α ≠ ±\sqrt{−1}. The normal form of the equation is

γ(u_{xx} + 2u_{y}u_{xy} + 2(u_{y}^{2} + u_{x})u_{yy}) +

+ (1 − γ^{2})(−u_{y}^{2} − 2u_{x})^{1/2}(u_{xy} + u_{y}u_{yy}) = (u_{y}^{2} + 2u_{x})^{2}, γ ≠ ±\sqrt{−1}.

The symmetry algebra is

sym(E) = ⟨2xp + yq, y^{2} − 2x^{2}p − 2xyq, p, q, 1, y − xq⟩.

4) 𝔤̄ = 𝔰𝔬(3, ℂ) ⋌ ℂ^{3},

𝔤̄ = ⟨xe_{1} + ye_{2} + ze_{3} + pe_{4} + qe_{5} + re_{6}⟩ = \{(\begin{matrix}0 & x & y \\ −x & 0 & z \\ −y & −z & 0 \end{matrix}) ⊕ (\begin{matrix}p \\ q \\ r \end{matrix})\} ,

x, y, z, p, q, r ∈ ℂ.

The isotropy subalgebra is 𝔤 = ⟨e_{1}⟩. The contact submodule is

C = ⟨e_{2} + 𝔤, e_{3} + 𝔤, e_{4} + 𝔤, e_{5} + 𝔤⟩.

The characteristic system is E = ⟨e_{3} + e_{4} + 𝔤, e_{2} − e_{5} + 𝔤⟩,

F = ⟨e_{3} − e_{4} + 𝔤, e_{2} + e_{5} + 𝔤⟩.

The normal form of the equation is

Hess u + (1 + u_{x}^{2} + u_{y}^{2})^{2} = 0.

The symmetry algebra is

sym(E) = ⟨yp − xq, x + up, y + uq, p, q, 1⟩.

5) 𝔤̄ = 𝔰𝔩(2, ℂ) × 𝔰𝔩(2, ℂ),

𝔤̄ = ⟨xe_{1} + ye_{2} + ze_{3} + pe_{4} + qe_{5} + re_{6}⟩ = \{(\begin{matrix}x & y \\ z & −x \end{matrix}) × (\begin{matrix}p & q \\ r & −p \end{matrix})\} ,

x, y, z, p, q, r ∈ ℂ.

The isotropy subalgebra is 𝔤 = ⟨e_{1} + e_{4}⟩. The contact submodule is

C = ⟨e_{2} + 𝔤, e_{3} + 𝔤, e_{5} + 𝔤, e_{6} + 𝔤⟩.

The characteristic system is E = ⟨e_{3} + e_{6} + 𝔤, e_{2} + αe_{5} + 𝔤⟩,

F = ⟨e_{2} + e_{5} + 𝔤, αe_{3} + e_{6} + 𝔤⟩,

where α ≠ 0, 1, and two systems with parameters α and 1/α are equivalent. The normal form of the equation is

(−u + (y − xu)u_{y} + xyu_{y}^{2})u_{xx} +

+ (1 + xu_{y} − yu_{x} − u(u − xu_{x} − yu_{y}) − 2xyu_{x}u_{y})u_{xy} +

+ (u + (−x − yu)u_{x} + xyu_{x}^{2})u_{yy} +

+γ ·((1+xu_{y})^{2}u_{xx}+2(1+xu_{y})(u−xu_{x})u_{xy} +(u−xu_{x})^{2}u_{yy}) = 0, γ ≠ ±\sqrt{−1}, (two equations with parameters γ and −γ are equivalent).

The symmetry algebra is

sym(E) = ⟨xp + yq,

1+u^{2}+(y−xu)p−(x+yu)q, q+up, p−uq, y+xu−x^{2}p−xyq, x−yu+xyp+y^{2}q⟩.

6)–8) These three Lie algebras contain no 1-dimensional subalgebras for which the isotropy representation is isomorphic to 𝔤_{1} or 𝔤_{2}.

9) When γ = −1, this Lie algebra contains the subalgebra

𝔤 = \{(\begin{matrix}x & 0 \\ 0 & −x \end{matrix}) × (\begin{matrix}x & 0 \\ 0 & −x \end{matrix})| x ∈ ℂ\} ,

for which the isotropy representation is isomorphic to 𝔤_{2}. However, in this case there exists no 4-dimensional submodule in 𝔤̄/𝔤 with nondegenerate curvature form.

Summing up the results, we get

Proposition 7. Let (𝔤̄, 𝔤, C, E) be the 4-tuple corresponding to some Monge– Ampère equation of type H_{(4,i)} with transitive symmetry algebra. Suppose that 𝔤 ≠ 0. Then the pair of Lie algebras (𝔤̄, 𝔤) is equivalent to one and only one of the following pairs:

1) 𝔤̄ = (𝔰𝔩(2, ℂ) ⋌ ℂ^{2}) × ℂ, 𝔤 = \{(\begin{matrix}0 & x \\ 0 & 0 \end{matrix})| x ∈ ℂ\};

2) 𝔤̄ = 𝔤𝔩(2, ℂ) ⋌ ℂ^{2}, 𝔤 = \{(\begin{matrix}0 & x \\ 0 & 0 \end{matrix})| x ∈ ℂ\};

3) 𝔤̄ = (\begin{matrix}A & 0 \\ 0 & 0 \end{matrix}) ⋌ 𝔫_{3}, 𝔤 = (\begin{matrix}0 & x & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}), where A ∈ 𝔰𝔩(2, ℂ), x ∈ ℂ;

4) 𝔤̄ = 𝔰𝔬(3, ℂ) ⋌ ℂ^{3}, 𝔤 = 𝔰𝔬(2, ℂ);

5) 𝔤̄ = 𝔰𝔩(2, ℂ) × 𝔰𝔩(2, ℂ), 𝔤 = \{(\begin{matrix}x & 0 \\ 0 & −x \end{matrix}) × (\begin{matrix}x & 0 \\ 0 & −x \end{matrix})| x ∈ ℂ\};

6) 𝔤̄ = 𝔰𝔩(3, ℂ), 𝔤 = (\begin{matrix}A & 0 \\ 0 & 0 \end{matrix}), where A ∈ 𝔰𝔩(2, ℂ).

The final classification of equations in terms of their normal forms looks like this:

Theorem 6. Every Monge–Ampère equation of type H_{(4,i)} whose algebra of contact symmetries is transitive, but not simply transitive is locally equivalent to one and only one of the following equations:

a) Hess u + u^{γ} = 0, γ ≠ 0;

b) Hess u + exp(4u) = 0;

c) γ(u_{xx} + 2u_{y}u_{xy} + 2(u_{y}^{2} + u_{x})u_{yy}) +

+ (1 − γ^{2})(−u_{y}^{2} − 2u_{x})^{1/2}(u_{xy} + u_{y}u_{yy}) = (u_{y}^{2} + 2u_{x})^{2}, γ ≠ ±\sqrt{−1};

d) Hess u + (1 + u_{x}^{2} + u_{y}^{2})^{2} = 0; e) (−u + (y − xu)u_{y} + xyu_{y}^{2})u_{xx} +

+ (1 + xu_{y} − yu_{x} − u(u − xu_{x} − yu_{y}) − 2xyu_{x}u_{y})u_{xy} +

+ (u + (−x − yu)u_{x} + xyu_{x}^{2})u_{yy} +

+γ ·((1+xu_{y})^{2}u_{xx}+2(1+xu_{y})(u−xu_{x})u_{xy} +(u−xu_{x})^{2}u_{yy}) = 0, γ ≠ ±\sqrt{−1},

(two equations corresponding to γ and −γ, respectively, are always equivalent).

All of these equations are of type H_{(4,4)}, except for the equation c) with γ = 0, which is of type H_{(4,1)}.

Equations d) and e) admit a geometric interpretation in terms of Riemannian geometry. Indeed, equation d) can be written as K = −1, where K is the Gaussian curvature for the surface z = u(x, y). In this interpretation, the solutions of the equation are the surfaces with constant curvature −1.

For equation e), we have to consider 3-dimensional Riemannian space of constant curvature. The surfaces lying in this space have two invariants \overline{K} and \overline{H}, which are analogous to the Gaussian and mean curvatures in 3-dimensional Euclidean space. Then equation e) can be written as \overline{K} = γ.

2.3. Monge–Ampère equations on 5-dimensional Lie groups. In this section we classify equations of type H_{(4,i)} with simply transitive contact symmetry algebra.

2.3.1. Classification of 5-dimensional Lie algebras with invariant contact structure. Let 𝔤 be a simply transitive Lie algebra of contact vector fields on M. Given a point o ∈ M, we identify the tangent space T_{o}M with 𝔤. The subspace C_{o} can be identified with a 4-dimensional subspace in 𝔤, which we shall denote by C. The pair (𝔤, C) will then be called contact.

Definition 10. Let D be a subspace in 𝔤. The map

\begin{aligned}Ω_{C}: & C × C & ⟶ 𝔤/C, \\ & (v_{1}, v_{2}) & ⟼ [v_{1}, v_{2}] + C \end{aligned}

is called the curvature form of D.

Since the restriction of the 2-form dω_{0} to the contact distribution C is nondegenerate, the curvature form of the subspace C_{o} ⊂ 𝔤 is also nondegenerate. Note that if 𝔞 is an ideal in 𝔤, and if 𝔞 ⊂ C for some subspace C, then 𝔞 lies in the kernel of the curvature form Ω_{C}.

Definition 11. Two contact pairs (𝔤_{1}, C_{1}) and (𝔤_{2}, C_{2}) are said to be equivalent if there exists an isomorphism ψ: 𝔤_{1} → 𝔤_{2} such that ψ(C_{1}) = C_{2}.

The following result is a special case of Proposition 6.

Proposition 8. Suppose that 𝔤_{1} and 𝔤_{2} are two simply transitive Lie algebras of contact vector fields, o_{1}, o_{2} ∈ J^{1}(ℂ^{2}, ℂ), and (𝔤_{1}, C_{1}), (𝔤_{2}, C_{2}) are the corresponding contact pairs. In order that there exist a contact diffeomorphism φ such that φ(o_{1}) = o_{2} and φ_{∗}(𝔤_{1}) = 𝔤_{2}, it is necessary and sufficient that the pairs (𝔤_{1}, C_{1}) and (𝔤_{2}, C_{2}) be equivalent.

Thus our initial problem splits up into the following two:

[A] classify all 5-dimensional Lie algebras 𝔤 over the complex numbers (see [8]);

[B] classify (up to the group Aut(𝔤)) all 4-dimensional subspace in 𝔤 with nondegenerate curvature form.

Since codim_{𝔤} C = 1, we shall define a 4-dimensional subspace in 𝔤 by specifying a 1-dimensional subspace U in the dual space 𝔤^{∗}. Then C = U^{⊥}, where U^{⊥} = {v ∈ 𝔤 | α(v) = 0 for all α ∈ U}.

Example 6. Suppose that a 5-dimensional Lie algebra 𝔤 is isomorphic to 𝔤 ≅ 𝔰𝔩(2, ℂ) × 𝔞𝔣𝔣(1, ℂ) and that, for a fixed basis {e_{1}, e_{2}, e_{3}, e_{4}, e_{5}} of 𝔤, the commutation table of 𝔤 has the form:

\begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & e_{2} & −e_{3} & 0 & 0 \\ e_{2} & −e_{2} & 0 & e_{1} & 0 & 0 \\ e_{3} & e_{3} & −e_{1} & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{5} \\ e_{5} & 0 & 0 & 0 & −e_{5} & 0 \end{matrix} .

Then Aut(𝔤) ≅ SL(2, ℂ) × Aff(1, ℂ). Up to the natural action of Aut(𝔤) in 𝔤^{∗}, every 1-dimensional subspace U of 𝔤^{∗} such that U^{⊥} contains no ideals of 𝔤, is conjugate to one of the following two subspaces: U_{1} = ⟨−e_{1}^{∗} + e_{5}^{∗}⟩ or U_{2} = ⟨−e_{2}^{∗} + e_{5}^{∗}⟩. The curvature form of U_{2}^{⊥} ⊂ 𝔤 is degenerate, and therefore the contact subspace C is equivalent to C = U_{1}^{⊥} = ⟨e_{1} + e_{5}, e_{2}, e_{3}, e_{4}⟩.

Similar reasoning in the remaining cases brings us to the following result:

Proposition 9. Every contact pair (𝔤, C) over the field of complex numbers is equivalent to one and only one of the following pairs:

𝔤_{1} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & e_{2} & −e_{3} & 0 & 0 \\ e_{2} & −e_{2} & 0 & e_{1} & 0 & 0 \\ e_{3} & e_{3} & −e_{1} & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{5} \\ e_{5} & 0 & 0 & 0 & −e_{5} & 0 \end{matrix}

𝔤_{2} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 2e_{2} & −2e_{3} & e_{4} & −e_{5} \\ e_{2} & −2e_{2} & 0 & e_{1} & 0 & e_{4} \\ e_{3} & 2e_{3} & −e_{1} & 0 & e_{5} & 0 \\ e_{4} & −e_{4} & 0 & −e_{5} & 0 & 0 \\ e_{5} & e_{5} & −e_{4} & 0 & 0 & 0 \end{matrix}

U_{1} = ⟨−e_{1}^{∗} + e_{5}^{∗}⟩, C_{1} = ⟨e_{1} + e_{5}, e_{2}, e_{3}, e_{4}⟩; U_{2} = ⟨−e_{3}^{∗} + e_{4}^{∗}⟩, C_{2} = ⟨e_{1}, e_{2}, e_{3} + e_{4}, e_{5}⟩;

135 HYPERBOLIC MONGE–AMPÈRE EQUATIONS

𝔤_{3} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 0 & (1 + α)e_{3} & (α − 1)e_{4} & αe_{5} \\ e_{2} & 0 & 0 & βe_{3} & (β + 1)e_{4} & (β − 1)e_{5} \\ e_{3} & −(1 + α)e_{3} & −βe_{3} & 0 & 0 & 0 \\ e_{4} & −(α − 1)e_{4} & −(β + 1)e_{4} & 0 & 0 & 0 \\ e_{5} & −αe_{5} & −(β − 1)e_{5} & 0 & 0 & 0 \end{matrix}

U_{3} = ⟨−e_{3}^{∗} − e_{4}^{∗} + e_{5}^{∗}⟩, C_{3} = ⟨e_{1}, e_{2}, e_{3} + e_{5}, e_{4} + e_{5}⟩, (α, β) ∼ (−α, α + β);

𝔤_{4} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 0 & 2e_{3} & e_{4} & e_{5} \\ e_{2} & 0 & 0 & 0 & e_{4} & −e_{5} \\ e_{3} & −2e_{3} & 0 & 0 & 0 & 0 \\ e_{4} & −e_{4} & −e_{4} & 0 & 0 & e_{3} \\ e_{5} & −e_{5} & e_{5} & 0 & −e_{3} & 0 \end{matrix}

U_{4} = ⟨−e_{2}^{∗} + e_{3}^{∗}⟩, C_{4} = ⟨e_{1}, e_{2} + e_{3}, e_{4}, e_{5}⟩;

𝔤_{5} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & (2 + 2α)e_{2} & (−2 + 2α)e_{3} & (1 + α + β)e_{4} & (1 + α − β)e_{5} \\ e_{2} & −(2 + 2α)e_{2} & 0 & 0 & 0 & 0 \\ e_{3} & −(−2 + 2α)e_{3} & 0 & 0 & 0 & 0 \\ e_{4} & −(1 + α + β)e_{4} & 0 & 0 & 0 & e_{2} \\ e_{5} & −(1 + α − β)e_{5} & 0 & 0 & −e_{2} & 0 \end{matrix}

U_{5} = ⟨−e_{2}^{∗} + e_{3}^{∗}⟩, C_{5} = ⟨e_{1}, e_{2} + e_{3}, e_{4}, e_{5}⟩, (α, β) ∼ (α, −β);

𝔤_{6} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 0 & (1 + α)e_{3} & (1 + α)e_{4} & (α − 1)e_{5} \\ e_{2} & 0 & 0 & 0 & e_{3} & e_{5} \\ e_{3} & −(1 + α)e_{3} & 0 & 0 & 0 & 0 \\ e_{4} & −(1 + α)e_{4} & −e_{3} & 0 & 0 & 0 \\ e_{5} & −(α − 1)e_{5} & −e_{5} & 0 & 0 & 0 \end{matrix}

U_{6} = ⟨−e_{3}^{∗} + e_{5}^{∗}⟩, C_{6} = ⟨e_{1}, e_{2}, e_{3} + e_{5}, e_{4}⟩;

𝔤_{7} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 0 & e_{3} & e_{3} + e_{4} & 0 \\ e_{2} & 0 & 0 & 0 & 0 & e_{5} \\ e_{3} & −e_{3} & 0 & 0 & 0 & 0 \\ e_{4} & −e_{3} − e_{4} & 0 & 0 & 0 & 0 \\ e_{5} & 0 & −e_{5} & 0 & 0 & 0 \end{matrix}

U_{7} = ⟨−e_{3}^{∗} + e_{5}^{∗}⟩, C_{7} = ⟨e_{1}, e_{2}, e_{3} + e_{5}, e_{4}⟩;

𝔤_{8} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & (2 + α)e_{2} & (1 + α)e_{3} & αe_{4} & e_{5} \\ e_{2} & −(2 + α)e_{2} & 0 & 0 & 0 & 0 \\ e_{3} & −(1 + α)e_{3} & 0 & 0 & 0 & e_{2} \\ e_{4} & −αe_{4} & 0 & 0 & 0 & e_{3} \\ e_{5} & −e_{5} & 0 & −e_{2} & −e_{3} & 0 \end{matrix}

O.P. Tchij 136

𝔤_{9} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & e_{5} & e_{3} & 0 & 0 \\ e_{2} & −e_{5} & 0 & 0 & e_{4} & 0 \\ e_{3} & −e_{3} & 0 & 0 & 0 & 0 \\ e_{4} & 0 & −e_{4} & 0 & 0 & 0 \\ e_{5} & 0 & 0 & 0 & 0 & 0 \end{matrix}

U_{9} = ⟨−e_{3}^{∗} − e_{4}^{∗} + e_{5}^{∗}⟩, C_{9} = ⟨e_{1}, e_{2}, e_{3} + e_{5}, e_{4} + e_{5}⟩;

𝔤_{10} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & (1 + α)e_{2} & (α − 1)e_{3} & e_{3} + (α − 1)e_{4} & 2e_{5} \\ e_{2} & −(1 + α)e_{2} & 0 & 0 & 0 & 0 \\ e_{3} & −(α − 1)e_{3} & 0 & 0 & 0 & 0 \\ e_{4} & −e_{3} − (α − 1)e_{4} & 0 & 0 & 0 & e_{2} \\ e_{5} & −2e_{5} & 0 & 0 & −e_{2} & 0 \end{matrix}

U_{10} = ⟨−e_{2}^{∗} + e_{3}^{∗}⟩, C_{10} = ⟨e_{1}, e_{2} + e_{3}, e_{4}, e_{5}⟩;

𝔤_{11} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & e_{1} & 0 & 0 \\ e_{3} & 0 & −e_{1} & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{1} \\ e_{5} & 0 & 0 & 0 & −e_{1} & 0 \end{matrix}

U_{11} = ⟨e_{1}^{∗}⟩, C_{11} = ⟨e_{2}, e_{3}, e_{4}, e_{5}⟩;

𝔤_{12} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & (1 + α)e_{2} & e_{2} + (1 + α)e_{3} & αe_{4} & e_{5} \\ e_{2} & −(1 + α)e_{2} & 0 & 0 & 0 & 0 \\ e_{3} & −e_{2} − (1 + α)e_{3} & 0 & 0 & 0 & 0 \\ e_{4} & −αe_{4} & 0 & 0 & 0 & e_{2} \\ e_{5} & −e_{5} & 0 & 0 & −e_{2} & 0 \end{matrix}

U_{12} = ⟨e_{2}^{∗}⟩, C_{12} = ⟨e_{1}, e_{3}, e_{4}, e_{5}⟩, α ∼ 1/α;

𝔤_{13} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & (2 + 2α)e_{2} & (−2 + 2α)e_{3} & (1 + α)e_{4} & e_{4} + (1 + α)e_{5} \\ e_{2} & −(2 + 2α)e_{2} & 0 & 0 & 0 & 0 \\ e_{3} & −(−2 + 2α)e_{3} & 0 & 0 & 0 & 0 \\ e_{4} & −(1 + α)e_{4} & 0 & 0 & 0 & e_{2} \\ e_{5} & −e_{4} − (1 + α)e_{5} & 0 & 0 & −e_{2} & 0 \end{matrix}

U_{13} = ⟨−e_{2}^{∗} + e_{3}^{∗}⟩, C_{13} = ⟨e_{1}, e_{2} + e_{3}, e_{4}, e_{5}⟩;

𝔤_{14} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 2e_{2} & e_{3} & e_{3} + e_{4} & e_{4} + e_{5} \\ e_{2} & −2e_{2} & 0 & 0 & 0 & 0 \\ e_{3} & −e_{3} & 0 & 0 & 0 & 0 \\ e_{4} & −e_{3} − e_{4} & 0 & 0 & 0 & e_{2} \\ e_{5} & −e_{4} − e_{5} & 0 & 0 & −e_{2} & 0 \end{matrix}

𝔤_{15} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & e_{1} & 0 \\ e_{4} & 0 & 0 & −e_{1} & 0 & e_{2} \\ e_{5} & 0 & −e_{1} & 0 & −e_{2} & 0 \end{matrix}

U_{15} = ⟨e_{1}^{∗}⟩, C_{15} = ⟨e_{2}, e_{3}, e_{4}, e_{5}⟩;

𝔤_{16} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & e_{2} & e_{3} & e_{2} + e_{4} & 0 \\ e_{2} & −e_{2} & 0 & 0 & 0 & 0 \\ e_{3} & −e_{3} & 0 & 0 & 0 & e_{2} \\ e_{4} & −e_{2} − e_{4} & 0 & 0 & 0 & e_{3} \\ e_{5} & 0 & 0 & −e_{2} & −e_{3} & 0 \end{matrix}

U_{16} = ⟨e_{2}^{∗}⟩, C_{16} = ⟨e_{1}, e_{3}, e_{4}, e_{5}⟩;

𝔤_{17} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & e_{2} & e_{2} + e_{3} & e_{3} + e_{4} & 0 \\ e_{2} & −e_{2} & 0 & 0 & 0 & 0 \\ e_{3} & −e_{2} − e_{3} & 0 & 0 & 0 & 0 \\ e_{4} & −e_{3} − e_{4} & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & 0 & 0 & −e_{2} & 0 \end{matrix}

U_{17} = ⟨e_{2}^{∗}⟩, C_{17} = ⟨e_{1}, e_{3}, e_{4}, e_{5}⟩;

𝔤_{18} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 2e_{2} & e_{2} + 2e_{3} & e_{4} & e_{4} + e_{5} \\ e_{2} & −2e_{2} & 0 & 0 & 0 & 0 \\ e_{3} & −e_{2} − 2e_{3} & 0 & 0 & 0 & 0 \\ e_{4} & −e_{4} & 0 & 0 & 0 & e_{2} \\ e_{5} & −e_{4} − e_{5} & 0 & 0 & −e_{2} & 0 \end{matrix}

U_{18} = ⟨e_{2}^{∗}⟩, C_{18} = ⟨e_{1}, e_{3}, e_{4}, e_{5}⟩;

𝔤_{19} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 3e_{2} & 2e_{3} & e_{4} & e_{4} + e_{5} \\ e_{2} & −3e_{2} & 0 & 0 & 0 & 0 \\ e_{3} & −2e_{3} & 0 & 0 & 0 & e_{2} \\ e_{4} & −e_{4} & 0 & 0 & 0 & e_{3} \\ e_{5} & −e_{4} − e_{5} & 0 & −e_{2} & −e_{3} & 0 \end{matrix}

U_{19} = ⟨−e_{2}^{∗} + e_{3}^{∗}⟩, C_{19} = ⟨e_{1}, e_{2} + e_{3}, e_{4}, e_{5}⟩;

𝔤_{20} \begin{matrix}[ , ] & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & e_{1} & e_{2} \\ e_{4} & 0 & 0 & −e_{1} & 0 & e_{3} \\ e_{5} & 0 & −e_{1} & −e_{2} & −e_{3} & 0 \end{matrix}

U_{20} = ⟨e_{1}^{∗}⟩, C_{20} = ⟨e_{2}, e_{3}, e_{4}, e_{5}⟩.

The equivalence relation for contact pairs gives rise to an equivalence relation on the set of parameters (α, β). In the cases 3, 5, and 12, this latter relation is nontrivial and it is written out right after the contact subspace.

The representation of all contact Lie algebras in terms of characteristic functions will be given in 2.3.4.

2.3.2. Algebraic classification of equations. Let 𝔤 be a simply transitive algebra of contact symmetries for some Monge–Ampère system. At a point o ∈ M, we identify the tangent space T_{o}M with 𝔤. The subspaces E_{o}, F_{o}, and C_{o} can be identified with subspaces in 𝔤, which we shall denote by E, F, and C, respectively. There is a one-to-one correspondence between Monge–Ampère systems with simply transitive symmetry algebras and triples (𝔤, C, E). The curvature form Ω_{C} determined by the subspace C in 𝔤 is nondegenerate, and it follows from the condition dω_{0}(E, F) = 0 that Ω_{C}(E_{o}, F_{o}) = 0.

Definition 12. Two triples (𝔤_{1}, C_{1}, E_{1}) and (𝔤_{2}, C_{2}, E_{2}) are said to be equivalent if there exists an isomorphism ψ: 𝔤_{1} → 𝔤_{2} such that ψ(C_{1}) = C_{2} and ψ(E_{1}) = E_{2}.

Locally, a triple (𝔤, C, E) determines a unique Monge–Ampère system with transitive symmetry algebra. Namely, similar to Proposition 6 we have:

Proposition 10. Let 𝔤_{1} and 𝔤_{2} be the simply transitive symmetry algebras of two characteristic Monge–Ampère systems H_{1} = (E_{1}, F_{1}) and H_{2} = (E_{2}, F_{2}). Let o_{1}, o_{2} ∈ M, and let (𝔤_{1}, C_{1}, E_{1}), (𝔤_{2}, C_{2}, E_{2}) be the triples corresponding to the systems H_{1} and H_{2}. In order that there exist a contact diffeomorphism φ such that φ(o_{1}) = o_{2}, φ_{∗}(𝔤_{1}) = 𝔤_{2} and φ_{∗}(H_{1}) = H_{2}, it is necessary and sufficient that the triples (𝔤_{1}, C_{1}, E_{1}) and (𝔤_{2}, C_{2}, E_{2}) be equivalent.

Thus, our problem splits up into the following two:

[A] classify (up to equivalence) all complex contact pairs (𝔤, C) with dim 𝔤 = 5;

[B] classify (up to the group Aut(𝔤, C)) all 2-dimensional subspaces E in C that are not Lagrangean relative to the curvature form Ω_{C}.

Example 7. Let 𝔤 = ⟨e_{1}, e_{2}, e_{3}, e_{4}, e_{5}⟩ be the 5-dimensional Heisenberg algebra, which has only two nonzero Lie bracket relations: [e_{2}, e_{3}] = e_{1} and [e_{4}, e_{5}] = e_{1}. Up to the equivalence of contact pairs, every 4-dimensional subspace C with nondegenerate curvature form is conjugate to the subspace ⟨e_{2}, e_{3}, e_{4}, e_{5}⟩. The restriction of the group Aut(𝔤, C) to C is isomorphic to SP (4, ℂ) × ℂ^{∗}. The group SP (4, ℂ) acts transitively on the set of non-Lagrangean 2-dimensional subspaces, so that we can assume that E = ⟨e_{2}, e_{3}⟩. The subspace F can be recovered in a unique way from Ω_{C}(E, F) = 0, and hence F = ⟨e_{4}, e_{5}⟩. Thus, up to equivalence, there exists a unique Monge– Ampère system with given symmetry algebra. This system is of type H_{(1,1)}.

We shall now provide a classification of type H_{(4,i)} systems with simply transitive symmetry algebras. The type of the distributions may depend on parameters α, β, γ if there are any. The distribution E will be of type 4 for almost all values of the parameters (i.e., the set of parameters α, β, γ for which the distribution has a different type is closed in the Zarisski topology on ℂ^{3}). We shall give the contact subspace C by specifying its basis {v_{1}, v_{2}, v_{3}, v_{4}}.

Proposition 11. Every triple (𝔤, C, E) with E having type 4 is equivalent to one and only one of the following triples:

𝔤_{1}

v_{1} = e_{2}, v_{2} = e_{3}, v_{3} = e_{1} + e_{5}, v_{4} = e_{4}, E_{1} = ⟨αv_{1} + v_{2} + v_{3}, βv_{1} + γv_{2} + v_{4}⟩, F_{1} = ⟨v_{1} − γv_{3} + v_{4}, v_{2} + βv_{3} − αv_{4}⟩; E_{2} = ⟨v_{3}, αv_{1} + v_{2} + v_{4}⟩, F_{2} = ⟨v_{1} − v_{3}, v_{2} + αv_{3}⟩;

𝔤_{2}

v_{1} = e_{5}, v_{2} = e_{3} + e_{4}, v_{3} = e_{1}, v_{4} = e_{2}, E_{1} = ⟨αv_{1} + v_{3}, βv_{1} − v_{2} + v_{4}⟩, F_{1} = ⟨3βv_{1} + αv_{2} + 3v_{4}, −3v_{1} + v_{3}⟩; E_{2} = ⟨v_{2}, αv_{1} + v_{3} + v_{4}⟩, F_{2} = ⟨αv_{1} + v_{4}, −3v_{1} + v_{2}⟩;

𝔤_{3}

\begin{matrix}v_{1} = e_{3} + e_{5}, v_{2} = 2e_{1} + e_{2}, v_{3} = e_{4} + e_{5}, v_{4} = −e_{1} + e_{2}, \\ E_{1} = ⟨v_{2}, v_{1} + γv_{3} + v_{4}⟩, F_{1} = ⟨v_{2} + v_{3}, γv_{3} + v_{4}⟩; \end{matrix}

𝔤_{4}

v_{1} = e_{4}, v_{2} = e_{5}, v_{3} = e_{1}, v_{4} = e_{2} + e_{3}, E_{1} = ⟨αv_{1} + v_{2} + v_{3}, βv_{1} + γv_{2} + v_{4}⟩, F_{1} = ⟨2v_{1} − γv_{3} + v_{4}, 2v_{2} + βv_{3} − αv_{4}⟩; E_{2} = ⟨v_{3}, αv_{1} + v_{2} + v_{4}⟩, F_{2} = ⟨2v_{1} − v_{3}, 2v_{2} + αv_{3}⟩;

𝔤_{5}

\begin{matrix}v_{1} = e_{2} + e_{3}, v_{2} = e_{1}, v_{3} = e_{4}, v_{4} = e_{5}, \\ E_{1} = ⟨v_{2} + γv_{3}, v_{1} + v_{3} + v_{4}⟩, F_{1} = ⟨v_{2} − 4v_{3}, −γv_{1} + 4v_{3} + 4v_{4}⟩; \end{matrix}

𝔤_{6}

v_{1} = e_{3} + e_{5}, v_{2} = e_{4}, v_{3} = e_{2}, v_{4} = e_{1}, E_{1} = ⟨v_{2} + v_{3}, βv_{1} + γv_{2} + v_{4}⟩, F_{1} = ⟨(γ − β)v_{1} + (1 + γ − β)v_{2} + v_{3}, βv_{1} + βv_{2} + v_{4}⟩;

𝔤_{7}

v_{1} = e_{4}, v_{2} = e_{3} + e_{5}, v_{3} = e_{1}, v_{4} = e_{2}, E_{1} = ⟨v_{2} + v_{3}, αv_{1} + βv_{2} + v_{4}⟩, F_{1} = ⟨(−1 − β)v_{1} + βv_{2} + v_{4}, (1 + α + β)v_{1} − (α + β)v_{2} + v_{3}⟩;

𝔤_{8}

v_{1} = e_{2} + e_{3}, v_{2} = e_{4}, v_{3} = e_{5}, v_{4} = e_{1}, E_{1} = ⟨v_{2} + v_{3}, βv_{1} + γv_{2} + v_{4}⟩, F_{1} = ⟨(γ − β)v_{1} + (1 + γ − β)v_{2} + v_{3}, βv_{1} + βv_{2} + v_{4}⟩;

𝔤_{9}

\begin{matrix}v_{1} = e_{3} + e_{5}, v_{2} = e_{1}, v_{3} = e_{4} + e_{5}, v_{4} = e_{2}, \\ E_{1} = ⟨αv_{2} + v_{3}, βv_{1} + v_{4}⟩, F_{1} = ⟨(1 + α)v_{1} + αv_{4}, v_{2} + (β − 1)v_{3}⟩; \end{matrix}

𝔤_{10}

\begin{matrix}v_{1} = e_{2} + e_{3}, v_{2} = e_{4}, v_{3} = e_{5}, v_{4} = e_{1}, \\ E_{1} = ⟨v_{2} + v_{3}, βv_{2} + v_{4}⟩, F_{1} = ⟨(1 − β)v_{1} + 2v_{2} + 2v_{3}, v_{3} − v_{4}⟩; \end{matrix}

𝔤_{12}

\begin{matrix}v_{1} = e_{3}, v_{2} = e_{1}, v_{3} = e_{4}, v_{4} = e_{5}, \\ E_{1} = ⟨v_{2} + βv_{3}, v_{1} + v_{3} + v_{4}⟩, F_{1} = ⟨v_{2} − v_{3}, −βv_{1} + v_{3} + v_{4}⟩; \end{matrix}

𝔤_{13}

\begin{matrix}v_{1} = e_{2} + e_{3}, v_{2} = e_{1}, v_{3} = e_{4}, v_{4} = e_{5}, \\ E_{1} = ⟨v_{2} + βv_{3}, v_{1} + v_{4}⟩, F_{1} = ⟨v_{2} − 4v_{3}, −βv_{1} + 4v_{4}⟩; \end{matrix}

𝔤_{14}

\begin{matrix}v_{1} = e_{2} + e_{3}, v_{2} = e_{4}, v_{3} = e_{5}, v_{4} = e_{1}, \\ E_{1} = ⟨v_{3}, αv_{2} + v_{4}⟩, F_{1} = ⟨αv_{1} − v_{3}, v_{3} − v_{4}⟩; \end{matrix}

𝔤_{16}

\begin{matrix}v_{1} = e_{3}, v_{2} = e_{5}, v_{3} = e_{4}, v_{4} = e_{1}, \\ E_{1} = ⟨v_{2} + v_{3}, αv_{1} + βv_{2} + v_{4}⟩, F_{1} = ⟨v_{1} + βv_{3} − v_{4}, v_{2} − αv_{3}⟩; \end{matrix}

𝔤_{17}

\begin{matrix}v_{1} = e_{3}, v_{2} = e_{5}, v_{3} = e_{1}, v_{4} = e_{4}, \\ E_{1} = ⟨v_{3}, αv_{1} + v_{2} + v_{4}⟩, F_{1} = ⟨αv_{2} + v_{3}, v_{2} + v_{4}⟩; \end{matrix}

𝔤_{18}

v_{1} = e_{3}, v_{2} = e_{1}, v_{3} = e_{4}, v_{4} = e_{5},

E_{1} = ⟨v_{2} + v_{3}, αv_{1} + v_{4}⟩, F_{1} = ⟨v_{1} − v_{4}, v_{2} − αv_{3}⟩; E_{2} = ⟨v_{2}, v_{1} + v_{4}⟩, F_{2} = ⟨v_{2} − v_{3}, v_{4}⟩;

𝔤_{19}

v_{1} = e_{2} + e_{3}, v_{2} = e_{4}, v_{3} = e_{5}, v_{4} = e_{1},

E_{1} = ⟨v_{3}, αv_{1} + βv_{2} + v_{4}⟩,

F_{1} = ⟨αv_{1} + αv_{2} + v_{4}, (β − α)v_{1} + (β − α)v_{2} + v_{3}⟩;

𝔤_{20}

\begin{matrix}v_{1} = e_{2}, v_{2} = e_{3}, v_{3} = e_{4}, v_{4} = e_{5}, \\ E_{1} = ⟨v_{3}, v_{2} + v_{4}⟩, F_{1} = ⟨v_{1} + v_{3}, v_{4}⟩. \end{matrix}

Note that in the case of the Lie algebras 𝔤_{11} and 𝔤_{15}, there exist no subspaces of type 4.

The corresponding Monge–Ampère equations are listed in 2.3.5.

2.3.3. Construction of invariant differential forms. In this section we offer a method for constructing a representation of an abstract Lie algebra in the Lie algebra of vector fields on a manifold. This method allows to find representations of 5-dimensional contact Lie algebras in the algebra of contact vector fields cont(M).

As can be seen from the above, the problem of finding an invariant Monge– Ampère equation reduces to finding an invariant 2-form ϑ. Towards the end of this section, we shall describe a method for constructing arbitrary invariant differential forms on Lie groups.

Let 𝔤 be a finite-dimensional Lie algebra, and let U(𝔤) be the universal enveloping algebra of 𝔤 (see [2]). Denote by U(𝔤)^{∗} the dual space of the vector space U(𝔤), which can be endowed with an algebra structure:

< f · g, u > = < f ⊗ g, c(u) > f, g ∈ U(𝔤)^{∗}, u ∈ U(𝔤),

where c is the coproduct in the algebra U(𝔤).

Fix a basis {e_{1}, …, e_{n}} for the Lie algebra 𝔤, and consider the algebra

ℝ[[X_{1}, …, X_{n}]]

of formal real power series in X_{1}, …, X_{n}. For I = (i_{1}, …, i_{n}) ∈ ℕ^{n}, we define

e_{I} = \frac{e_{1}^{i_{1}} · . . . · e_{n}^{i_{n}}}{i_{1}! · . . . · i_{n}!}

and let X^{I} = X_{1}^{i_{1}} · . . . · X_{n}^{i_{n}}. By the Poincare–Birkhoff–Witt theorem ([2]), the elements e_{I} form a basis of the vector space U(𝔤). Consequently, specifying an element f ∈ U(𝔤)^{∗} is equivalent to specifying its values at the basis elements. It follows that the map

s : U(𝔤)^{∗} ⟶ ℝ[[X_{1}, …, X_{n}]]
f ⟼ ∑_{I} f(e_{I}) · X^{I}

is an isomorphism of the algebras U(𝔤)^{∗} and ℝ[[X_{1}, …, X_{n}]].

Consider the representation of the Lie algebra 𝔤 in the space U(𝔤). A 𝔤module structure in U(𝔤) can then be defined in two ways:

x_{L}.u = xu and x_{R}.u = −ux,

where x ∈ 𝔤, u ∈ U(𝔤), which corresponds to the left and right regular representations. A 𝔤-module structure in U(𝔤)^{∗} is introduced accordingly:

(x.f)(u) = −f(x.u),

where x ∈ 𝔤, f ∈ U(𝔤)^{∗}. Then the map

x_{U(𝔤)} : f ⟼ x.f,

with x ∈ 𝔤, f ∈ U(𝔤)^{∗} is a derivation of the algebra U(𝔤)^{∗} (see [2]). By virtue of the isomorphism s, any element x ∈ 𝔤 determines a derivation v(x) of the algebra ℝ[[X_{1}, …, X_{n}]]:

v(x).s(f) = s(x.f),

where f ∈ U(𝔤)^{∗}, so that the derivation v(x) may be viewed as a formal vector field v(x) in n variables. Thus,

v(x) = f_{1} \frac{∂}{∂x_{1}} + ⋯ + f_{n} \frac{∂}{∂x_{n}} ,  (10)

where

f_{i} = − ∑_{I} ε_{i}(x.e_{I}) · X^{I},  (11)

and ε_{i} ∈ U(𝔤)^{∗} are such that s(ε_{i}) = X_{i} (i = 1, …, n). As a result, we obtain two representations of 𝔤 in terms of the vector fields on ℝ^{n}:

ρ_{L} : 𝔤 ⟶ D(ℝ^{n})
x ⟼ v_{L}(x)

and

ρ_{R} : 𝔤 ⟶ D(ℝ^{n})
x ⟼ v_{R}(x),

which correspond to the left and right regular representations of 𝔤 in U(𝔤).

Lemma 4. For any x, y ∈ 𝔤,

[v_{L}(x), v_{R}(y)] = 0.

Proof. Since x_{L} ∘ y_{R} = y_{R} ∘ x_{L} for all f ∈ U(𝔤)^{∗}, we have [v_{L}(x), v_{R}(y)].s(f) = 0, and the desired equality now follows immediately from the fact that s is an isomorphism.

For every point a ∈ ℝ^{n}, the vectors v_{R}(e_{1})_{a}, …, v_{R}(e_{n})_{a} form a basis of the vector space T_{a}(ℝ^{n}). If therefore g(a) : T_{0}(ℝ^{n}) → T_{a}(ℝ^{n}) is a map such that for x = ∑_{i=1}^{n} c_{i} · v_{R}(e_{i})_{0} ∈ T_{0}(ℝ^{n}) we have g(a) · x = ∑_{i=1}^{n} c_{i} · v_{R}(e_{i})_{a}, then g(a) is an isomorphism of T_{0}(ℝ^{n}) onto T_{a}(ℝ^{n}) and g(0) = id_{T_{0}(ℝ^{n})}. Consider the dual map g^{∗}(a) of g^{∗}(a), which is an isomorphism of T_{a}^{∗}(ℝ^{n}) onto T_{0}^{∗}(ℝ^{n}). Suppose that v_{R}^{∗}(e_{1})_{0}, …, v_{R}^{∗}(e_{n})_{0} is the dual basis, and define a differential form on ℝ^{n} by

ω_{i}(a) = g^{∗}(a)^{−1} · (v_{R}^{∗}(e_{i})_{0}).

Thus for each point a ∈ ℝ^{n}, the covectors {ω_{1}(a), …, ω_{n}(a)} form a dual basis of the space T_{a}(ℝ^{n}), so that < ω_{i}(a), v_{R}(e_{j})_{a} >= δ_{ij}.

Proposition 12. Consider the action of 𝔤 on ℝ^{n} given by the Lie algebra homomorphism

ρ_{L} : 𝔤 ⟶ D(ℝ^{n}).

Then the forms {ω_{1}, …, ω_{n}} constitute a basis for the space of the differential forms that are invariant under this action.

Proof. It suffices to show that for any i, j = 1, …, n we have L_{v_{L}(e_{i})}ω_{j} = 0. Since the vectors {v_{R}(e_{1})_{a}, …, v_{R}(e_{n})_{a}}q form a basis of the space T_{a}(ℝ^{n}), we can require only that the equality

(L_{v_{L}(e_{i})}ω_{j})(v_{R}(e_{k})) = 0

be satisfied for all i, j, k = 1, …, n. We have

(L_{v_{L}(e_{i})}ω_{j})(v_{R}(e_{k})) = d(ω_{j}(v_{L}(e_{i})))(v_{R}(e_{k})) + (v_{L}(e_{i})⌟dω_{j})(v_{R}(e_{k})) =

= (v_{R}(e_{k}))(ω_{j}(v_{L}(e_{i}))) + dω_{j}(v_{L}(e_{i}), v_{R}(e_{k})) =
= (v_{R}(e_{k}))(ω_{j}(v_{L}(e_{i}))) + v_{L}(e_{i})ω_{j}(v_{R}(e_{k})) −

− (v_{R}(e_{k}))(ω_{j}(v_{L}(e_{i}))) − ω_{j}[v_{L}(e_{i}), v_{R}(e_{k})] = v_{L}(e_{i}) δ_{jk} = 0,

which proves the proposition.

Let S be the vector space consisting of all invariant differential forms of degree k on ℝ^{n}. The elements ω_{i_{1}}(0) ∧ ⋯ ∧ ω_{i_{k}}(0) ∈ Λ^{k}(T_{0}(ℝ^{n})) with k ⩽ n, 1 ⩽ i_{1} < ⋯ < i_{k} ⩽ n form a basis of the space Λ^{k}(T_{0}(ℝ^{n})). Given ω ∈ S, we define an element ω′ ∈ Λ^{k}(ℝ^{n}) by letting

ω′ = ∑ c_{i_{1}...i_{k}} · ω_{i_{1}} ∧ ⋯ ∧ ω_{i_{k}},

where the coefficients c_{i_{1}...i_{k}} are such that

ω(0) = ∑ c_{i_{1}...i_{k}} · ω_{i_{1}}(0) ∧ ⋯ ∧ ω_{i_{k}}(0).

Corollary. For every ω ∈ S, the following equality holds:

ω = ω′.

Proof. Since for any X ∈ D(ℝ^{n}) the operator L_{X} is a derivation of the algebra of forms on ℝ^{n}, we have ω′ ∈ S. Since an invariant differential form is uniquely determined by its value at the point 0, the equality ω(0) = ω′(0) proves what we wanted.

2.3.4. Representation of contact algebras in terms of characteristic functions. Now we shall construct a vector field representation for the 5-dimensional Heisenberg algebra 𝔤 = 𝔤_{11}.

Let (u, p, x, q, y) denote coordinates in 𝔤 corresponding to a fixed basis. The following relations are satisfied within the universal enveloping algebra U(𝔤):

e_{2}e_{3} − e_{3}e_{2} = e_{1}, e_{4}e_{5} − e_{5}e_{4} = e_{1}.

We choose the vectors of the form e_{I} = \frac{e_{1}^{α_{1}}e_{3}^{α_{2}}e_{2}^{α_{3}}e_{5}^{α_{4}}e_{4}^{α_{5}}}{α_{1}!α_{2}!α_{3}!α_{4}!α_{5}!} as a basis for U(𝔤). Consider the decomposition of the element e_{1}.e_{I} relative basis. Elements of the first degree occur in this decomposition only in the expression e_{1}.1 = e_{1}. Thus, according to the formulas (10) and (11), we get ρ_{L}(e_{1}) = −\frac{∂}{∂u}. Similarly,

e_{2}.1 = e_{2}, e_{2}.e_{3} = e_{1} + e_{3}e_{2} ⟹ ρ_{L}(e_{2}) = −x\frac{∂}{∂u} − \frac{∂}{∂p};

e_{3}.1 = e_{3} ⟹ ρ_{L}(e_{3}) = −\frac{∂}{∂x};

e_{4}.1 = e_{4}, e_{4}.e_{5} = e_{1} + e_{5}e_{4} ⟹ ρ_{L}(e_{4}) = −y \frac{∂}{∂u} − \frac{∂}{∂q};

e_{5}.1 = e_{5} ⟹ ρ_{L}(e_{5}) = −\frac{∂}{∂y} .

Given a contact symmetry X ∈ cont(M), we can consider its characteristic function ω_{0}(X), where ω_{0} = du−pdx−qdy. As a result, we obtain the following representation of our Lie algebra 𝔤: ⟨−1, −x, p, −y, q⟩.

Similar computations for the rest of the contact Lie algebras yield the following results:

Proposition 13. Any simply transitive Lie algebra of contact vector fields on a 5-dimensional complex manifold is locally equivalent to one and only one of the following algebras:

\begin{matrix}𝔤_{1} & ⟨−1 + xp, p, x − \frac{x^{2}}{2} p, q, e^{−y}⟩ \\ 𝔤_{2} & ⟨p + 2yq, q, −e^{2x} − yp − y^{2}q, e^{−x}, −ye^{−x}⟩ \\ 𝔤_{3} & ⟨−αu + (1 + α)xp + (α − 1)yq, (1 − β)u + βxp + (1 + β)yq, p, q, −1⟩ \\ 𝔤_{4} & ⟨−2u + yq, p + yq, −1, q, y⟩ \\ 𝔤_{5} & ⟨2(1 − α)u + 2(1 + α)xp + (1 + α + β)yq, p, −1, q, −yp⟩ \\ 𝔤_{6} & ⟨(1 − α)u + (1 + α)xp + (1 + α)yq, −u + yp, p, q, −1⟩ \\ 𝔤_{7} & ⟨(x + y)p + yq, −u, p, q, −1⟩ \\ 𝔤_{8} & ⟨−(2 + α)u + p + yq, −1, e^{x} − y, −\frac{y^{2}}{2} + e^{x}y, q⟩ \\ 𝔤_{9} & ⟨p, x + q, e^{−x}, e^{−y}, −1⟩ \\ 𝔤_{10} & ⟨−(1 + α)u + ((α − 1)x + y)p + (α − 1)yq, −1, p, q, y⟩ \\ 𝔤_{11} & ⟨−1, −x, p, −y, q⟩ \\ 𝔤_{12} & ⟨−x − (1 + α)u + (1 + α)xp + αyq, −1, p, q, y⟩ \\ 𝔤_{13} & ⟨−2(1 + α)u + 2(α − 1)xp + (1 + α)yq − \frac{q^{2}}{2} , −1, p, q, y⟩ \\ 𝔤_{14} & ⟨−\frac{x^{2}e^{2x}}{2} + e^{x}y − 2u + p + (y − e^{x}x)q − \frac{q^{2}}{2} , −1, e^{x}, q, y + \frac{x^{2}e^{x}}{2} ⟩ \\ 𝔤_{15} & ⟨−1, −x, −y, −\frac{x^{2}}{2} + q, p⟩ \\ 𝔤_{16} & ⟨−x − u + xp + yq, −1, q, p, y − xq⟩ \\ 𝔤_{17} & ⟨−x − u + (x + y)p + yq, −1, p, q, y⟩ \\ 𝔤_{18} & ⟨−x − 2u + 2xp + yq − \frac{q^{2}}{2} , −1, p, q, y⟩ \\ 𝔤_{19} & ⟨−\frac{y^{3}}{6} + \frac{e^{x}y^{2}}{2} − 3u + p + yq, −1, −y + e^{x}, −\frac{y^{2}}{2} + ye^{x}, q⟩ \\ 𝔤_{20} & ⟨−1, −x, −y − \frac{x^{2}}{2} , q, p + \frac{x^{2}y}{2} + \frac{y^{2}}{2} ⟩ \end{matrix}

2.3.5. Classification results. We continue with the example considered in the previous section and find normal forms for the equations that are invariant with respect to the 5-dimensional Heisenberg algebra 𝔤.

Let 𝔤 = ⟨−1, −x, p, −y, q⟩ be the representation of 𝔤 in terms of characteristic functions. Compute the representation ρ_{R}(𝔤) corresponding to the right regular representation of 𝔤 in U(𝔤).

ρ_{R}(e_{1}) = \frac{∂}{∂u};

ρ_{R}(e_{2}) = \frac{∂}{∂p};

ρ_{R}(e_{3}) = \frac{∂}{∂x} + p\frac{∂}{∂u};

ρ_{R}(e_{4}) = \frac{∂}{∂q};

ρ_{R}(e_{5}) = \frac{∂}{∂y} + q \frac{∂}{∂u}.

The forms ω_{1} = du − pdx − qdy, ω_{2} = dp, ω_{3} = dx, ω_{4} = dq, ω_{5} = dy constitute a basis of the space of invariant differential forms. According to example 7, the characteristic distributions have the form E = ⟨ρ_{R}(e_{2}), ρ_{R}(e_{3})⟩ and F = ⟨ρ_{R}(e_{4}), ρ_{R}(e_{5})⟩. By formula (4), where H can now be assumed to be the unit matrix, we get

ϑ = ϑ_{0} = dx ∧ dp + dq ∧ dy.

It follows that the wave equation u_{xy} = 0 is the only equation that admits the 5-dimensional Heisenberg algebra as its transitive Lie algebra of contact symmetries.

Similar computations for the triples (𝔤, C, E) from Proposition 11 yield the following result:

Theorem 7. Any hyperbolic Monge–Ampère equation of type H_{(4,i)} with simply transitive algebra of contact symmetries is locally equivalent to one and only one of the following equations:

𝔤_{1 1}

2e^{u+u_{y}}(αu_{y} + β)u_{xx} + 2(1 − αγ + β)u_{xy} + (2e^{−2(u+u_{y})} + αu_{x}^{2})e^{u+u_{y}}u_{yy} +

+ (2e^{−2(u+u_{y})}(u_{y} + γ) + u_{x}^{2}(αu_{y} + β))e^{u+u_{y}} + 2αe^{u+u_{y}} Hess u = 0;

𝔤_{1 2}
2αe^{u+u_{y}}u_{xx} + 2u_{xy} + (2e^{−2(u+u_{y})} + αu_{x}^{2})e^{u+u_{y}} = 0;
𝔤_{2 1}

\frac{2}{27}(−27u_{x} + 9αu_{x} + 6u_{x}u^{2} + 6u_{x}^{2}u + 9αu + 27β − 27u + 2u^{3} + 2u_{x}^{3})e^{−2x}u_{xx} −

− \frac{2}{3}(2u^{2} + 4uu_{x} + 2u_{x}^{2} − 6u_{y}e^{2x} − 9 + 3α)u_{xy} + 2e^{2x}(2u + u_{x})u_{yy} +

+ \frac{2}{27}(−54βu − 9αu^{2} − 27βu_{x} − 2u^{3}u_{x} + 81u_{y}e^{2x} − 9αuu_{x} + 2uu_{x}^{3} − u^{4} + u_{x}^{4} +

+ 27u^{2} − 81α + 27uu_{x} − 36u_{y}u_{x}ue^{2x} − 18u_{y}u^{2}e^{2x} − 18u_{y}u_{x}^{2}e^{2x} −

− 27αu_{y}e^{2x} + 27u_{y}^{2}e^{4x})e^{−2x} − 2e^{2x} Hess u = 0;

𝔤_{2 2}

− \frac{2}{9}e^{−2x}(u + u_{x})(u − 3 + u_{x})u_{xx} + 2(−1 + \frac{2}{3}u + \frac{2}{3}u_{x})u_{xy} − 2e^{2x}u_{yy} +

+(−\frac{2}{3}u^{2}+ \frac{4}{27}u^{3}− \frac{2}{27}u_{x}^{3}− \frac{2}{3}u_{x}u+ \frac{2}{9}u_{x}u^{2}+2α)e^{−2x}−2u_{y} + \frac{4}{3}u_{y}u_{x}+ \frac{4}{3}u_{y}u = 0;

𝔤_{3}

u_{xy} + γ \frac{u_{x}}{u_{y}} u_{yy} + 3ux^{\frac{2+2α+β}{3}} uy^{\frac{2−α+β}{3}} = 0;

𝔤_{4 1}

−\frac{e^{−x}}{u_{x}} u_{xx} + \frac{γα − β − 2}{u_{x}^{1/2}}u_{xy} − 2βe^{x}u_{yy} − 2γe^{−x} − α\frac{e^{x}}{u_{x}} Hess u = 0;

𝔤_{4 2}

u_{xy} + αe^{x}u_{x}^{1/2}u_{yy} + e^{−x}u_{x}^{1/2} = 0;

𝔤_{5}

u_{y}(4ux^{\frac{α−β−7}{4}} u_{y} + (γ − 4)u_{x}^{−1})u_{xx} + (4 − γ − 8ux^{\frac{α−β−3}{4}} u_{y})u_{xy} +

+ 4ux^{\frac{α−β+1}{4}} u_{yy} − 4ux^{\frac{α+β+5}{4}} − γu_{x}^{− \frac{α+β+7}{4}} Hess u = 0;

𝔤_{6}

\frac{βu_{x}^{2} − (β + γ)u_{x}u_{y} + (γ + 2)u_{y}^{2}}{u_{x}^{2}} u_{xx} + \frac{(β + γ)u_{x} − 2(γ + 2)u_{y}}{u_{x}} u_{xy} +

+ (2 + γ)u_{yy} + 2ux^{\frac{α+3}{2}} e^{\frac{uy(1+α)}{2ux}} + βu_{x}^{− \frac{α+3}{2}} e^{− \frac{uy(1+α)}{2ux}} Hess u = 0;

𝔤_{7}

\frac{−βu_{x}^{2} + (α − β − 1)u_{x}u_{y} + αu_{y}^{2}}{u_{x}^{2}} u_{xx} + \frac{(1 + β − α)u_{x} − 2αu_{y}}{u_{x}} u_{xy} +

+ αu_{yy} + e^{\frac{uy}{ux}} u_{x} + \frac{α}{u_{x}} e^{− \frac{uy}{ux}} Hess u = 0;

𝔤_{8}

(e^{(1+α)x} + e^{−x}u_{x} + u_{y})e^{−(2+α)x}u_{xx} +

+ (2u_{x} + 2e^{x}u_{y} − γe^{(2+α)x} + βe^{(2+α)x})e^{−(2+α)x}u_{xy} −

− (u_{x} − βe^{(2+α)x})e^{−(1+α)x}u_{yy} − (−e^{(α−1)x}u_{x}γ + 2e^{−3x}u_{x}^{2} + 3e^{−2x}u_{x}u_{y} − e^{αx}u_{y}γ +

+ e^{−x}u_{y}^{2} + e^{(−1+α)x}u_{x} − e^{(1+2α)x}β)e^{−αx} + e^{−(1+α)x} Hess u = 0;

𝔤_{9}

αu_{y}u_{xx}+(βα−α+1)u_{xy}+α(y+u_{x})u_{yy}+(−β+1+αyu_{y}+αu_{x}u_{y})+α Hess u = 0;

𝔤_{10}

ln(u_{x})(ln(u_{x}) − β + 1)u_{xx} + 2(2 ln(u_{x}) − β + 1)u_{xy} + 4u_{yy} + 4ux^{\frac{3−α}{2}} = 0;

𝔤_{12}

(1 − β)u_{xy} + e^{1−u_{x}}u_{yy} + e^{α(1−u_{x})} + βe^{α(u_{x}−1)} Hess u = 0;

𝔤_{13}

(4 − β)u_{xy} + ln(u_{x})ux^{\frac{3−α}{4}} u_{yy} + 4ux^{\frac{3−α}{4}} + βux^{\frac{α−3}{4}} Hess u = 0;

𝔤_{14}

−e^{−x}u_{xx}+(2x+2x^{2}−α)u_{xy}+x(e^{2x}α+xe^{2x}+x^{2}e^{2x}+u_{x}+e^{x}u_{y}+e^{x}xu_{y})e^{−x}u_{yy}+

+(2xe^{2x} +3x^{2}e^{2x} +x^{3}e^{2x} +u_{x} +e^{x}u_{y} +e^{x}xu_{y} −αxe^{2x})e^{−x} −xe^{−x} Hess u = 0;

𝔤_{16}

βu_{xx} +(1+2βu_{y} −α)u_{xy} +u_{y}(1−α+βu_{y})u_{yy} +e^{− \frac{2ux+u^{2}_{y}}{2}} +αe^{\frac{2ux+u^{2}_{y}}{2}} Hess u = 0;

𝔤_{17}
(α + u_{x})u_{x}u_{xx} + (α + 2u_{x})u_{xy} + u_{yy} + e^{−u_{x}} = 0;

𝔤_{18}
(1 − α)u_{xy} + u_{x}e^{−u_{x}}u_{yy} + e^{−u_{x}} + αe^{u_{x}} Hess u = 0;

𝔤_{19}

(xe^{3x}+u_{x}+e^{x}u_{y})e^{−2x}u_{xx}+(2u_{x}+2e^{x}u_{y}−e^{3x}β+αe^{3x})e^{−x}u_{xy}−(u_{x}−αe^{3x})u_{yy}+

+e^{−2x}(u_{x}e^{3x}β−2u_{x}^{2}−3u_{x}e^{x}u_{y}+e^{4x}u_{y}β−e^{2x}u_{y}^{2}−u_{x}xe^{3x}+αe^{6x}x)+Hess u = 0;

𝔤_{20}

2xu_{xx} + 2(1 − x^{2})u_{xy} + (−2u_{y} − \frac{x^{3}}{3} )u_{yy} − 2xu_{y} − \frac{5x^{4}}{6} + x^{2} + 2 Hess u = 0.

Some of the above families of equations can be equivalent to each other under the action of a discrete group on the set of parameters (α, β, γ).

3. Equations of type H_{(3,i)}

We shall assume in this section that all function that occur in our discussion are complex analytic functions. This assumption allows to use the existing classification of symmetry algebras on a 3-dimensional contact manifold and the equivalence criterion for G-structures. An essential feature of the equations of type H_{(3,i)} is that the algebra sym(E) of contact symmetries can in this case be infinite-dimensional.

3.1. Symmetry algebras of characteristic distributions of type 3. If a 2-dimensional characteristic distribution E on an arbitrary 5-dimensional manifold is of type (3), then it is locally equivalent to the contact distribution on J^{3}(ℂ, ℂ) (see [23]). Here J^{3}(ℂ, ℂ) is the space of third jets of analytic maps from ℂ to ℂ. It follows that there is a local coordinate system (x, y, y_{1}, y_{2}, y_{3}) on the manifold in which E is generated by the vector fields

E = ⟨\frac{∂}{∂x} + y_{1} \frac{∂}{∂y} + y_{2} \frac{∂}{∂y_{1}} + y_{3} \frac{∂}{∂y_{2}} , \frac{∂}{∂y_{3}} ⟩ .

Indeed, in this case

E^{(1)} = ⟨\frac{∂}{∂x} + y_{1} \frac{∂}{∂y} + y_{2} \frac{∂}{∂y_{1}} , \frac{∂}{∂y_{2}} , \frac{∂}{∂y_{3}} ⟩;

E^{(2)} = ⟨\frac{∂}{∂x} + y_{1} \frac{∂}{∂y} , \frac{∂}{∂y_{1}} , \frac{∂}{∂y_{2}} , \frac{∂}{∂y_{3}} ⟩;

E^{(3)} = ⟨\frac{∂}{∂x}, \frac{∂}{∂y} , \frac{∂}{∂y_{1}} , \frac{∂}{∂y_{2}} , \frac{∂}{∂y_{3}} ⟩ ,

and E has type (3).

Any diffeomorphism J^{3}(ℂ, ℂ) that preserves E is called contact in the sense of contact structure on J^{3}(ℂ, ℂ).

By the Lie–Bäcklund theorem [20], every contact transformation of J^{3}(ℂ, ℂ) is a lift of some transformation from J^{1}(ℂ, ℂ). A classification of all symmetry algebras on 3-dimensional contact manifold can be found in the works of Sophus Lie (see [29, 31]).

Symmetry algebras of 3-dimensional contact manifold can be divided into the following two classes: reducible, i.e. those that can be derived by lifting transformations from ℂ^{2}, and irreducible, i.e. those that cannot be so derived. For the algebras in the first class, we specify the corresponding subalgebra in the algebra of vector fields on the plane, whereas for those in the second class, we give their representation in terms of characteristic functions. Thus, Lie algebras of the form 𝔤̄ ⊂ D(ℂ^{2}) will be written as

⟨A(x, y)\frac{∂}{∂x} + B(x, y)\frac{∂}{∂y} ⟩ ,

while for the second class algebras 𝔤̄ ⊂ ⟨f(x, y, y_{1})⟩ we shall have

X_{f} = −\frac{∂f}{∂y_{1}} \frac{∂}{∂x} + (f − y_{1} \frac{∂f}{∂y_{1}} ) \frac{∂}{∂y} + (\frac{∂f}{∂x} + y_{1} \frac{∂f}{∂y} ) \frac{∂}{∂y_{1}} ,

where (x, y, y_{1}) is the natural coordinate system in J^{1}(ℂ, ℂ).

An algebra of vector fields on the plane can also be written in terms of characteristic functions for f(x, y, y_{1}) = B(x, y) − y_{1}A(x, y). The natural prolongation of the contact symmetry X_{f} to the space J^{3}(ℂ, ℂ) has the form:

X_{f}^{(3)} = X_{f} + D^{2}(f)\frac{∂}{∂y_{2}} + D^{3}(f)\frac{∂}{∂y_{3}} ,

where D = \frac{∂}{∂x} + y_{1} \frac{∂}{∂y} + y_{2} \frac{∂}{∂y_{1}} + y_{3} \frac{∂}{∂y_{2}} .

Among all symmetry algebras of J^{1}(ℂ, ℂ), we now consider only those that correspond to Monge–Ampère equations of type H_{(3,i)} with transitive symmetry algebra. A first obvious requirement to such an algebra 𝔤̄ ⊂ sym(J^{1}(ℂ, ℂ)) should be that its prolongation 𝔤̄^{(3)} ⊂ sym(J^{3}(ℂ, ℂ)) be transitive on J^{3}(ℂ, ℂ). We shall now determine what conditions should be satisfied for 𝔤̄ in order that there exist no invariant 4-dimensional distribution on J^{3}(ℂ, ℂ) with nondegenerate curvature form. The nonexistence of such a distribution implies that J^{3}(ℂ, ℂ) admits no invariant contact structure in the sense of J^{1}(ℂ^{2}, ℂ).

Proposition 14. Let 𝔤̄ be a transitive Lie algebra on a 5-dimensional manifold M, and let 𝔤 = 𝔤̄_{o} be the isotropy subalgebra for some point o ∈ M of general position. Assume that 𝔤̄ contains a subalgebra 𝔟 such that

(1) 𝔤̄ = 𝔤 ⊕ 𝔟;

(2) 𝔟 contains a 4-dimensional commutative subalgebra 𝔥. Then M admits no 𝔤̄-invariant 4-dimensional distribution with nondegenerate curvature form.

Proof. Since the dimension of the homogeneous space M is equal to the codimension of 𝔤 in 𝔤̄, it follows from (1) that dim 𝔟 = 5, and the subalgebra 𝔟 determines a simply transitive action on M. Since 𝔟 ⊂ 𝔤̄, every 𝔤̄-invariant distribution on M will be invariant under 𝔤̄. Every 𝔟-invariant distribution C can be uniquely recovered from the subspace C_{o} ⊂ 𝔟. If dim C=4, then either C_{o} = 𝔥 or C_{o} ≠ 𝔥. Since 𝔥 is commutative, the curvature of C in the first case is identically equal to 0. In the second case we find that dim C_{o} ∩ 𝔥 = dim C_{o} + dim 𝔥 − dim 𝔟 = 3, and the subspace C_{o} ∩ 𝔥 is isotropic relative to the curvature form of C. Thus, the curvature form is degenerate, for otherwise the dimension of the isotropic subspace would be at most equal to 2.

Corollary. Let 𝔤̄ be a subalgebra of the Lie algebra of vector fields on the plane, and suppose that 𝔤̄ contains the subalgebra

𝔟 = ⟨\frac{∂}{∂x}, \frac{∂}{∂y} , x\frac{∂}{∂y} , x^{2} \frac{∂}{∂y} , x^{3} \frac{∂}{∂y} ⟩ .  (12)

Then the space J^{3}(ℂ, ℂ) admits no 𝔤̄^{(3)}-invariant 4-dimensional distribution with nondegenerate curvature form.

Proof. The Lie algebra 𝔟^{(3)} has the form

𝔟^{(3)} = ⟨\frac{∂}{∂x}, \frac{∂}{∂y} , x\frac{∂}{∂y} + \frac{∂}{∂y_{1}} , x^{2} \frac{∂}{∂y} + 2x\frac{∂}{∂y_{1}} + 2\frac{∂}{∂y_{2}} ,

x^{3} \frac{∂}{∂y} + 3x^{2} \frac{∂}{∂y_{1}} + 6x\frac{∂}{∂y_{2}} + 6\frac{∂}{∂y_{3}} ⟩

and acts simply transitively on J^{3}(ℂ, ℂ). The subspace in 𝔟 generated by the last four vectors is a commutative subalgebra. So that the conditions of Proposition 14 are satisfied.

Consequently, those symmetry algebras of 3-dimensional contact space that contain the subalgebra of the form (12) or, in the infinite-dimensional case, the subalgebra ⟨\frac{∂}{∂x}, f(x)\frac{∂}{∂y}⟩, cannot by symmetry algebras for Monge–Ampère equations of type H_{(3,i)}, so that from now on we can rule them out.

3.2. Classification of equations whose symmetry groups are not simply transitive. If E is an equation whose symmetry algebra sym(E) is not simply transitive, then dim sym(E) ⩾ 6. Based on the classification of symmetry algebras for 3-dimensional contact space [29, 31], we obtain the following result:

Proposition 15. Let 𝔤̄ be an algebra of contact symmetries of the space J^{1}(ℂ, ℂ) such that

(a) dim 𝔤̄ ⩾ 6;

(a) the prolongation 𝔤̄^{(3)} is transitive in J^{3}(ℂ, ℂ);

(b) 𝔤̄ contains no subalgebras 𝔟 specified in Proposition 14. Then 𝔤̄ is locally equivalent to one and only one of the following algebras:

1) ⟨\frac{∂}{∂x}, \frac{∂}{∂y}, x\frac{∂}{∂x}, y \frac{∂}{∂x}, x\frac{∂}{∂y}, y \frac{∂}{∂y}⟩;

2) ⟨\frac{∂}{∂x}, \frac{∂}{∂y}, x\frac{∂}{∂x}, y \frac{∂}{∂y}, y \frac{∂}{∂x}, x\frac{∂}{∂y}, x^{2} \frac{∂}{∂x} + xy \frac{∂}{∂y}, xy \frac{∂}{∂x} + y^{2} \frac{∂}{∂y}⟩;

3) ⟨\frac{∂}{∂x}, x\frac{∂}{∂x}, x^{2} \frac{∂}{∂x}, \frac{∂}{∂y}, y \frac{∂}{∂y}, y^{2} \frac{∂}{∂y}⟩;

4) ⟨\frac{∂}{∂x}, x\frac{∂}{∂x}, y \frac{∂}{∂y}, \frac{∂}{∂y}, x\frac{∂}{∂y}, x^{2} \frac{∂}{∂y}⟩;

5) ⟨\frac{∂}{∂x}, 2x\frac{∂}{∂x} + 2y \frac{∂}{∂y}, x^{2} \frac{∂}{∂x} + 2xy \frac{∂}{∂y}, \frac{∂}{∂y}, x\frac{∂}{∂y}, x^{2} \frac{∂}{∂y}⟩;

6) ⟨\frac{∂}{∂x}, x\frac{∂}{∂x}, x^{2} \frac{∂}{∂x} + xy \frac{∂}{∂y}, y \frac{∂}{∂y}, \frac{∂}{∂y}, x\frac{∂}{∂y}⟩;

7) ⟨\frac{∂}{∂x}, x\frac{∂}{∂x}, x^{2} \frac{∂}{∂x} + 2xy \frac{∂}{∂y}, y \frac{∂}{∂y}, \frac{∂}{∂y}, x\frac{∂}{∂y}, x^{2} \frac{∂}{∂y}⟩;

8) ⟨1, x, x^{2}, y_{1}, xy_{1}, y_{1}^{2}⟩;

9) ⟨1, x, x^{2}, y, y_{1}, xy_{1}, y_{1}^{2}⟩;

10) ⟨1, x, x^{2}, y, y_{1}, xy_{1}, x^{2}y_{1} − 2xy, y_{1}^{2}, 2yy_{1} − xy_{1}^{2}, 4xyy_{1} − 4y^{2} − x^{2}y_{1}^{2}⟩;

11) ⟨\frac{∂}{∂x}, f(y)\frac{∂}{∂y}⟩;

12) ⟨\frac{∂}{∂x}, x\frac{∂}{∂x}, f(y)\frac{∂}{∂y}⟩;

13) ⟨\frac{∂}{∂x}, x\frac{∂}{∂x}, x^{2} \frac{∂}{∂x}, f(y)\frac{∂}{∂y}⟩;

14) ⟨f(x)\frac{∂}{∂x} − f′(x)y \frac{∂}{∂y}⟩;

15) ⟨f(x)\frac{∂}{∂x} + (f′(x)y + f′′′(x)y^{3})\frac{∂}{∂y}⟩;

16) ⟨f(x)\frac{∂}{∂x} + (f′(x)y + \frac{1}{2}f′′(x)y)\frac{∂}{∂y}⟩;

17) ⟨f(x)\frac{∂}{∂x} + \frac{1}{2}f′(x)y \frac{∂}{∂y}, y \frac{∂}{∂y}⟩.

The Lie algebras under the numbers 11 through 17 are precisely the infinite dimensional subalgebras in sym(J^{1}(ℂ, ℂ)) whose third prolongations are transitive and which contain no subalgebra of the form ⟨\frac{∂}{∂x}, f(x)\frac{∂}{∂y}⟩.

We shall now show that, of the above Lie algebras, only those under 11, 14, and 16 can serve as symmetry algebras for equations of type H_{(3,i)}.

We shall now consider in turn all the cases, except the above three. In doing so, for every finite dimensional Lie algebra we fix a basis {e_{1}, …, e_{n}} consisting of the vectors listed in Proposition 15. We shall view the space J^{3}(ℂ, ℂ) as a homogeneous manifold relative to the action of the Lie algebra 𝔤̄^{(3)}.

1) Let o = (0, 0, 0, 1, 0). The stabilizer 𝔤 = 𝔤̄_{o} of the point o is ⟨e_{3} + 2e_{6}⟩. Let (e_{1} + 𝔤, e_{2} + 𝔤, e_{3} + 𝔤, e_{4} + 𝔤, e_{5} + 𝔤) be a basis for the quotient space 𝔤̄/𝔤. Relative to this basis, the isotropic 𝔤-module 𝔤̄/𝔤 has the form

\{(\begin{matrix}x & 0 & 0 & 0 & 0 \\ 0 & 2x & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −x & 0 \\ 0 & 0 & 0 & 0 & x \end{matrix})| x ∈ ℂ\} .

The submodule corresponding to the invariant distribution E of type 3 is ⟨e_{1} +e_{5} +𝔤, e_{4} +𝔤⟩, and E^{(1)} = ⟨e_{1} +e_{5} +𝔤, e_{3} +𝔤, e_{4} +𝔤⟩. Every 4-dimensional submodule C containing E but not E^{(1)} coincides with the submodule C = ⟨e_{1} + 𝔤, e_{2} + 𝔤, e_{4} + 𝔤, e_{5} + 𝔤⟩. The curvature form of the latter submodule is degenerate, and hence this submodule cannot define a the contact distribution. It follows that the homogeneous space in question admits no invariant Monge– Ampère equation of type H_{(3,i)}.

2) The Lie algebra under the number 2 contains the Lie algebra under the number 1. Consequently, the corresponding homogeneous space again admits no invariant Monge–Ampère equation of type H_{(3,i)}.

3) Let o = (0, 0, 1, 1, 0). Consider the stabilizer 𝔤 = 𝔤̄_{o} = ⟨e_{3} + e_{6}⟩. Let (e_{1} + 𝔤, e_{2} + 𝔤, e_{3} + 𝔤, e_{1} + e_{4} + 𝔤, e_{2} + e_{5} + 𝔤) be a basis of the quotient space 𝔤̄/𝔤. Relative to this basis, the isotropic 𝔤-module 𝔤̄/𝔤 has the form

\{(\begin{matrix}0 & 0 & 0 & 0 & 0 \\ 2x & 0 & 0 & 0 & 0 \\ 0 & x & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2x & 0 \end{matrix})| x ∈ ℂ\} .

It can be seen from the structure of this isotropic module, that the homogeneous space in question admits only two invariant 1-dimensional distributions. On the other hand, for every Monge–Ampère equation of type H_{(3,i)}, we have at least three invariant 1-dimensional distributions: E^{(1)} ∩ F^{(1)}, E^{(2)} ∩ F, and the kernel of the map X ↦ [X, E^{(1)}]/E^{(1)} with X ∈ E.

4) Let o = (0, 0, 0, 0, 1). Then we have 𝔤 = 𝔤̄_{o} = ⟨e_{2} + 3e_{3}⟩. Let (e_{1} + 𝔤, e_{2} + 𝔤, e_{4} + 𝔤, e_{5} + 𝔤, e_{6} + 𝔤) be a basis of 𝔤̄/𝔤. Relative to this basis, the isotropic 𝔤-module 𝔤̄/𝔤 has the form

\{(\begin{matrix}x & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 3x & 0 & 0 \\ 0 & 0 & 0 & 2x & 0 \\ 0 & 0 & 0 & 0 & x \end{matrix})| x ∈ ℂ\} .

To the invariant distribution E of type 3 corresponds the submodule ⟨2e_{1} + e_{6}+𝔤, e_{2}+𝔤⟩, and E^{(1)} = ⟨e_{1}+𝔤, e_{2}+𝔤, e_{6}+𝔤⟩. Any 2-dimensional submodule F such that E^{(1)} ⊕ F = 𝔤̄/𝔤, coincides with the submodule ⟨e_{4} + 𝔤, e_{5} + 𝔤⟩. Furthermore, the subspace ⟨e_{4} + 𝔤⟩ is the kernel of the curvature form of the distribution C = E ⊕ F. Consequently, the homogeneous space in question admits no Monge–Ampère equation of type H_{(3,i)}.

5) Let o = (0, 0, 0, 0, 1). Consider the stabilizer 𝔤 = 𝔤̄_{o} = ⟨e_{3}⟩. We fix the basis (e_{1} + 𝔤, e_{2} + 𝔤, e_{4} + 𝔤, e_{5} + 𝔤, e_{6} + 𝔤) in 𝔤̄/𝔤. Relative to this basis, the isotropic 𝔤-module 𝔤̄/𝔤 has the form

\{(\begin{matrix}0 & 0 & 0 & 0 & 0 \\ 2x & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2x & 0 & 0 \\ 0 & 0 & 0 & x & 0 \end{matrix})| x ∈ ℂ\} .

It can be seen from the structure of this isotropic module that the corresponding homogeneous space admits only two invariant homogeneous distributions. Similar to case 3, we conclude that in this case there exist no Monge–Ampère equations of type H_{(3,i)}.

6) Let o = (0, 0, 0, 1, 0). Then 𝔤 = 𝔤̄_{o} = ⟨e_{2} + 2e_{4}⟩ is the stabilizer of o. Fix the basis (e_{1} +𝔤, e_{3} +𝔤, e_{4} +𝔤, e_{5} +𝔤, e_{6} +𝔤) of the quotient space 𝔤̄/𝔤. Relative to this basis, the isotropic 𝔤-module 𝔤̄/𝔤 has the form

\{(\begin{matrix}x & 0 & 0 & 0 & 0 \\ 0 & −x & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2x & 0 \\ 0 & 0 & 0 & 0 & x \end{matrix})| x ∈ ℂ\} .

To the invariant distribution E of type 3 corresponds the submodule ⟨e_{1} + e_{6} + 𝔤, e_{3} + 𝔤⟩, and E^{(1)} = ⟨e_{1} + e_{6} + 𝔤, e_{3} + 𝔤, e_{4} + 𝔤⟩. Any 4-dimensional submodule C containing E but not E^{(1)}, coincides with the submodule C = ⟨e_{1} + 𝔤, e_{3} + 𝔤, e_{5} + 𝔤, e_{6} + 𝔤⟩. The latter has a degenerate curvature form and hence cannot define the contact distribution. Consequently, the corresponding homogeneous space admits no invariant Monge–Ampère equation of type H_{(3,i)}.

7) This Lie algebra is contained in the Lie algebra under the number 5. Therefore, the corresponding homogeneous space admits no invariant Monge– Ampère equation H_{(3,i)}.

8) Let o = (0, 0, 0, 0, 1). Then 𝔤 = 𝔤̄_{o} = ⟨e_{6}⟩ is the stabilizer of o. Fix the basis (e_{1} + 𝔤, e_{2} + 𝔤, e_{3} + 𝔤, e_{4} + 𝔤, e_{5} + 𝔤) in quotient space 𝔤̄/𝔤. Relative to this basis, the isotropic 𝔤-module 𝔤̄/𝔤 has the form

\{(\begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & x & 0 & 0 & 0 \\ 0 & 0 & 2x & 0 & 0 \end{matrix})| x ∈ ℂ\} .

To the invariant distribution E of type 3 corresponds the submodule ⟨e_{3} − 2e_{4} + 𝔤, e_{5} + 𝔤⟩. We have E^{(1)} = ⟨e_{3} + 𝔤, e_{4} + 𝔤, e_{5} + 𝔤⟩, and there exist no 2-dimensional submodule F such that E^{(1)} ⊕ F = 𝔤̄/𝔤. It follows that the given homogeneous space admits no invariant Monge–Ampère equation of type H_{(3,i)}.

9) and 10) These two Lie algebras are both contained in the Lie algebra under the number 8, and it follows that the corresponding homogeneous spaces admit invariant Monge–Ampère equations of type H_{(3,i)}.

We have thereby shown that an equation E of type H_{(3,i)} with dim sym(E) ⩾ 6 cannot have a finite dimensional symmetry algebra.

12) The stabilizer of the point o = (0, 0, 1, 0, 1) has the form

𝔤 = 𝔤̄_{o} = ⟨3x\frac{∂}{∂x} + (y^{3} + 3y)\frac{∂}{∂y} , O(y^{4})\frac{∂}{∂y} ⟩.

In a suitable basis of the quotient space 𝔤̄/𝔤, the isotropic 𝔤-module 𝔤̄/𝔤 has the form

\{(\begin{matrix}x & 0 & 0 & 0 & 0 \\ 0 & 2x & y & 0 & 0 \\ 0 & 0 & −x & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & −x \end{matrix})| x, y ∈ ℂ\} .

To the invariant distribution E of type 3 corresponds the submodule generated by the second and third vectors in the above basis of 𝔤̄/𝔤. There exists a unique submodule F of the isotropic module such that E^{(1)} ⊕ F = 𝔤̄/𝔤. The curvature form of the submodule C = E ⊕ F is degenerate, and hence the corresponding homogeneous space admits no Monge–Ampère equation of type H_{(3,i)}.

13) This Lie algebra is contained as subalgebra in the Lie algebra under the number 12, so that the corresponding homogeneous space admits no invariant Monge–Ampère equation of type H_{(3,i)}.

Similar to case 12, it can be shown that in cases 15 and 17 there exist no invariant Monge–Ampère equations of type H_{(3,i)}.

We now turn to cases 11, 14, and 16.

11) Let o = (0, 0, 1, 0, 0). The stabilizer of o is

𝔤 = 𝔤̄_{o} = ⟨O(y^{4})\frac{∂}{∂y} ⟩.

In a suitable basis of the quotient space 𝔤̄/𝔤, the isotropic 𝔤-module 𝔤̄/𝔤 has the form

\{(\begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & x & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix})| x ∈ ℂ\} .

To the invariant distribution E of type 3 corresponds the submodule generated by the second and third vectors in the above-mentioned basis of 𝔤̄/𝔤. In the coordinates of the space J^{3}(ℂ, ℂ), we have

E = ⟨\frac{∂}{∂x} + y_{1} \frac{∂}{∂y} + y_{2} \frac{∂}{∂y_{1}} + y_{3} \frac{∂}{∂y_{2}} , \frac{∂}{∂y_{3}} ⟩ .

It follows from the structure of the isotropic module, the corresponding homogeneous space admits a 4-dimensional space of invariant vector fields. The corresponding invariant distribution in J^{3}(ℂ, ℂ) is generated by the following vectors:

⟨y_{1} \frac{∂}{∂y_{2}} + 3y_{2} \frac{∂}{∂y_{3}} , y_{1} \frac{∂}{∂y_{3}} , y_{1} \frac{∂}{∂y_{1}} + 2y_{2} \frac{∂}{∂y_{2}} + 3y_{3} \frac{∂}{∂y_{3}} , \frac{∂}{∂x}⟩ .

The 1-dimensional distribution E^{(1)} ∩ F^{(1)} is not contained entirely in the distribution E. Consequently,

E^{(1)} ∩ F^{(1)} = ⟨y_{1} \frac{∂}{∂y_{2}} + (3y_{2} + k_{1}y_{1})\frac{∂}{∂y_{3}} ⟩

for some constant k_{1}.

The distribution F satisfies the condition E^{(1)} ⊕ F = T M, and hence

F = ⟨y_{1} \frac{∂}{∂y_{1}} + (2y_{2} + k_{2}y_{1})\frac{∂}{∂y_{2}} + (3y_{3} + 3k_{2}y_{2} + k_{3}y_{1})\frac{∂}{∂y_{3}} ,

\frac{∂}{∂x} + k_{4}y_{1} \frac{∂}{∂y_{2}} + (3k_{4}y_{2} + k_{5}y_{1})\frac{∂}{∂y_{3}} ⟩

for some constants k_{2}, k_{3}, k_{4}, k_{5}.

Using [E, F] ⊂ C and F^{(1)} = F ⊕ (E^{(1)} ∩ F^{(1)}), we obtain a system of equations for the unknown constants. Solving this system, we get k_{1} = 2k_{2}, k_{3} = k_{2}^{2} − k_{4}, k_{5} = k_{2}k_{4}. Moreover, we must have k_{4} ≠ 0, for otherwise the curvature form of the distribution C = E ⊕ F would be degenerate.

A suitable change of coordinates transforms the distribution C = E ⊕ F to the standard contact structure in J^{1}(ℂ^{2}, ℂ). Indeed, let (x, y, u, p, q) be a new coordinate system, and suppose that these new coordinates satisfy the following relationships together with the coordinates (x, y, y_{1}, y_{2}, y_{3}):

x̄ = y, ȳ = y_{1},

ū = x − \frac{y_{2}}{k_{4}y_{1}} ,

p̄ = \frac{2y_{2}^{2} + k_{2}y_{1}y_{2} + k_{4}y_{1}^{2} − y_{1}y_{3}}{k_{4}y_{1}^{3}} ,

q̄ = −\frac{y_{2} + k_{2}y_{1}}{k_{4}y_{1}^{2}} .

In the new coordinate system, the characteristic distributions E and F are generated by the following vector fields:

E = ⟨\frac{∂}{∂p}, y (\frac{∂}{∂x} + p\frac{∂}{∂u}) − y(k_{4}yq + k_{2}) (\frac{∂}{∂y} + q \frac{∂}{∂u}) + \frac{yp − 1}{y} \frac{∂}{∂q} ⟩ ,

F = ⟨\frac{yp − 1}{y} \frac{∂}{∂p} + y (\frac{∂}{∂y} + q \frac{∂}{∂u}) , (k_{4}yq + k_{2})\frac{∂}{∂p} + \frac{∂}{∂q} ⟩ .

Let us now find the Monge–Ampére equation corresponding to these characteristic distributions. The corresponding matrix H has the form

H = (\begin{matrix}0 & y & 0 & 0 \\ 1 & 0 & \frac{yp−1}{y} & k_{4}yq + k_{2} \\ 0 & −y(k_{4}yq + k_{2}) & y & 0 \\ 0 & \frac{yp−1}{y} & 0 & 1 \end{matrix}) .

Using formula (4), we get

ϑ = dx ∧ dp + dq ∧ dy − 2(k_{4}yq + k_{2}) · dx ∧ dq + 2\frac{1 − yp}{y^{2}} · dx ∧ dy.

Thus, the most general form of a Monge–Ampère equation E of type H_{(3,i)} with the given symmetry algebra is

u_{xy} + (−k_{4}yq − k_{2})u_{yy} + \frac{1 − yp}{y^{2}} = 0.

Its algebra of contact symmetries has the following representation in terms of characteristic functions:

sym(E) = ⟨\frac{f′′(x)}{k_{4}} + f′(x)yq + f(x)p, 1⟩ .

Since k_{4} ≠ 0, we can perform the following coordinate transformation: x̄ = \sqrt{−k_{4}}x, ȳ = y, ū = \sqrt{−k_{4}}u, and as a result, our equation assumes the form

u_{xy} + (yq + γ)u_{yy} + \frac{1 − yp}{y^{2}} = 0,  (13)

with the following symmetry algebra:

sym(E) = ⟨−f′′(x) + f′(x)yq + f(x)p, 1⟩ .

Later on we shall find out how essential the parameter γ is for this family of equations.

14) Consider the stabilizer of the point o = (0, 1, 0, 0, 0):

𝔤 = 𝔤̄_{o} = ⟨f(x)\frac{∂}{∂x} − f′(x)y \frac{∂}{∂y} ⟩,

where f(x) ∈ ⟨O(x^{5})⟩. In a suitable basis of the quotient space 𝔤̄/𝔤, the isotropic 𝔤-module 𝔤̄/𝔤 has the form

\{(\begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & x & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix})| x ∈ ℂ\} .

To the invariant distribution E of type 3, corresponds the submodule generated by the second and third vectors in the above-mentioned basis of 𝔤̄/𝔤. Similar to the previous case, we find that the corresponding homogeneous space admits a 4-dimensional space of invariant vector fields. The corresponding invariant distribution in J^{3}(ℂ, ℂ) is generated by the following vectors:

⟨y^{3} \frac{∂}{∂y_{2}} + 6y^{2}y_{1} \frac{∂}{∂y_{3}} , y^{4} \frac{∂}{∂y_{3}} , y^{2} \frac{∂}{∂y_{1}} + 3yy_{1} \frac{∂}{∂y_{2}} + (4yy_{2} + 3y_{1}^{2})\frac{∂}{∂y_{3}} ,

y \frac{∂}{∂y} + y_{1} \frac{∂}{∂y_{1}} + y_{2} \frac{∂}{∂y_{2}} + y_{3} \frac{∂}{∂y_{3}} ⟩ .

The 1-dimensional distribution E^{(1)} ∩ F^{(1)} is not entirely contained in the distribution E. Hence

E^{(1)} ∩ F^{(1)} = ⟨y^{3} \frac{∂}{∂y_{2}} + (6y^{2}y_{1} + k_{1}y^{4})\frac{∂}{∂y_{3}} ⟩

for some constant k_{1}.

The distribution F satisfies the condition E^{(1)} ⊕ F = T M. Thus,

F = ⟨y^{2} \frac{∂}{∂y_{1}} + (3yy_{1} + k_{2}y^{3})\frac{∂}{∂y_{2}} + (4yy_{2} + 3y_{1}^{2} + 6k_{2}y^{2}y_{1} + k_{3}y^{4})\frac{∂}{∂y_{3}} ,

y \frac{∂}{∂y} + y_{1} \frac{∂}{∂y_{1}} + (y_{2} + k_{4}y^{3})\frac{∂}{∂y_{2}} + (y_{3} + 6k_{4}y^{2}y_{1} + k_{5}y^{4})\frac{∂}{∂y_{3}} ⟩

for some constants k_{2}, k_{3}, k_{4}, k_{5}.

Taking into account that [E, F] ⊂ C and F^{(1)} = F ⊕ (E^{(1)} ∩ F^{(1)}), we find a system of equation which must be satisfied by our unknown constants. Solving this system, we obtain: k_{1} = 2k_{2}, k_{3} = k_{2}^{2} + k_{4}, k_{5} = k_{2}k_{4}. Moreover, we must have k_{2} ≠ 0, because otherwise the curvature form of the distribution C = E ⊕ F would be degenerate.

We perform a coordinate transformation that takes the distribution C = E⊕ F into the standard contact structure in J^{1}(ℂ^{2}, ℂ). Namely, let (x, y, u, p, q) be a new coordinate system given by

x̄ = x, ȳ = y,

ū = y_{2} − \frac{3yy_{1}^{2} + 2k_{2}y^{3}y_{1}}{2y^{2}} ,

p̄ = \frac{(k_{2}y^{2}y_{1} + 3y_{1}^{2} − yy_{2} − k_{4}y^{2})y_{1} − (3yy_{1} + k_{2}y^{3})y_{2} + y^{2}y_{3}}{y^{2}} ,

q̄ = \frac{2k_{4}y^{4} − 4k_{2}y^{2}y_{1} + 2yy_{2} − 3y_{1}^{2}}{2y^{2}} .

In this new coordinates, we find the most general form of a Monge–Ampère equation of type H_{(3,i)} with the given symmetry algebra:

u_{xy} + \frac{u + k_{4}y^{3} − yu_{y}}{k_{2}y} u_{yy} +

+ \frac{1}{2k_{2}y^{3}} (2k_{2}^{2}k_{4}y^{6} + 4k_{2}^{2}y^{3}u + 4k_{4}y^{3}u + 5y^{2}u_{y}^{2} + 5u^{2} − 2k_{2}^{−}2y^{4}u_{y} −

− 10yu − u_{y} − 4k_{4}y^{4}u_{y} − 2k_{2}y^{2}u_{x} − k_{4}^{2}y^{6}) = 0,

where k_{2} ≠ 0, while k_{4} is an arbitrary constant.

The coordinate transformation x̄ = k_{2}x, ȳ = y, ū = \frac{1}{k_{2}^{2}} u gives the following equation:

u_{xy} + \frac{u + γy^{3} − yu_{y}}{y} u_{yy} +

+ \frac{1}{2y^{3}} (2γy^{6} + 4y^{3}u + 4γy^{3}u + 5y^{2}u_{y}^{2} + 5u^{2} − 2y^{4}u_{y} −

− 10yuu_{y} − 4γy^{4}u_{y} − 2y^{2}u_{x} − γ^{2}y^{6}) = 0, (14)

for which

sym(E) = ⟨y^{2}f′′(x) − yf′′′(x) − 3uf′(x) − f(x)p + f′(x)yq⟩.

16) Consider the point o = (0, 1, 0, 0, 0) and its stabilizer

𝔤 = 𝔤̄_{o} = ⟨f(x)\frac{∂}{∂x} + (f′(x)y + \frac{1}{2}f′′(x)y)\frac{∂}{∂y} ⟩ ,

where f(x) ∈ ⟨2x^{5} − 5x^{4} + 10x^{3} − 15x^{2} + 15x, O(x^{6})⟩. In a suitable basis of 𝔤̄/𝔤, the isotropic 𝔤-module 𝔤̄/𝔤 has the form

\{(\begin{matrix}3x & 0 & 0 & 0 & 0 \\ 0 & 4x & y & 0 & 0 \\ 0 & 0 & −x & 0 & 0 \\ 0 & 0 & 0 & 2x & 0 \\ 0 & 0 & 0 & 0 & x \end{matrix})| x, y ∈ ℂ\} .

To the invariant distribution E^{(1)} corresponds the submodule generated by the first three vectors in the above-mentioned basis of 𝔤̄/𝔤. It can be seen from the structure of the isotropic module that the submodule F is uniquely determined by the condition E^{(1)} ⊕ F = 𝔤̄/𝔤. Thus, the corresponding equation E does not involve any parameters. The invariant distribution on J^{3}(ℂ, ℂ) corresponding to F has the form

F = ⟨y^{2} \frac{∂}{∂y_{1}} + 2y(2y_{1} + 3)\frac{∂}{∂y_{2}} + 2(3yy_{2} + 3y_{1}^{2} + 14y_{1} + 14)\frac{∂}{∂y_{3}} ,

y^{3} \frac{∂}{∂y} + 2y^{2}(y_{1} + 1)\frac{∂}{∂y_{1}} + 2y(yy_{2} + y_{1}^{2} + 3y_{1} + 2)\frac{∂}{∂y_{2}} +

+2(y^{2}y_{3} + (3y_{1} + 5)yy_{2} + 2y_{1}^{2} + 6y_{1} + 4)\frac{∂}{∂y_{3}} ⟩ .

We perform a coordinate change that takes the distribution C = E ⊕ F to the standard contact structure in the space J^{1}(ℂ^{2}, ℂ). Let (x, y, u, p, q) be a new coordinate system defined by

x̄ = x, ȳ = y,

ū = y_{2} − \frac{2y_{1}^{2} + 6y_{1}}{y} ,

p̄ = \frac{y^{2}y_{3} − 6yy_{1}y_{2} − 6yy_{2} + 6y_{1}^{3} + 14y_{1}^{2} + 8y_{1}}{y^{2}} ,

q̄ = \frac{2yy_{2} − 4y_{1}^{2} − 8y_{1} − 8}{y^{2}}.

In these new coordinates, we recover the equation from the characteristic distributions. As a result, we obtain the most general form of a Monge–Ampère equation of type H_{(3,i)} with given symmetry algebra:

2u_{xy} + (\frac{y^{2}u_{y}}{2} − yu + 4) u_{yy} +

+ \frac{3y^{3}uu_{y} + 40yu − 160 − 28y^{2}u_{y} − 2y^{2}u^{2} − y^{4}u_{y}^{2} − 4y^{2}u_{x}}{y^{3}} = 0. (15)

Its symmetry algebra has the form

sym(E) = ⟨\frac{y^{2}f′′′′(x)}{2} − 2yf′′′(x) − 6f′′(x)+

+u(yf′′(x) − f′(x)) − f(x)p − (yf′(x) + \frac{y^{2}f′′(x)}{2})q⟩ .

Note that no two of the equations (13), (14), and (15) are equivalent. Indeed, the image of the isotropy subalgebra is 2-dimensional in the case of the equation (15), and is 1-dimensional for the other two equations. On the other hand, the symmetry algebras for the symmetry algebras of the equations (13) and (14) contain the finite dimensional subalgebras 𝔤𝔩(2, ℂ) and 𝔰𝔩(2, ℂ), respectively, which are not isomorphic to each other.

To complete the classification, we need to determine whether the parameter occurring in equations 13 and 14 is essential.

Proposition 16. (a) Two equations (13) are equivalent to each other if and only if the corresponding values of the parameter γ are the same apart from the sign (i.e., the two equations corresponding to γ and −γ are equivalent);

(b) No two equations of the form (14) are equivalent to each other.

Proof. The equation (13) is of type H_{(3,1)} when γ = 0 and is of type H_{(3,2)} whenever γ ≠ 0. It will therefore suffice to consider the nonzero values of the parameter. In this case, we can associate with the equations (13) and (14) a G-structure P such that the restriction of the canonical form ω to P has the following structure equations:

dω^{1} = −ω^{2} ∧ ω^{3} − ω^{4} ∧ ω^{5} mod ⟨ω^{1}, θ⟩,
dω^{2} = −ω^{1} ∧ ω^{4} mod ⟨ω^{2}, ω^{3}, θ⟩,
dω^{3} = 0 mod ⟨ω^{1}, ω^{2}, ω^{3}, θ⟩,
dω^{4} = −ω^{1} ∧ ω^{3} mod ⟨ω^{4}, ω^{5}, θ⟩,
dω^{5} = −ω^{3} ∧ ω^{4}.

The Lie algebra 𝔤 of the structure group of the fiber bundle P has the form

𝔤 = \{(\begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & x & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix})| x ∈ ℂ\} .

The above correspondence is in agreement with the group of contact transformations, and two equations are equivalent if and only if the corresponding G-structures are isomorphic.

Every G-structure with the given structure group is involutive in the sense of the definition given in [24, 37]. Indeed, if it is viewed as a linear Lie algebra, its first prolongation 𝔤^{(1)} is 1-dimensional, the corresponding Cartan numbers are σ_{1} = 1, σ_{2} = 0, and the Cartan test for involutive G-structures is satisfied: dim 𝔤^{(1)} = σ_{1}. Two involutive G-structures with constant structure functions are locally equivalent if and only if these constants coincide ([37], Theorem 11.8). In the case of equation (13), this constant depends on γ^{2}, and in the case of equation (14) it depends on γ. Thus, two equations (13) with parameters γ and −γ are equivalent (the corresponding coordinate transformation is x̄ = −x, ȳ = y, ū = −u), while no two equations (14) are equivalent to each other as long as they have different values of γ.

Summing up the above results, we obtain the following theorem:

Theorem 8. Every Monge–Ampère equation of type H_{(3,i)} with transitive (but not simply transitive) algebra of contact symmetries is locally equivalent to one and only one of the following equations:

a) u_{xy} + (yu_{y} + γ)u_{yy} + \frac{1−yu_{x}}{y^{2}} = 0 (equations with parameters γ and −γ are equivalent);

b)

u_{xy} + \frac{u + γy^{3} − yu_{y}}{y} u_{yy} +

+ \frac{1}{2y^{3}} (2γy^{6} + 4y^{3}u + 4γy^{3}u + 5y^{2}u_{y}^{2} + 5u^{2} − 2y^{4}u_{y} −

− 10yuu_{y} − 4γy^{4}u_{y} − 2y^{2}u_{x} − γ^{2}y^{6}) = 0;

c)

2u_{xy} + (\frac{y^{2}u_{y}}{2} − yu + 4) u_{yy} +

+ \frac{3y^{3}uu_{y} + 40yu − 160 − 28y^{2}u_{y} − 2y^{2}u^{2} − y^{4}u_{y}^{2} − 4y^{2}u_{x}}{y^{3}} = 0.

All of these equations have a finite-dimensional symmetry algebra and are of type H_{(3,2)}, except for the equation a) when γ = 0, which has the type H_{(3,1)}.

3.3. Classification of equation with simply transitive symmetry group. The classification of equations of type H_{(3,i)} with simply transitive symmetry algebra is similar to the same classification in the case H_{(4,i)}. Examining the classification in 2.3.5 and selecting only those values of parameters for which the characteristic distribution E is of type 3, we obtain the following result:

Theorem 9. Every hyperbolic Monge–Ampère equation of type H_{(3,i)} with simply transitive contact symmetry algebra is locally equivalent to one and only one of the following equations:

𝔤_{1 1}

2e^{u+u_{y}}(αu_{y} + β)u_{xx} + 2(1 − αγ + β)u_{xy} + (2e^{−2(u+u_{y})} + αu_{x}^{2})e^{u+u_{y}}u_{yy} +

+ (2e^{−2(u+u_{y})}(u_{y} + γ) + u_{x}^{2}(αu_{y} + β))e^{u+u_{y}} + 2αe^{u+u_{y}} Hess u = 0

with (β − αγ)^{2} − 2βγ + (1 + 2α) = 0 and α ≠ −\frac{1}{2};

𝔤_{1 2}
2αe^{u+u_{y}}u_{xx} + 2u_{xy} + (2e^{−2(u+u_{y})} + αu_{x}^{2})e^{u+u_{y}} = 0

with α = \frac{1}{2};

𝔤_{2 1}

\frac{2}{27}(−27u_{x} + 9αu_{x} + 6u_{x}u^{2} + 6u_{x}^{2}u + 9αu + 27β − 27u + 2u^{3} + 2u_{x}^{3})e^{−2x}u_{xx} −

− \frac{2}{3}(2u^{2} + 4uu_{x} + 2u_{x}^{2} − 6u_{y}e^{2x} − 9 + 3α)u_{xy} + 2e^{2x}(2u + u_{x})u_{yy} +

+ \frac{2}{27}(−54βu − 9αu^{2} − 27βu_{x} − 2u^{3}u_{x} + 81u_{y}e^{2x} −

− 9αuu_{x} + 2uu_{x}^{3} − u^{4} + u_{x}^{4} + 27u^{2} − 81α + 27uu_{x} − 36u_{y}u_{x}ue^{2x} −

− 18u_{y}u^{2}e^{2x} − 18u_{y}u_{x}^{2}e^{2x} − 27αu_{y}e^{2x} + 27u_{y}^{2}e^{4x})e^{−2x} − 2e^{2x} Hess u = 0

with α = 0 and β = 1;

𝔤_{4 1}

−\frac{e^{−x}}{u_{x}} u_{xx} + \frac{γα − β − 2}{u_{x}^{1/2}}u_{xy} − 2βe^{x}u_{yy} − 2γe^{−x} − α\frac{e^{x}}{u_{x}} Hess u = 0

with 1) α = −1 and β = −γ; 2) α = −3 and β = −3γ − 2;

𝔤_{8}

(e^{(1+α)x} + e^{−x}u_{x} + u_{y})e^{−(2+α)x}u_{xx} +

+ (2u_{x} + 2e^{x}u_{y} − γe^{(2+α)x} + βe^{(2+α)x})e^{−(2+α)x}u_{xy} −
− (u_{x} − βe^{(2+α)x})e^{−(1+α)x}u_{yy} − (−e^{(α−1)x}u_{x}γ +

+ 2e^{−3x}u_{x}^{2} + 3e^{−2x}u_{x}u_{y} − e^{αx}u_{y}γ + e^{−x}u_{y}^{2} + e^{(−1+α)x}u_{x} − e^{(1+2α)x}β) −

− e^{−αx} + e^{−(1+α)x} Hess u = 0

with 1) α = −1 and β = −γ^{2}/4, γ ≠ 0; 2) α = \frac{γ^{2}+2γ+4}{2(γ+2)} and β = γ + 2; 3)

α = \frac{γ^{2}+2γ+2}{2(γ+1)} and β = γ + 1.

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Received July 27, 1999