The Diffiety Institute Preprint Series

Preprint DIPS–1/2005 April 8, 2005

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The Diffiety Institute

Differential invariants of generic hyperbolic Monge–Ampère equations

by

M. Marvan, A. M. Vinogradov, V. A. Yumaguzhin

To appear in Russian Acad. Sci. Dokl. Math.

Differential invariants of generic hyperbolic Monge–Ampère equations

M. MARVAN, A. M. VINOGRADOV, V. A. YUMAGUZHIN

Abstract. In this paper, we construct differential invariants of generic hyperbolic Monge–Ampère equations with respect to contact transformations. We give a solution of the equivalence problem for these equations.

Introduction. In this paper we look for differential invariants of classical Monge– Ampère equations of hyperbolic type with respect to the group of contact transformations by interpreting them as a geometrical structure on 5-dimensional contact manifolds. We limit ourselves to the case of generic equations and solve the equivalence problem for them by the approach developed by the second author, see [10],[1]. In spite of more than 200 years of history of Monge–Ampère equations and numerous publications dedicated to them it would be an exaggeration to say that their nature is completely understood. Establishing the existence and uniqueness theorems for this class of equations (see [4, 3] for local aspects and [9] for global ones) was an important achievement of the classical theory. For a modern exposition of elements of the classical theory, say, of Monge’s method of integration, see [5, 6] and Morimoto [7]. Differential invariants of non-generic equations and some applications will be given in subsequent papers. Below, all manifolds and maps are supposed to be smooth. By [f]_{p}^{k} , k = 0, 1, 2, … denote the k-jet of a map f at a point p. As usually, ℝ^{n} stands for the n-dimensional arithmetic space, ℝ being the real numbers, and ℝP^{n} for the n-dimensional projective space.

Bundle of Monge–Ampère equations. Consider the trivial bundle

τ : ℝ^{2} × ℝ → ℝ^{2}

and the manifold M of all 1-jets of its sections. Natural coordinates in M are denoted by (x, y, z, z_{x}, z_{y}). Classically Monge–Ampère equations are defined to be equations of the form

N(z_{xx}z_{yy} − z_{xy}^{2}) + Az_{xx} + Bz_{xy} + Cz_{yy} + D = 0 ,  (1)

Key words and phrases. Monge–Ampère equation, contact transformation, differential invariant.

M. Marvan acknowledges the support from GAČR under grant 201/04/0538 and from MŠMT under grant MSM: J10/98:192400002.

V.A. Yumaguzhin acknowledges the support from MŠMT under grant MSM: J10/98:192400002.

where the coefficients N, A, B, C, D are functions of x, y, z, z_{x}, z_{y}. Geometrically such an equation may be seen as the section

p ↦ [N(p) : A(p) : B(p) : C(p) : D(p)]

of the trivial bundle

π′ : ℝP^{4} × M → M .

Equation (1) is called hyperbolic if

B^{2} − 4AC + 4ND > 0 ,

see [2]. This condition defines the open subset E ⊂ ℝP^{4} × M. Put

π = π′|_{E}: E → M .

Recall that M has a canonical contact structure, see [8]. Contact transformations of M preserve the class of hyperbolic Monge–Ampère equations. In particular, such a transformation φ is lifted to a diffeomorphism φ^{(0)} of E such that φ ∘ π = φ^{(0)} ∘ π. Let

π_{k} : J^{k}π → M , π_{k} : [S]_{p}^{k} ↦ p ,

be the bundle of all k-jets of sections S of π, k = 0, 1, 2, … φ^{(0)} can be lifted to the diffeomorphism φ^{(k)} of J^{k}π by the formula

φ^{(k)}( [S]_{p}^{k}) = [ φ^{(0)} ∘ S ∘ φ^{−1} ]_{φ(p)}^{k} .

Let Γ be the pseudogroup of all contact transformations of M. It acts on J^{k}π’s by lifted diffeomorphisms.

Proposition 1. (1) Γ acts transitively on J^{0}π and J^{1}π.

(2) Generic orbits of Γ on J^{2}π are of codimension 2, and on J^{3}π of codimension 29.

Let I be a tensor on J^{k}π, say, a function, vector field or a differential form. It is a differential invariant of order k if φ^{(k)} preserves I for any φ ∈ Γ. Functions that are differential invariants are also called scalar differential invariants, see [1].

Corollary 1. (1) The algebra of scalar differential invariants on J^{2}π is generated

by 2 functionally independent invariants.

(2) The algebra of scalar differential invariants on J^{3}π is generated by 29 functionally independent invariants.

Let ℰ be equation (1), S the corresponding section of π, and I a differential invariant of order k. Denote by I_{ℰ} the restriction of I to the image of j_{k}S. Since j_{k}S is a diffeomorphism on its image, I_{ℰ} may be viewed as a tensor on M. Below we describe differential invariants I in terms of their restrictions I_{ℰ}, i.e., as tensors on M. The subscript ℰ will be usually omitted.

Skew-orthogonal distributions associated with a hyperbolic Monge–Ampère equation. Let

τ_{k} : J^{k}τ → ℝ^{2} , τ_{k} : [S]_{x}^{k} ↦ x ,

be the bundle of all k-jets of sections of τ. A natural projection

τ_{k,l} : J^{k}τ → J^{l}τ, k ≥ l ,

sends [S]_{x}^{k} to [S]_{x}^{l}. Sections j_{k}S of τ_{k}, k ≥ 0, associated with a section S of τ, are defined as

j_{k}S : x ↦ [S]_{x}^{k} .

Put L_{S}^{k} = Im j_{k}S. Let [S]_{x}^{2} = q ∈ J^{2}τ. Recall that q is naturally identified with the tangent space K_{q} to L_{S}^{1} at [S]_{x}^{1}. The canonical contact 1-form on M = J^{1}τ is denoted by U_{1} and the distribution of contact hyperplanes on M by

𝒞 : p ↦ 𝒞_{p} .

Below, equation (1) is considered to be a submanifold ℰ of J^{2}τ, as usual. Let P be a one-dimensional subspace of 𝒞_{p}, p ∈ M, such that (τ_{1})_{∗}P ≠ 0. Put

l(P) = { q ∈ (τ_{2,1})^{−1}(p) | P ⊂ K_{q} } .

Denote by 𝒬_{p} the span of all one-dimensional subspaces P of 𝒞_{p} such that τ_{∗}P ≠ 0 and l(P) is tangent to ℰ at least at one point.

Proposition 2. Let ℰ be a hyperbolic Monge–Ampère equation. Then 𝒬_{p} is the union of two-dimensional subspaces 𝒟_{p}^{1} and 𝒟_{p}^{2} such that

(1) 𝒞_{p} = 𝒟_{p}^{1} ⊕ 𝒟_{p}^{2},

(2) 𝒟_{p}^{1} and 𝒟_{p}^{2} are skew-orthogonal with respect to the symplectic form dU_{1}|_{p}.

In this way ℰ determines a pair of 2-dimensional skew-orthogonal subdistributions

𝒟^{1} : p ↦ 𝒟_{p}^{1} and 𝒟^{2} : p ↦ 𝒟_{p}^{2}

of the contact distribution 𝒞.

Proposition 3. Let ℰ be a hyperbolic Monge–Ampère equation and q ∈ J^{2}τ. Then q ∈ ℰ iff one of the following equivalent conditions holds:

(1) dim K_{q} ∩ 𝒟_{p}^{1} = 1,

(2) dim K_{q} ∩ 𝒟_{p}^{2} = 1. In its turn, a pair of 2-dimensional skew-orthogonal subdistributions 𝒟^{1} and 𝒟^{2} of 𝒞 determines a submanifold ℰ = {q ∈ J^{2}τ | dim K_{q} ∩ 𝒟^{i} = 1, i = 1, 2} ∈ J^{2}π which is a hyperbolic Monge–Ampère equation.

So, there exists a natural bijection between hyperbolic Monge–Ampère equations and pairs of 2-dimensional skew-orthogonal non-lagrangian subdistributions 𝒟^{1}, 𝒟^{2} of 𝒞 on M. In particular, the equivalence problem for hyperbolic Monge–Ampère equations with respect to contact transformations is equivalent to that for such pairs of subdistributions.

Projections. For a distribution 𝒟 on M denote by 𝒟^{(1)} the distribution generated by all vector fields X, Y ∈ 𝒟 and their commutators [X, Y]. Put 𝒟^{(2)} = (𝒟^{(1)})^{(1)}. For a hyperbolic Monge–Ampère equation ℰ we have

dim(𝒟^{1})^{(1)} = dim(𝒟^{2})^{(1)} = 3 and D^{3} = (𝒟^{1})^{(2)} ∩ (𝒟^{2})^{(2)}

is a one-dimensional distribution not belonging to 𝒞. In this way one gets the decomposition

T (M) = 𝒟^{1} ⊕ 𝒟^{2} ⊕ 𝒟^{3}

and the corresponding projections

𝒫_{i} : T (M) → 𝒟^{i}, i = 1, 2, 3, 𝒫_{j}^{(1)} : T (M) → 𝒟^{j} ⊕ 𝒟^{3}, j = 1, 2 .

Interpreted as vector-valued 1-forms these projections have the following description. Let X_{1}, …, X_{5} be vector fields on M such that

𝒟^{1} = ⟨X_{1}, X_{2}⟩ , 𝒟^{2} = ⟨X_{3}, X_{4}⟩ , and 𝒟^{3} = ⟨X_{5}⟩ .

Denote by ω^{1}, …, ω^{5} the dual 1-forms, i.e., X_{j} ω^{i} = δ_{j}^{i}, i, j = 1, …, 5. Then

\begin{matrix}𝒫_{1} = ω^{1} ⊗ X_{1} + ω^{2} ⊗ X_{2} , & 𝒫_{2} = ω^{3} ⊗ X_{3} + ω^{4} ⊗ X_{4} , \\ 𝒫_{3} = ω^{5} ⊗ X_{5} , & 𝒫_{j}^{(1)} = 𝒫_{j} + 𝒫_{3} , j = 1, 2 . \end{matrix}

These vector-valued differential 1-forms are differential invariants of ℰ with respect to contact transformations. For a generic ℰ,

dim(𝒟^{1})^{(2)} = dim(𝒟^{2})^{(2)} = 5 .

Curvature forms. Let P : T M → T M be a projection. Then the curvature ℛ of P is a 2-form on M defined by the formula

ℛ(X, Y) = (id_{T M} −P)( [P (X), P (Y)] ) ,

where id_{T M} is the identity map. The curvatures of projections 𝒫_{1}, 𝒫_{2}, 𝒫_{1}^{(1)}, 𝒫_{2}^{(1)} are:

ℛ_{1} = ω^{1} ∧ ω^{2} ⊗ X_{5} , ℛ_{2} = ω^{3} ∧ ω^{4} ⊗ X_{5} ,
ℛ_{1}^{1} = (b_{15}^{3}ω^{1} + b_{25}^{3}ω^{2}) ∧ ω^{5} ⊗ X_{3} + (b_{15}^{4}ω^{1} + b_{25}^{4}ω^{2}) ∧ ω^{5} ⊗ X_{4} ,
ℛ_{2}^{1} = (b_{35}^{1}ω^{3} + b_{45}^{1}ω^{4}) ∧ ω^{5} ⊗ X_{1} + (b_{35}^{2}ω^{3} + b_{45}^{2}ω^{4}) ∧ ω^{5} ⊗ X_{2}

respectively, with functions b_{jk}^{i} coming from relations

[X_{j}, X_{k}] = ∑_{i=1}^{5} b_{jk}^{i}X_{i} .

Clearly, these curvatures are differential invariants of ℰ.

Scalar invariants on J^{2}π. From invariant 5-forms:

\frac{1}{2}(ℛ_{2}^{1} ℛ_{1}) (ℛ_{2}^{1} ℛ_{1}) = λ_{1} · ω^{1} ∧ … ∧ ω^{5} ⊗ X_{5} ,

\frac{1}{2}(ℛ_{1}^{1} ℛ_{2}) (ℛ_{1}^{1} ℛ_{2}) = λ_{2} · ω^{1} ∧ … ∧ ω^{5} ⊗ X_{5} ,
(ℛ_{2}^{1} ℛ_{1}) (ℛ_{1}^{1} ℛ_{2}) = λ_{12} · ω^{1} ∧ … ∧ ω^{5} ⊗ X_{5}

with

\begin{matrix}λ_{1} = b_{35}^{2}b_{45}^{1} − b_{35}^{1}b_{45}^{2} , λ_{2} = b_{15}^{4}b_{25}^{3} − b_{15}^{3}b_{25}^{4} , \\ λ_{12} = b_{15}^{3}b_{35}^{1} + b_{15}^{4}b_{45}^{1} + b_{25}^{3}b_{35}^{2} + b_{25}^{4}b_{45}^{2} \end{matrix}

one immediately derives scalar differential invariants

I^{1} = λ_{12}/λ_{1} , I^{2} = λ_{12}/λ_{2}

on J^{2}π due to the fact that λ_{1} ≠ 0 and λ_{2} ≠ 0 for a generic ℰ.

Theorem 1. As smooth function algebra, the algebra of scalar differential invariants on J^{2}π is generated by I^{1} and I^{2}.

The complete parallelism. Consider the invariant 1-forms:

Ω^{1} = 𝒫_{1} dI^{1} = X_{1}(I^{1})ω^{1} + X_{2}(I^{1})ω^{2} ,
Ω^{2} = 𝒫_{1} dI^{2} = X_{1}(I^{2})ω^{1} + X_{2}(I^{2})ω^{2} ,
Ω^{3} = 𝒫_{2} dI^{1} = X_{3}(I^{1})ω^{3} + X_{4}(I^{1})ω^{4} ,
Ω^{4} = 𝒫_{2} dI^{2} = X_{3}(I^{2})ω^{3} + X_{4}(I^{2})ω^{4} ,
Ω^{5} = 𝒫_{3} dI^{1} = X_{5}(I^{1})ω^{5} , Ω̃^{5} = 𝒫_{1} dI^{2} = X_{5}(I^{2})ω^{5} .

If ℰ is generic, then

\begin{matrix}X_{5}(I^{1}) ≠ 0 , X_{5}(I^{2}) ≠ 0 , \\ Δ_{1} = |\begin{matrix}X_{1}(I^{1}) & X_{2}(I^{1}) \\ X_{1}(I^{2}) & X_{2}(I^{2}) \end{matrix}| ≠ 0 , Δ_{2} = |\begin{matrix}X_{3}(I^{1}) & X_{4}(I^{1}) \\ X_{3}(I^{2}) & X_{4}(I^{2}) \end{matrix}| ≠ 0 . \end{matrix}

So, {Ω^{1}, …, Ω^{4}, Ω^{5}} and {Ω^{1}, …, Ω^{4}, Ω̃^{5}} are invariant coframes on M. Each of them defines an invariant complete parallelism on M.

Scalar invariants on J^{3}π. Coefficient of proportionality

I^{3} = X_{5}(I^{1})/X_{5}(I^{2})

between invariant 1-forms Ω^{5} and Ω̃^{5} is a scalar differential invariant on J^{3}π. By applying natural operations of linear algebra and tensor analysis to already obtained differential invariants one gets numerous new scalar differential invariants of ℰ. For instance, further invariants I^{4} and I^{5} are obtained as factors connecting invariant 2forms ℛ_{1} dI^{1}, ℛ_{2} dI^{1} and Ω^{i} ∧ Ω^{j} on J^{3}π:

ℛ_{1} dI^{1} = I^{4}Ω^{1} ∧ Ω^{2}, ℛ_{2} dI^{1} = I^{5}Ω^{3} ∧ Ω^{4}.

More exactly,

I^{4} = Δ_{1}/X_{5}(I^{1}) , I^{5} = Δ_{2}/X_{5}(I^{1}) .

Theorem 2. Scalar differential invariants I^{1}, …, I^{5} on J^{3}π are functionally independent.

The equivalence problem. For a generic hyperbolic Monge–Ampère equation ℰ functions I_{ℰ}^{1}, …, I_{ℰ}^{5} form a natural invariant chart on M. Put

Ω_{ℰ}^{i} = ∑_{j=1}^{5} Ω_{j}^{i}(I_{ℰ}^{1}, …, I_{ℰ}^{5})dI_{ℰ}^{j} , i = 1, …, 5 .

Theorem 3. The equivalence class of a generic equation ℰ with respect to contact transformations is uniquely determined by the functions Ω_{j}^{i}(I_{ℰ}^{1}, …, I_{ℰ}^{5}), i, j = 1, …, 5.

Invariant operators. Invariant vector fields Y_{j} are defined by duality relations Y_{j} Ω^{i} = δ_{j}^{i}, i, j = 1, …, 5. Consider invariant endomorphisms of vector bundles

□_{1} = Y_{5} ℛ_{1}^{1} : T M → 𝒟^{2} and □_{2} = Y_{5} ℛ_{2}^{1} : T M → 𝒟^{1}

and their compositions

∇_{1} = □_{2}|_{𝒟^{2}} ∘ □_{1}|_{𝒟^{1}} : 𝒟^{1} → 𝒟^{1} and ∇_{2} = □_{1}|_{𝒟^{1}} ∘ □_{2}|_{𝒟^{2}} : 𝒟^{2} → 𝒟^{2}

which are invariant as well.

Theorem 4. The endomorphism ∇_{i} of 𝒟_{i}, i = 1, 2, has two, one, or zero different real eigenvalues, if I > 0, I = 0, I < 0, respectively, where I = 4 − I^{1}I^{2}.

Thus, generic hyperbolic Monge–Ampère equations are subdivided locally into three subclasses according to the sign of I.

References

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[7] T. Morimoto, Monge–Ampère equations viewed from contact geometry. in: Symplectic Singularities and Geometry of Gauge Fields (Warsaw, 1995), 105–121, Banach Center Publ., 39, Polish Acad. Sci., Warsaw, 1997.

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Mathematical Institute of the Silesian University at Opava, Na Rybnicku 1, 746 01 Opava, Czech Republic.

E-mail address: Michal.Marvan@math.slu.cz

University of Salerno, Dipartimento di Matematica ed Informatica, Facolta’ di Scienze Matematiche, Fisiche e Naturali, Universita’ degli Studi di Salerno, Via Ponte del Melillo, 84084 Fisciano (SA), Italy, and Diffiety Institute, Russia.

E-mail address: vinograd@unisa.it

Program Systems Institute of RAS, 152020, Pereslavl’-Zalesskiy, m.Botik, Russia.

E-mail address: yuma@diffiety.botik.ru