The Diffiety Institute Preprint Series

Preprint DIPS–3/2008 November 26, 2008

Available via INTERNET: http://diffiety.ac.ru; http://diffiety.org

The Diffiety Institute

Differential invariants of generic parabolic Monge-Ampere equations

by

D. Catalano Ferraioli and A. M. Vinogradov

DIFFERENTIAL INVARIANTS OF GENERIC PARABOLIC MONGE-AMPERE EQUATIONS

D. CATALANO FERRAIOLI AND A. M. VINOGRADOV

Abstract. Some new results on geometry of classical parabolic Monge-Ampere equations (PMA) are presented. PMAs are either integrable, or nonintegrable according to integrability of its characteristic distribution. All integrable PMAs are locally equivalent to the equation u_{xx} = 0. We study nonintegrable PMAs by associating with each of them a 1-dimensional distribution on the corresponding first order jet manifold, called the directing distribution. According to some property of these distributions, nonintegrable PMAs are subdivided into three classes, one generic and two special ones. Generic PMAs are uniquely characterized by their directing distributions. To study directing distributions we introduce their canonical models, projective curve bundles (PCB). A PCB is a 1-dimensional subbundle of the projectivized cotangent bundle to a 4-dimensional manifold. Differential invariants of projective curves composing such a bundle are used to construct a series of contact differential invariants for corresponding PMAs. These give a solution of the equivalence problem for PMAs with respect to contact transformations.

1. Introduction

Since Monge’s "Application de l’Analyse à la Géométrie" Monge-Ampere equations periodically attract attention of geometers. This is not only due to the numerous applications to geometry, mechanics and physics. Geometry of these equations being tightly related with various parts of the modern differential geometry has all merits to be studied as itself. Last 2-3 decades manifested a return of interest to geometry of Monge-Ampere equations, mostly to elliptic and hyperbolic ones. The reader will find an account of recent results together with an extensive bibliography in [5]. In this article we study geometry of classical parabolic Monge-Ampere equations (PMAs) on the basis of a new approach sketched in [8]. According to it, a PMA E ⊂ J^{2}(π), π being a 1-dimensional fiber bundle over a bidimensional manifold, is completely characterized by its characteristic distribution D_{E} which is a 2-dimensional Lagrangian distribution on J^{1}(π), and vice versa. Such distributions and, accordingly, the corresponding to them PMAs, are naturally subdivided into four classes, integrable, generic and two types of special ones (see [8] and sec.4). All integrable Lagrangian foliations are locally contact equivalent. A consequence of it is that a PMA E is locally

2000 Mathematics Subject Classification. 34A26, 35K55, 53A55, 53D10, 53A20. Key words and phrases. Geometric Methods in Differential Equations, Differential Invariants, Par-

abolic Equations, Monge-Ampere Equations, Projective Differential Geometry, Contact Geometry.

contact equivalent to the equation u_{xx} = 0 iff the distribution D_{E} is integrable. This exhausts the integrable case. On the contrary, nonintegrable Lagrangian distributions are very diversified and our main goal here is to describe their multiplicity, i.e., more precisely, equivalence classes of PMAs with respect to contact transformations. With this purpose we associate with a Lagrangian distribution a projective curve bundle (shortly, PCB) over a 4-dimensional manifold N. A PCB over N is a 1-dimensional smooth subbundle of the "projectivized" cotangent bundle P T^{∗}(N) of N. Under some regularity conditions such a bundle possesses a canonical contact structure and, as a consequence, a canonically inscribed in it Lagrangian distribution. There exists a oneto-one correspondence between generic Lagrangian distributions and regular PCBs. This way the equivalence problem for generic PMAs is reformulated as the equivalence problem for PCBs (see [8]) and this is the key point of our approach. The fiber of such a bundle over a point x ∈ N is a curve γ_{x} in the projective space P T_{x}^{∗}(N). The curve γ_{x} can be characterized by its scalar differential invariants with respect to the group of projective transformations. By putting such invariants for single curves γ_{x} together for all x ∈ N one obtains scalar differential invariants for the considered PCB and, consequently, for the corresponding PMA. This kind of invariants resolve the equivalence problem for generic PMAs on the basis of the "principle of n−invariants" (see [1, 10]). We have chosen them among others for their transparent geometrical meaning. It should be stressed, however, that there are other choices, maybe, less intuitive but more efficient in practice. We shall discuss this point separately. Special Lagrangian distributions admit a similar interpretation in terms of 2dimensional distributions on 4-dimensional manifolds supplied with additional structures, called fringes (see [8]). Differential invariants of fringes, also coming from projective differential geometry, allow to construct basic scalar differential invariants for special PMAs. They will be discussed in a separate paper. It is worth mentioning that all linear PMAs are special. Our approach is based on the theory of solution singularities for nonlinear PDEs (see [9]). Indeed, a Lagrangian distribution or, equivalently, the associated PCB, represents equations that describe fold type singularities of multivalued solutions of the corresponding PMA. An advantage of this point of view is that it allows a similar analysis of higher order PDEs and, in particular, to understand what are higher order analogues of Monge-Ampere equations. These topics will be discussed in a forthcoming joint paper by M. Bachtold and the second author. The paper is organized as follows. The notations and generalities concerning jet spaces and Monge-Ampere equations, we need throughout the paper, are collected in sections 2 and 3, respectively. In particular, the interpretation of PMAs as Lagrangian distributions is presented here. Section 4 contains some basic facts on geometry of Lagrangian distributions. The central of them is the notion of the directing distribution of a Lagrangian distribution. The above mentioned subdivision of nonitegrable PMAs into generic and special types reflects some contact properties of this distribution. Projective curve bundles that are canonical models of directing distributions are introduced and studied subsequently in section 6. For completeness in section 5 we give a short proof of the known fact that integrable PMAs are locally equivalent one to another. Finally, basic scalar differential invariants of generic PCBs and hence of generic PMAs are constructed and discussed in section 7. Throughout the paper we use the following notations and conventions:

• all objects in this paper, e.g., manifolds, mappings, functions, vector fields, etc, are supposed to be smooth;

• C^{∞}(M) stands for the algebra of smooth functions on the manifold M and C^{∞}(M)-modules of all vector fields and differential k-forms are denoted by D(M) and Λ^{k}(M), respectively;

• the evaluation of X ∈ D(M) (resp., of α ∈ Λ^{k}(M)) at p ∈ M is denoted by X|_{p} ( resp., α|_{p});

• d_{p}f : T_{p}M ⟶ T_{f(p)}N stands for the differential of the map f : M → N,M and N being two manifolds;

• For X, Y ∈ D(M) and α ∈ Λ^{k}(M) we shall use the shorten notation X^{r}(Y) for L_{X}^{r}(Y) and X^{r}(α) for L_{X}^{r}(α) (with X^{0}(Y) = Y and X^{0}(α) = α) for the r-th power of the Lie derivative L_{X}.

• depending on the context by a distribution on a manifold M we understand either a subbundle D of the tangent bundle T M, whose fiber over p ∈ M is denoted by D(p), or the C^{∞}(M)-module of its sections. In particular, X ∈ D means that X_{p} ∈ D(p), ∀p ∈ M;

• we write D = ⟨X_{1}, ..., X_{2}⟩ if the distribution D is generated by vector fields X_{1}, ..., X_{r} ∈ D(M); similarly, D = Ann(α_{1}, ..., α_{s}) means that D is constituted by vector fields annihilated by forms α_{1}, ..., α_{s} ∈ Λ^{1}(M);

• if M is equipped with a contact distribution C and S ⊂ C is a subdistribution of C, then S^{⊥} denotes the C-orthogonal complement to S.

2. Preliminaries

In this section the notations and basic facts we need throughout the paper are collected. The reader is referred to [1, 3, 4] for further details.

2.1. Jet bundles. Let E be an (n + m)-dimensional manifold. The manifold of kth order jets, k ≥ 0, of n-dimensional submanifolds of E is denoted by J^{k}(E, n) and π_{k,l} : J^{k}(E, n) ⟶ J^{l}(E, n), k ≥ l, stands for the canonical projection. If E is fibered by a map π : E → M over an n-dimensional manifold M, then J^{k}π denotes the k-th order jet manifold of local sections of π. J^{k}π is an open domain in J^{k}(E, n). The k−th order jet of an n−dimensional submanifold L ⊂ E at a point z ∈ L is denoted by [L]_{z}^{k}. Similarly, if σ is a (local) section of π and x ∈ M, then [σ]_{x}^{k} = [σ(U)]_{σ(x)}^{k}, U being the domain of σ, stands for the k−th order jet of σ at x. The correspondence z ↦ [L]_{z}^{k} defines the k−th lift of L

j_{k}L : L ⟶ J^{k}(E, n).

Similarly, the k−th lift of a local section σ of π

j_{k}σ : U → J^{k}π

sends x ∈ U to [σ]_{x}^{k}, i.e., j_{k}σ = j_{k}(σ(U)) ∘ σ. Put

L^{(k)} = Im(j_{k}L), M_{σ}^{k} = Im(j_{k}σ).

Let θ_{k+1} = [L]_{z}^{k+1} be a point of J^{k+1}(E, n). Then the R-plane associated with θ_{k+1} is the subspace

R_{θ_{k+1}} = T_{θ_{k}}(L^{(k)})

of T_{θ_{k}}(J^{k}(E, n)) with θ_{k} = [L]_{z}^{k}. The correspondence θ_{k+1} ↦ R_{θ_{k+1}} is biunique. Put

V_{θ_{k+1}} = T_{θ_{k}}(J^{k}(E, n))/R_{θ_{k+1}}.

and denote by

pr_{θ_{k+1}} : T_{θ_{k}}(J^{k}(E, n)) ⟶ V_{θ_{k+1}}

the canonical projection. The vector bundle

ν_{k+1} : V_{(k+1)} ⟶ J^{k+1}(E, n), k ≥ 0,

whose fiber over θ_{k+1}, is V_{θ_{k+1}} is naturally defined. By ν_{k,r} denote the pullback of ν_{k} via π_{r,k}, r ≥ k. Let C(θ_{k}) ⊂ T_{θ_{k}}(J^{k}π) be the span of all R−planes at θ_{k}. Then θ_{k} ↦ C(θ_{k}) is the Cartan distribution on J^{k}(E, n) denoted by C_{k}. This distribution can be alternatively defined as the kernel of the ν_{k}-valued Cartan form U_{k} on J^{k}(E, n):

U_{k}(ξ) = pr_{θ_{k}}(d_{θ_{k}}π_{k,k−1}(ξ)) ∈ V_{θ_{k}}, ξ ∈ T_{θ_{k}}(J^{k}(E, n)).

A diffeomorphism φ : J^{k}(E, n) → J^{k}(E, n) is called contact if it preserves the Cartan distribution. Similarly, a vector field Y on J^{k}(E, n) is called contact if [Y, C_{k}] ⊂ C_{k}. A contact diffeomorphism φ (respectively, a contact field Y) canonically lifts to a contact diffeomorphism φ^{(l)} (respectively, a contact field Y^{(l)}) on J^{k+l}(E, n)). Below the above constructions will be mainly used for n = 2, m = 1, k = 1, 2. In this case C_{1} is the canonical contact structure on J^{1}(E, 2), dim E = 3, and the bundle ν_{1} is 1-dimensional. ν_{1} is canonically isomorphic to the bundle whose fiber over θ ∈ J^{1}(E, 2) is T_{θ}(J^{1}(E, 2))/C(θ). A vector field X on E defines a section s_{X} ∈ Γ(ν_{1}), s_{X}(θ) = pr_{θ}(X). Since ν_{1} is 1-dimensional, the ν_{1}-valued form U_{1} can be presented as

(1)  U_{1} = U_{X} · s_{X}, U_{X} ∈ Λ^{1}(J^{1}(E, 2)),

in the domain where s_{X} ≠ 0.

Let M be a manifold supplied with a contact distribution C. An almost everywhere nonvanishing differential form U ∈ Λ^{1}(M) is called contact if it vanishes on C. A vector field Y ∈ D(M) is contact iff

(2)  L_{X}(U) = λU, λ ∈ C^{∞}(M),

for a contact form U. For instance, U_{X} (see (1)) is a contact form. If a vector field X ∈ D(J^{1}(E, 2) is contact, then f = X U_{1} ∈ Γ(ν_{1}) is called the generating function of X. X is completely determined by f and is denoted by X_{f} in order to underline this fact. If U is a contact form on a contact manifold (M, C) and X ∈ D(M) is contact, then f = X U ∈ C^{∞}(M) is the generating function of X with respect to U. Vector fields X, Y ∈ D(M) belonging to C are called C-orthogonal if [X, Y] also belongs to C. Obviously, this is equivalent to dU(X, Y) = 0 for a contact form U. Observe that C-orthogonality is a C^{∞}(M)-linear property. A subdistribution D of C is called Lagrangian if any two fields X, Y ∈ D are C-orthogonal and D is not contained in another distribution of bigger dimension possessing this property. If dim M = 2n + 1, then dim D = n. A local chart (x, y, u) in E, where (x, y) are interpreted as independent variables and u as the dependent one, extends canonically to a local chart

(3)  (x, y, u, u_{x} = p, u_{y} = q, u_{xx} = r, u_{xy} = s, u_{yy} = t)

on J^{2}(E, 2). Functions (x, y, u, p, q) form a (standard) chart in J^{1}(E, 2). The local contact form U = U_{∂_{u}} (see (1)) in this chart reads

U = du − pdx − qdy.

Accordingly, in this chart the contact vector field corresponding to the generating with respect to U function f reads

(4)  X_{f} = −f_{p}∂_{x} − f_{q}∂_{y} + (f − pf_{p} − qf_{q})∂_{u} + (f_{x} + pf_{x})∂_{p} + (f_{y} + qf_{x})∂_{q}.

In the sequel we shall use C and U for the contact distribution and a contact form in the current context, respectively.

3. Parabolic MA equations

3.1. MA equations. Let E be a 3-dimensional manifold. A k-th order differential equation imposed on bidimensional submanifolds of E is a hypersurface E ⊂ J^{k}(E, 2). In a standard jet chart it is seen as a k-th order equation for one unknown function in two variables. In the sequel we shall deal only with second order equations of this kind. In a jet chart (3) on J^{2}(E, 2) such an equation reads

(5)  F (x, y, u, p, q, r, s, t) = 0.

The standard subdivision of equations (5) into hyperbolic, parabolic and elliptic ones is intrinsically characterized by the nature of singularities of their multi-valued solutions (see [9]). Some elementary facts from solution singularity theory we need in this paper are brought below. Let θ_{2} ∈ J^{2}(E, 2), θ_{1} = π_{2,1}(θ_{2}) and F_{θ_{1}} = π_{2,1}^{−1}(θ_{1}). Recall that an R-plane at θ_{1} is a Lagrangian plane in C(θ_{1}), i.e., a bidimensional subspace R ⊂ C(θ_{1}) such that dω|_{R} = 0 for a contact 1-form ω on J^{1}(E, 2). If P ⊂ C(θ_{1}) is a 1-dimensional subspace, then

(6)  l(P) = {θ ∈ F_{θ_{1}}|R_{θ} ⊃ P } ⊂ F_{θ_{1}}

is the 1-ray corresponding to P. A local chart on F_{θ_{1}} is formed by restrictions of r, s, t to F_{θ_{1}}. By abusing the notation we shall use r, s, t for these restrictions as well. Denote by (r̃, s̃, t̃) coordinates in T_{θ_{2}}(F_{θ_{1}}) with respects to the basis ∂_{r}|_{θ_{2}}, ∂_{s}|_{θ_{2}}, ∂_{t}|_{θ_{2}}. Cones

V_{θ_{2}} = {r̃t̃− s̃^{2} = 0} ⊂ T_{θ_{2}}(F_{θ_{1}}), θ_{2} ∈ F_{θ_{1}},

define a distribution of cones V : θ_{2} ↦ V_{θ_{2}} on F_{θ_{1}} called the ray distribution. This distribution is invariant with respect to contact transformations.

Lemma 1. If P is spanned by the vector

w = ζ_{1}∂_{x} + ζ_{2}∂_{y} + μ∂_{u} + η_{1}∂_{p} + η_{2}∂_{q} ∈ T_{θ_{1}}(J^{1}(E, 2))

then the 1-ray l(P) is described by equations

(7)  \{ \begin{matrix}ζ_{1}r + ζ_{2}s = η_{1}, \\ ζ_{1}s + ζ_{2}t = η_{2}. \end{matrix}

In particular, l(P) is tangent to V.

Proof. Put p_{0} = p(θ_{1}), q_{0} = q(θ_{1}). Since P ⊂ C(θ_{1}) we have

(8)  w U = 0 ⇔ μ = ζ_{1}p_{0} + ζ_{2}q_{0},

and hence

w = ζ_{1}(∂_{x} + p_{0}∂_{u}) + ζ_{2}(∂_{y} + q_{0}∂_{u}) + η_{1}∂_{p} + η_{2}∂_{q}.

Moreover, R_{θ_{2}} ⊃ P iff

w (dp − rdx − sdy)_{θ_{2}} = w (dq − sdx − tdy)_{θ_{2}} = 0

and these relations are identical to (7). Obviously, the components of the tangent to l(P) vector at θ_{1} are

(9)  (r̃, s̃, t̃) = (ζ_{2}^{2}, −ζ_{1}ζ_{2}, ζ_{1}^{2})

and manifestly satisfy the equation r̃t̃− s̃^{2} = 0. □

Put

E_{θ_{1}} = E ∩ F_{θ_{1}}

An equation E is of principal type if it intersects transversally fibers of the projection π_{2,1}. In such a case E_{θ_{1}} is a bidimensional submanifold of F_{θ_{1}}, ∀ θ_{1} ∈ J^{1}(E, 2). Further on we assume E to be of principal type.

The symbol of E at θ_{2} ∈ E is the bidimensional subspace

Smbl_{θ_{2}}(E) := T_{θ_{2}}E_{θ_{1}}

of T_{θ_{2}} (F_{θ_{1}}). A point θ_{2} ∈ E is elliptic (resp., parabolic, or hyperbolic) if V_{θ_{2}} intersects Smbl_{θ_{2}}(E) in its vertex only (resp., along a line, or along two lines). So, if θ_{2} is parabolic, then Smbl_{θ_{2}}(E) is a tangent to the cone V_{θ_{2}} plane. In other words, in this case E_{θ_{1}} is tangent to the ray distribution on F_{θ_{1}}.

Definition 1. An equation E is called elliptic (resp., parabolic, or hyperbolic) if all its points are elliptic (resp., parabolic, or hyperbolic).

Lemma 2. If a 1-ray l(P) is tangent to a parabolic equation E at a point θ_{2}, then l(P) ⊂ E_{θ_{1}}.

Proof. The 1-rays distribution on F_{θ_{1}} may be viewed as the distribution of Monge’s cones of a first order PDE for one unknown function in two variables. As it is easy to see, this equation in terms of coordinates x_{1} = r, x_{2} = t, y = s on F_{θ_{1}} is \frac{∂y}{∂x_{1}} · \frac{∂y}{∂x_{2}} = \frac{1}{4}. A banal computation then shows that characteristics of this equation are exactly 1-rays and hence its solutions are ruled surfaces composed of 1-rays. □

Corollary 1. If E is a parabolic equation, then E_{θ_{1}} is a ruled surface in F_{θ_{1}} composed of 1-rays.

For a parabolic equation E and a point θ_{1} ∈ J^{1}(E, 2) consider all 1-dimensional subspaces P ⊂ C(θ_{1}) such that l(P) ⊂ E_{θ_{1}}. This is a 1-parametric family of lines and, so, their union is a bidimensional conic surface W_{θ_{1}} in C(θ_{1}). Then

θ_{1} ↦ W_{θ_{1}}, θ_{1} ∈ J^{1}(E, 2),

is the Monge distribution of E. Integral curves of this distribution are curves along which multivalued solutions of E fold up. It is worth mentioning that tangent planes to a surface W_{θ_{1}} are all Lagrangian. We omit the proof but note that Lagrangian planes are simplest surfaces possessing this property. Given type singularities of multivalued solutions of a PDE are described by corresponding subsidiary equations. If E is a parabolic equation, then integral curves of the corresponding to E Monge distribution describe loci of fold type singularities of its solutions. Intrinsically, the class of Monge-Ampere (MA) equations is characterized by the property that these subsidiary equations are as simple as possible. More precisely, this means that conic surfaces W_{θ_{1}}s’ must be geometrically simplest. As we have already noticed, for parabolic equations the simplest are Lagrangian planes. Thus parabolic MA equations (PMAs) are conceptually defined as parabolic equations whose Monge distributions are distributions of Lagrangian planes. It will be shown below that this definition coincides with the traditional one.

Recall that, according to the traditional descriptive point of view, MA equations are defined as equations of the form

(10)  N(rt − s^{2}) + Ar + Bs + Ct + D = 0

with N, A, B, C and D being some functions of variables x, y, u, p, q. MA equations with N = 0 are called quasilinear. Δ = B^{2} − 4AC + 4ND is the discriminant of (10).

Proposition 1. Equation (10) is elliptic (resp., parabolic, or hyperbolic) if Δ < 0 (resp., Δ = 0, or Δ > 0).

Proof. As it is easy to see, the symbol of equation (10) at a point θ_{2} of coordinates (r, s, t) is described by the equation

(11)  N(tr̃ + rt̃− 2ss̃) + Ar̃ + Bs̃ + Ct̃= 0.

with (r̃, s̃, t̃) subject to the relation r̃t̃− s̃^{2} = 0. Now one directly extracts the result from this relation and (11). □

Finally, we observe that all above definitions and constructions are contact invariant.

3.2. Parabolic MA equations as Lagrangian distributions. In this section it will be shown that the conceptual definition of PMA equations coincides with the traditional one. First of all, we have

Proposition 2. Equation (10) is parabolic in the sense of definition 1 if and only if Δ = 0. The Monge distribution of parabolic equation (10) is a distribution of Lagrangian planes, i.e., a Lagrangian distribution.

Proof. First, let E be equation (10). A simple direct computation shows that E is tangent to the 1-ray distribution iff Δ = 0. Second, coefficients of equation (10) may be thought as functions on J^{1}(E, 2). Let A_{0}, …, N_{0} be their values at a point θ_{1} ∈ J^{1}(E, 2). Then

N_{0}(tr̃ + rt̃− 2ss̃) + A_{0}r̃ + B_{0}s̃ + C_{0}t̃= 0

is the equation of E_{θ_{1}} in F_{θ_{1}}. If N_{0} ≠ 0 this equation describes a standard cone with the vertex at the point θ_{2} of coordinates r = −C_{0}/N_{0}, s = B_{0}/2N_{0}, t = −A_{0}/N_{0}. So, E_{θ_{1}} is the union of 1-rays l(P) passing through θ_{2}. By definition this implies that P ⊂ R_{θ_{2}} and hence W_{θ_{1}} is the union of lines P that belong to R_{θ_{2}}. This shows that W_{θ_{1}} = R_{θ_{2}}. Finally, note that the case N_{0} = 0 can be brought to the previous one by a suitable choice of jet coordinates. □

From now on we shall denote by D_{E} the Monge distribution of a PMA equation E. From the proof of the above Proposition we immediately extract geometrical meaning of the correspondence E ↦ D_{E}.

Corollary 2. The distribution D_{E} associates with a point θ_{1} ∈ J^{1}(E, 2) the R-plane R_{θ_{2}} with θ_{2} being the vertex of the cone E_{θ_{1}}.

A coordinate description of D_{E} is as follows.

Proposition 3.

(12)  D_{E} = ⟨∂_{x} + p∂_{u} − \frac{C}{N} ∂_{p} + \frac{B}{2N} ∂_{q}, ∂_{y} + q∂_{u} + \frac{B}{2N} ∂_{p} − \frac{A}{N} ∂_{q}⟩ , if N ≠ 0

or

(13)  D_{E} = ⟨A∂_{x} + \frac{B}{2} ∂_{y} + (Ap + Bq)∂_{u} − D∂_{p}, \frac{B}{2} ∂_{p} − A∂_{q}⟩ , if N = 0.

Proof. If (x_{0}, …, t_{0}) are coordinates of θ_{2} ∈ J^{2}(E, 2), then the R-plane R_{θ_{2}} is generated by vectors

∂_{x} + p_{0}∂_{u} + r_{0}∂_{p} + s_{0}∂_{q} and ∂_{y} + q_{0}∂_{u} + s_{0}∂_{p} + t_{0}∂_{q}

at the point θ_{1}. By Corollary 2 one gets the needed result for N ≠ 0 just by specializing coordinates of θ_{2} in these expressions to that of the vertex of the cone E_{θ_{1}} (see the proof of Proposition 2). The case N = 0 is reduced to the previous one by a suitable transformation of jet coordinates. □

In its turn, the distribution D_{E} completely determines the equation E. More exactly, we have

Proposition 4. Let D be a Lagrangian subdistribution of C. Then the submanifold

E_{D} = {θ_{2} ∈ J^{2}(E, 2) : dim(R_{θ_{2}} ∩ D(θ_{1})) > 0}

of J^{2}(E, 2) is a parabolic Monge-Ampere equation and D = D_{E_{D}}.

Proof. There is the only one point θ_{2} ∈ (E_{D})_{θ_{1}} such that R_{θ_{2}} = D(θ_{1}). With this exception dim(R_{θ_{2}} ∩ D(θ_{1})) = 1. Hence (E_{D})_{θ_{1}} is the union of all 1-rays l(P) such that P ⊂ D(θ_{1}). They all pass through the exceptional point θ_{2} and hence constitute a cone in F_{θ_{1}}. But cones composed of 1-rays are tangent to the 1-ray distribution on F_{θ_{1}} and, so, all their points are parabolic. Finally, the last assertion directly follows from Corollary 2. □

Results of this section are summed up in the following Theorem which is the starting point of our subsequent discussion of parabolic Monge-Ampere equations.

Theorem 1. The correspondence D ⟼ E_{D} between Lagrangian distributions on J^{1}(E, 2) and parabolic Monge-Ampere equations is one-to-one.

The meaning of this Theorem is that it decodes the geometrical problem hidden under analytical condition (10). Namely, this problem is to find Legendrian submanifolds S of a 5-dimensional contact manifold (M, C) that intersect a given Lagrangian distribution D ⊂ C in a nontrivial manner, i.e.,

dim{T_{θ}(S) ∩ D(θ)} > 0, ∀ θ ∈ S.

The triple E = (M, C, D) encodes this problem. By this reason, in the rest of this paper the term “parabolic Monge-Ampere equation” will refer to such a triple. In particular, equivalence and classification problems for PMAs are interpreted as such problems for Lagrangian distributions on 5-dimensional contact manifolds.

4. Geometry of Lagrangian distributions

In this section we deduce some basic facts about Lagrangian distributions on 5dimensional contact manifolds which allow to reveal four natural classes of them. We fix the notation (M, C) for the considered contact manifold and D for a Lagrangian distribution on it. First of all, Lagrangian distributions are subdivided onto integrable and nonitegrable ones. Accordingly, the corresponding parabolic Monge-Ampere equations are called integrable, or non-integrable. In the subsequent section it will be shown that all integrable PMAs are locally contact equivalent to the equation u_{xx} = 0 and we shall concentrate on nonintegrable PMAs. If D is nonintegrable, then its first prolongation D_{(1)}, i.e., the span of all vector fields belonging to D and their commutators, is 3-dimensional. Moreover, we have

Lemma 3.

(1) D_{(1)} ⊂ C;

(2) the C-orthogonal complement R of D_{(1)} is a 1-dimensional subdistribution of D.

Proof. (1) If (locally) D = ⟨X, Y ⟩, then (locally) D_{(1)} = ⟨X, Y, [X, Y]⟩. But, by definition of C-orthogonality, [X, Y] ∈ C.

(2) The form dU restricted to C is nondegenerate. So, the assertion follows from the fact that in a symplectic linear space a Lagrangian subspace in a hyperplane contains the skew-orthogonal complement of the hyperplane. □

The 1-dimensional distribution R will be called the directing distribution of D (alternatively, of E). This way one gets the following flag of distributions

R ⊂ D ⊂ D_{(1)} ⊂ C.

The directing distribution R uniquely defines D_{(1)}, which is its C-orthogonal complement, and the distribution

D′ = {X ∈ D_{(1)}| [X, R] ⊂ D_{(1)}} .

Since [X, R] ⊂ D_{(1)} for any X ∈ D, we see that D ⊆ D′ ⊆ D_{(1)}. So, by obvious dimension arguments, only one of the following two possibilities may occur locally: either D′ = D, or D′ = D_{(1)}. A non-integrable Lagrangian distribution D is called generic in the first case and special in the second. Accordingly, the corresponding PMAs are called generic, or special. The following assertion directly follows from the above definitions.

Proposition 5. A generic Lagrangian distribution D is completely determined by its directing distribution R. It is no longer so for a special distribution and in this case R is the characteristic distribution of D_{(1)}.

In this paper we shall concentrate on generic PMA equations with a special attention to the equivalence problem. In view of Proposition 5 this problem takes part of the equivalence problem for 1-dimensional subdistributions of C. To localize this part a criterion allowing to distinguish directing distributions of generic PMA equations from other 1-dimensional subdistributions of C. The introduced below notion of the type of a 1-dimensional subdistribution of C gives such a criterion. Fix a 1-dimensional distribution S ⊂ C and put

S_{r} = {Y ∈ C | X^{s}(Y) ∈ C, ∀ X ∈ S and 0 ≤ s ≤ r} .

In the following lemma we list without a proof some obvious properties of S_{r}.

Lemma 4.

• S_{r+1} ⊂ S_{r} and S ⊂ S_{r};

• If (locally) S = ⟨X⟩, then (locally)

S_{r} = {Y ∈ C | X^{s}(Y) ∈ C, 0 ≤ s ≤ r}

• S_{r} is a finitely generated C^{∞}(M)-module;

• If (locally) C = Ann(U), U ∈ Λ^{1}(M), and (locally) S = ⟨X⟩, then

S_{r} = Ann(U, X(U), ..., X^{r}(U)).

Let T be a C^{∞}(M)-module. An open domain B ⊂ M is called regular for T if the localization of T to B is a projective C^{∞}(B)-module. If T is finitely generated (see [7]), then this localization is isomorphic to the C^{∞}(B)-module of smooth sections of a finite dimensional vector bundle over B. In such a case the dimension of this bundle is called the rank of T on B and denoted by rank_{B}T . Moreover, the manifold M is subdivided into a number of open domains that are regular for T and the set of its singular points which is closed and thin. In particular, vector bundles representing localizations of S_{r} to its regular domains are distributions contained in C and containing S (Lemma 4).

Lemma 5. Let B ⊂ M be a common regular domain for modules S_{r} and S_{r+1}. Then either rank_{B}S_{r} = rank_{B}S_{r+1} + 1, or rank_{B}S_{r} = rank_{B}S_{r+1}. In the latter case B is regular for S_{p}, p ≥ r, and rank_{B}S_{p} = rank_{B}S_{r}.

Proof. The first alternative takes place iff X^{r+1}(U) is C^{∞}(M)-independent of U, X(U), …, X^{r}(U) as it is easily seen from the last assertion of Lemma 4. In the second case the equality of ranks implies that localizations of S_{r} and S_{r+1} to B coincide. This shows that the localization of S_{p} to B stabilizes by starting from p = r. □

Corollary 3. With the exception of a thin closed set the manifold M is subdivided into open domains each of them is regular for all S_{p}, p ≥ 0. Moreover, in such a domain B, rank_{B}S_{p} = 4 − p, if p ≤ r, and rank_{B}S_{p} = 4 − r, if p ≥ r, for an integer r = r(B), 1 ≤ r ≤ 3.

Proof. It immediately follows from Lemma 5 that the function p ↦ rank_{B}S_{p} steadily decreases up to the instance, say p = r, when rank_{B}S_{r} = rank_{B}S_{r+1} occurs for the first time, and stabilizes after. Since S ⊂ S_{p}, the last assertion of Lemma 4 shows that this instance happens at most for p = 3, i.e., that r ≤ 3. On the other hand, forms U and X(U) are independent. Indeed, the equality X(U) = fU for a functionf means that X is a contact field with generating function i_{X}(U) ≡ 0. But only the zero field is such one. Hence r ≥ 1. Finally, observe that regular domains for S_{p+1} are obtained from those for S_{p} by removing from latters some thin subsets. Since, as we have seen before, the situation stabilizes after at most four steps by starting from p = 0, the existence of common regular domains for all S_{p}’s whose union is everywhere dense in M is guaranteed. □

Definition 2. If M is the only regular domain for all S_{p}’s, then S is called regular on M and the integer r by starting from which the rank_{B}S_{r} stabilizes is called the type of S and also the type of X, if S = ⟨X⟩.

Hence Lemma 5 tells that the type of S can be only one of the numbers 1, 2, or 3, and that the union of domains in which S is regular is everywhere dense in M. It is not difficult to exhibit vector fields of each these three types. For instance, vector fields of the form X_{f} − fX_{1} on M = J^{1}(E, 2) with everywhere non-vanishing function f are of type 1. Fields ∂_{x} + p∂_{z} + q∂_{p} and ∂_{x} + p∂_{z} + (xy + q)∂_{p} are of type 2 and 3, respectively. Now we can characterize directing distributions of generic PMA equations

Proposition 6. A 1-dimensional regular distribution S ⊂ C is the directing distribution of a generic PMA equation iff it is of type 3.

Proof. Let S be a regular distribution of type 3 and (locally) S =< X >. Then by definition S_{1} = S^{⊥} and D = S_{2} is a bidimensional subdistribution of C. D is Lagrangian, since D ⊂ S_{1} = S^{⊥} and S ⊂ D. If (locally) D = ⟨X, Y ⟩, then [X, Y] ∉ D. Indeed, assuming that [X, Y] ∈ D one sees that X^{p}(Y) ∈ D ⊂ C, ∀p, and hence Y ∈ S_{p}, ∀p. In particular, this implies that D = ⟨X, Y ⟩ ⊂ S_{3} in contradiction with the fact that dim S_{3} = 1 if S is of type 3. So, dim ⟨X, Y, [X, Y]⟩ = 3, and ⟨X, Y, [X, Y]⟩ ⊂ S^{⊥}, since X(Y), X^{2}(Y) ∈ C. Now, by dimension arguments, we conclude that ⟨X, Y, [X, Y]⟩ = S^{⊥} and, therefore, S is the directing distribution of D = S_{2}. Conversely, assume that (locally) R = ⟨X⟩ is the directing distribution of a generic PMA equation corresponding to the Lagrangian distribution D = ⟨X, Y ⟩ (locally). Then, by definition, R_{1} = R^{⊥}. Moreover, D = D′ implies that X^{2}(Y) ∉ R^{⊥} and hence fields X, Y, X(Y), X^{2}(Y) form a local basis of C. This shows that X^{3}(Y) ∉ C. Indeed, the assumption X^{3}(Y) ∈ C implies [X, C] ⊂ C, i.e., that X is a nonzero contact field with zero generating function. Hence Y ∉ R_{3} ⇔ [X, Y] ∉ R_{2}. From one side, this shows that D = R_{2} and, from other side, that R_{2} ≠ R_{3}. So, R_{3} = ⟨R⟩ ⇒ rank R = 3. □

From now on we denote by Z the directing distribution of the considered PMA E. A coordinate description of D = D_{E}, D_{1} and Z is easily obtained by a direct computation (see Proposition 3):

Proposition 7. Let E be a quasilinear nonintegrable PMA of the form (10) with A ≠ 0. By normalizing its coefficients to A = 1 one has: D = ⟨X_{1}, X_{2}⟩ and D_{(1)} = ⟨X_{1}, X_{2}, X_{3}⟩ with

X_{1} := ∂_{x} + p∂_{u} + \frac{B}{2} (∂_{y} + q∂_{u}) − D∂_{p},

X_{2} := \frac{B}{2} ∂_{p} − ∂_{q},

X_{3} := [X_{1}, X_{2}] = M_{1} (∂_{y} + q∂_{u}) + M_{2}∂_{p},

where

M_{1} = −\frac{1}{2}X_{2}(B), M_{2} = \frac{1}{2}X_{1}(B) + X_{2}(D),

and R = ⟨Z⟩ with

Z := M_{1}X_{1} − M_{2}X_{2}.

Proposition 8. Let E be a nonintegrable PMA of the form (10) with N ≠ 0. By normalizing its coefficients to N = 1 one has: D = ⟨X_{1}, X_{2}⟩ and D_{(1)} = ⟨X_{1}, X_{2}, X_{3}⟩ with

X_{1} := ∂_{x} + p∂_{u} − C∂_{p} + \frac{B}{2} ∂_{q},

X_{2} := ∂_{y} + q∂_{u} + \frac{B}{2} ∂_{p} − A∂_{q},

X_{3} := [X_{1}, X_{2}] = M_{1}∂_{q} + M_{2}∂_{p},

where

M_{1} = −X_{1}(A) − \frac{1}{2}X_{2}(B), M_{2} = \frac{1}{2}X_{1}(B) + X_{2}(C),

and R = ⟨Z⟩ with

Z := M_{1}X_{1} − M_{2}X_{2}.

5. Classification of integrable PMAs

For completeness we shall prove here the following, essentially known, result in a manner that illustrate the idea of our further approach.

Theorem 2. With the exception of singular points all integrable PMA equations are locally contact equivalent each other and, in particular, to the equation u_{xx} = 0.

Proof. Let E = (M, C, D) be an integrable PMA equation, i.e., the Lagrangian distribution D is integrable and, as such, define a 2-dimensional Legendrian foliation of M. Locally this foliation can be viewed as a fibre bundle Π : M → W over a 3-dimensional manifold W. The differential d_{θ}(Π) : T_{θ}M → T_{y}W, y = Π(θ), sends C(θ) to a bidimensional subspace P_{θ} ⊂ T_{y}W, since ker d_{θ}(Π) = D(θ). This way one gets the map Π_{y} : Π^{−1}(y) → G_{3,2}(y), θ ↦ P_{θ}, where G_{3,2}(y) is the Grassmanian of 2-dimensional subspaces in T_{y}W. Note that dim Π^{−1}(y) = dim G_{3,2}(y) = 2 and, so, the local rank of Π_{y} may vary from 0 to 2. We shall show that, with the exception of a thin set of singular points, Π_{y}’s are of rank 2, i.e., Π_{y}’s are local diffeomorphisms. First, assume that this rank is zero for all y ∈ W, i.e., Π_{y}’s are locally constant maps. In this case P_{θ} does not depend on θ ∈ Π^{−1}(y) and we can put P(y) = P_{θ}, for a θ ∈ Π^{−1}(y). Hence y ↦ P(y) is a distribution on W and C is its pullback via Π. This shows that the distribution tangent to fibers of Π, i.e., D, is characteristic for C. But a contact distribution does not admit nonzero characteristics. Second, if the rank of Π_{y}’s equals to one for all y ∈ W, then M is foliated by curves

γ_{P} = {θ ∈ Π^{−1}(y)|d_{θ}Π(C(θ)) = P }

with P being a bidimensional subspace of T_{y}W. Locally, this foliation may be seen as a fibre bundle Π_{0} : M → N over a 4-dimensional manifold N, and Π factorizes into the composition

C →^{Π_{0}} N →^{Π_{1}} W

with Π_{1} uniquely defined by Π and Π_{0}. By construction the 3-dimensional subspace d_{θ}Π_{0}(C(θ)) ⊂ T_{z}N, z = Π_{0}(θ), does not depend on θ ∈ Π_{0}^{−1}(z) = γ_{P} and one can put Q(z) = d_{θ}Π_{0}(C(θ)) for a θ ∈ Π_{0}^{−1}(z). As before we see that C is the pullback via Π_{0} of the 3-dimensional distribution z ↦ Q(z) in contradiction with the fact that C does not admit nonzero characteristics. Thus, except singular points, Π is of rank 2 and hence a local diffeomorphism. So, locally, Π_{y} identifies Π^{−1}(y) and an open domain in G_{3,2}(y). By observing that G_{3,2}(y) = π_{1,0}^{−1}(y), with π_{1,0} : J^{1}(W, 2) → W being a natural projection, one gets a local identification of M with an open domain in J^{1}(W, 2). It is easy to see that this identification is a contact diffeomorphism. In other words, we have proven that any integrable Lagrangian distribution on a 5-dimensional contact manifold is locally equivalent to the distribution of tangent planes to fibers of the projection J^{1}(ℝ^{3}, 2) → ℝ^{3}. Finally, we observe that D_{E} = ⟨∂_{x}, ∂_{y}⟩ for the equation E = {u_{xx} = 0} and hence this equation is integrable. □

6. Projective curve bundles and non-integrable generic PMAs

In this section non-integrable generic Lagrangian distributions and, therefore, the corresponding PMAs are represented as 4-parameter families of curves in the projective 3-space or, more exactly, as projective curve bundles. Differential invariants of single curves composing such a bundle (say, projective curvature, torsion,etc) put together give differential invariants of the whole bundle and consequently of the corresponding PMA. This basic geometric idea is developed in details in the subsequent section. Let N be a 4-dimensional manifold. Denote by P T_{a}^{∗}N the 3-dimensional projective space of all 1-dimensional subspaces of the cotangent to N space T_{a}^{∗}N at the point a ∈ N. The projectivization pτ^{∗} : P T^{∗}N → N of the cotangent to N bundle τ^{∗} : T^{∗}N → N is the bundle whose total space is P T^{∗}N = ⋃_{a∈N}P T_{a}^{∗}N and the fiber over a ∈ N is P T_{a}^{∗}N, i.e., (pτ^{∗})^{−1}(a) = P T_{a}^{∗}N. A projective curve bundle (PCB) over N is a 1-dimensional subbundle π : K → N of pτ^{∗}:

K ↪ P T^{∗}N
π ↓  ↓ pτ^{∗}
N ⟶^{id} N

The fiber π^{−1}(y), y ∈ N, is a smooth curve in the projective space P T_{y}^{∗}N. A diffeomorphism Φ : N → N′ lifts canonically to a diffeomorphism P T^{∗}N → P T^{∗}N′. This lift sends a PCB π over N to a PCB over N′, denoted by Φπ. A PCB π over N and a PCB π′ over N′ are equivalent if there exist a diffeomorphism Φ : N → N′ such that π′ = Φπ. Let π : K → N be a PCB and θ =< ρ >∈ K with ρ ∈ T_{π(θ)}^{∗}N. Denote by W_{θ} the 3-dimensional subspace of T_{π(θ)}N annihilated by θ, i.e.,

W_{θ} = {ξ ∈ T_{π(θ)}N | ρ(ξ) = 0} .

Two distributions are canonically defined on K. First of them is the 1-dimensional distribution R_{π} formed by all vertical with respect to π vectors. The second one, denoted by C_{π}, is defined by

(C_{π})_{θ} = {η ∈ T_{θ}K | d_{θ}π(η) ∈ W_{θ}} .

Obviously, dim C_{π} = 4 and R_{π} ⊂ C_{π}. If, locally, R_{π} = ⟨Z⟩ , Z ∈ D(K), and C_{π} = Ann(U_{π}), U_{π} ∈ Λ^{1}(K), then the osculating distributions of π are defined as

Z_{s}^{π} = Ann(U_{π}, Z(U_{π}), ..., Z^{s}(U_{π})), s = 0, 1, 2, 3.

It is easy to see that this definition does not depend on the choice of Z and U_{π}. Note that C_{π} = Z_{0}^{π}. Also, R_{π} ⊂ Z_{s}^{π}, ∀ s ≥ 0, as it easily follows from [i_{Z}, L_{Z}] = 0 and U_{π}(Z) = 0. Moreover, generically forms U_{π}, Z(U_{π}), ..., Z^{3}(U_{π}) are independent and so, by dimension arguments, R_{π} = Z_{3}^{π}. We say that π is a regular PCB iff the following two conditions are satisfied: (i) R_{π} = Z_{3}^{π} and (ii) C_{π} is a contact structure on K. We emphasize that regularity is a generic condition. Moreover, conditions (i)-(ii) are equivalent to the fact that R_{π} is of type 3 with respect to the contact distribution C_{π}. So, by Proposition 6, the distribution

D_{π} = {X ∈ R_{π}^{⊥}|L_{X}(R_{π}) ⊂ R_{π}^{⊥}}.

with R_{π}^{⊥} being the C_{π}-orthogonal complement of R_{π} is bidimensional and Lagrangian for a regular PCB π. Thus we have

Theorem 3. If π is a regular PCB, then D_{π} is a Lagrangian subdistribution of C_{π} and (K, C_{π}, D_{π}) is a generic PMA whose directing distribution is R_{π}. Conversely, a generic PMA locally determines a regular PCB.

Proof. The first assertion of the Theorem is already proved. It remains to represent a generic PMA (M, C, D) as a regular PCB. Integral curves of its directing distribution R foliate M. Locally, this foliation may be considered as a fiber bundle π : M → N over a 4-dimensional manifold N. Since R(θ) ⊂ C(θ), the subspace V_{θ} = d_{θ}π(C(θ)) ⊂ T_{y}N, y = π(θ), is 3-dimensional. Put Γ_{y} = π^{−1}(y). The map π_{y} : Γ_{y} → G_{4,3}(y), θ ↦ V_{θ}, with G_{4,3}(y) being the Grassmanian of 3-dimensional subspaces in T_{y}N, is almost everywhere of rank 1. Indeed, the assumption that locally this rank is zero leads, as in the proof of Theorem 2, to conclude that locally the contact distribution C is the pullback via π of a 3-dimensional distribution on N. The correspondence ι_{y} : G_{4,3}(y) → P T_{y}^{∗}N that sends a 3-dimensional subspace V ⊂ T_{y}N to Ann(V) is, obviously, a diffeomorphism. Hence the composition ι_{y} ∘ π_{y} is a local embedding with exception of a thin subset of singular points. Now, it is easy to see that images of Γ_{y}’s via ι_{y} ∘ π_{y}’s give the required PCB. □

The above construction associating a PCB with a given PMA is manifestly functorial, i.e., an equivalence F : (M, C, D) ⟶ (M′, C′, D′) of PMAs induces an equivalence Φ : (N, K, π) ⟶ (N′, K′, π′) of associated PCBs. Indeed, F sends R to R′ and hence integral curves of R (locally, fibers of π) to integral curves of R′ (locally, fibers of π′). This defines a map Φ of the variety N of fibers of π to the variety N′ of fibers of π′, etc. Thus we have

Corollary 4. The problem of local contact classification of generic PMAs is equivalent to the problem of local classification of regular PCBs with respect to diffeomorphisms of base manifolds.

Now we observe that there is another, in a sense, dual PCB associated with a given PMA equation. Namely, associate with a point θ ∈ K the line L_{θ} = d_{θ}π(D_{π}(θ)) ⊂ T_{y}N, y = π(θ). The correspondence θ ↦ L_{θ} ∈ P T_{y}N, where P T_{y}N denotes the projective space of lines in T_{y}N, defines a map of π^{−1}(y) to P T_{y}N, i.e., a (singular) curve in P T_{y}N. As before this defines locally a 1-dimensional subbundle in the projectivization P T N of T N. It will be called the second PCB associated with the considered PMA. PCBs may be considered as canonical models of PMAs. Besides other they suggest a geometrically transparent construction of scalar differential invariants of PMAs. Let I be a scalar projective differential invariant of curves in ℝP^{3}, say, the projective curvature (see [11, 2]), θ ∈ K and y = π(θ). The value of this invariant for the curve Γ_{y} = π^{−1}(y) in P T_{y}^{∗} is a function on Γ_{y}. Denote it by I_{π,y} and put I_{π}(θ) = I_{π,y}(θ) if θ ∈ Γ_{y} ⊂ K. Then, obviously, I_{π} ∈ C^{∞}(K) is a differential invariant of the PCB π and hence of the PMA represented by π.

Theorem 4. The differential invariants of the form I_{Ψ} are sufficient for a complete classification of generic PMA equations on the basis of the "principle of n-invariants".

Proof. According to the "principle of n-invariants", it is sufficient to construct n = dim M = 5 independent differential invariants of PMAs in order to solve the classification problem. Such invariants of the required form will be constructed in the next section. □

For the "principle of n-invariants" the reader is referred to [1, 10].

7. Differential invariants of generic PCBs

Let π : K → N be a regular PCB and, as before, R_{π} =< Z >, D_{π} =< Z, X > and C_{π} = Ann(U). Here the vector field Z and the 1-form U are unique up to a functional nowhere vanishing factor, while X is unique up to a transformation X ⟼ gX + φZ, g, φ ∈ C^{∞}(K) with nowhere vanishing g. Since the considered PCB is regular we have the following flag of distributions

R_{π} ⊂ D_{π} ⊂ R_{π}^{⊥} ⊂ C_{π} ⊂ D(K)

of dimensions increasing from 1 to 5, respectively. In terms of X, Z and U they are described as follows:

Proposition 9. Locally, with the exception of a thin set of singular points we have: (1) R_{π} = Ann(Z^{i}(U) : i = 0, 1, 2, 3) = ⟨Z⟩;

(2) D_{π} = Ann(U, Z(U), Z^{2}(U)) = ⟨Z, X⟩;

(3) R_{π}^{⊥} = Ann(U, Z(U)) =< Z, X, Z(X) >;

(4) C_{π} = Ann(U) =< Z, Z^{i}(X) : i = 0, 1, 2 >;

(5) {Z, X, Z(X), Z^{2}(X), Z^{3}(X)} is a base of the C^{∞}(K)-module D(K);

(6) for some functions r_{i} ∈ C^{∞}(K)

(14)  Z^{4}(U) + r_{1}Z^{3}(U) + r_{2}Z^{2}(U) + r_{3}Z(U) + r_{4}U = 0

Proof. Assertions (1)-(3) are direct consequences of Lemma 4, Proposition 6 and definitions. Assertion (4) follows from independence of Z^{2}(X) from Z, X, Z(X). This is so because otherwise Z would be a characteristic of R_{π}^{⊥} = < Z, X, Z(X) > in contradiction with the fact that R_{π} is of rank 3. Similarly, Z^{3}(X) is independent of Z, X, Z(X), Z^{2}(X). Indeed, otherwise Z would be a characteristic of C_{π}. This proves (5). Finally, forms Z^{s}(U), s ≥ 0, are annihilated by Z. Since dim M = 5 this implies that Z^{4}(U) depends on Z^{s}(U), s ≤ 3. But this is equivalent to (14). □

Corollary 5. If 0 ≤ k, l ≤ 3 and k + l = 3, then

Z^{k}(X)⌟Z^{l}(U) = (−1)^{k+1}Z^{3}(X)⌟U ≠ 0.

Proof. Assertions (5) and (6) of the above Proposition show that Z^{3}(X) completes a basis of C to a basis of D(K). So, Z^{3}(X)⌟U is a nowhere vanishing function.

It follows from the standard formula [i_{X}, L_{Y}] = i_{[X,Y]} that

Z^{k}(X)⌟Z^{l}(U) = [i_{Z^{k}(X)}, L_{Z}] (Z^{l−1}(U)) − L_{Z}(Z^{k}(X)⌟Z^{l−1}(U)) =

− Z^{k+1}(X)⌟Z^{l−1}(U) − L_{Z}(Z^{k}(X)⌟Z^{l−1}(U)) (15)

According to Proposition 9, (1)-(3), Z^{r}(X)⌟Z^{s}(U) = 0 if r + s = 2. So, for k + l = 3 relation (15) becomes

Z^{k}(X)⌟Z^{l}(U) = −Z^{k+1}(X)⌟Z^{l−1}(U)
□

By decomposing Z^{4}(X) with respect to the base in (5) of Proposition 9 we get

(16)  Z^{4}(X) + ρ_{1}Z^{3}(X) + ρ_{2}Z^{2}(X) + ρ_{3}Z(X) + ρ_{4}X + ρ_{5}Z = 0

for some functions ρ_{i} ∈ C^{∞}(K) and hence

(17) Z^{4}(X ∧ Z) + ρ_{1}Z^{3}(X ∧ Z) + ρ_{2}Z^{2}(X ∧ Z) + ρ_{3}Z(X ∧ Z) + ρ_{4}X ∧ Z = 0.

The last is a relation binding the bivector X ∧ Z, which generates the distribution D_{π}. This bivector is unique up to a functional factor.

Proposition 10. Functions r_{i}’s in decomposition (14) are expressed in terms of iterated Lie derivatives Z^{i}(U), Z^{j}(X) as follows:

r_{1} = −\frac{X⌟Z^{4}(U)}{X⌟Z^{3}(U)}, r_{2} = −\frac{Z(X)⌟Z^{4}(U) + r_{1}Z(X)⌟Z^{3}(U)}{Z(X)⌟Z^{2}(U)},

r_{3} = −\frac{Z^{2}(U)⌟Z^{4}(U) + r_{1}Z^{2}(X)⌟Z^{3}(U) + r_{2}Z^{2}(X)⌟Z^{2}(U)}{Z^{2}(X)⌟Z(U)},

r_{4} = −\frac{Z^{3}(X)⌟Z^{4}(U) + r_{1}Z^{3}(X)⌟Z^{3}(U) + r_{2}Z^{3}(X)⌟Z^{2}(U) + r_{3}Z^{3}(X)⌟Z(U)}{Z^{3}(X)⌟U}.

Proof. By subsequently inserting fields Z^{s}(X), 0 ≤ s ≤ 3, in (14) one easily gets the result by taking into account Proposition 9 and Corollary 5. □

Similarly, we have

Proposition 11. Functions ρ_{i} in decomposition (17) are expressed in terms of iterated Lie derivatives Z^{i}(U), Z^{j}(X) as follows:

ρ_{1} = −\frac{Z^{4}(X)⌟U}{Z^{3}(X)⌟U} , ρ_{2} = −\frac{Z^{4}(X)⌟Z(U) + ρ_{1}Z^{3}(X)⌟Z(U)}{Z^{2}(X)⌟Z(U)},

ρ_{3} = −\frac{Z^{4}(U)⌟Z^{2}(U) + ρ_{1}Z^{3}(X)⌟Z^{2}(U) + ρ_{2}Z^{2}(X)⌟Z^{2}(U)}{Z(X)⌟Z^{2}(U)},

ρ_{4} = −\frac{Z^{4}(X)⌟Z^{3}(U) + ρ_{1}Z^{3}(X)⌟Z^{3}(U) + ρ_{2}Z^{2}(X)⌟Z^{3}(U) + ρ_{3}Z(X)⌟Z^{3}(U)}{X⌟Z^{3}(U)}.

Proof. As before we get the desired result by subsequently inserting the vector field in the left hand side of (16) into 1-forms Z^{s}(U), 0 ≤ s ≤ 3. □

Remark 1. By introducing functions α_{kl} = Z^{k}(X)⌟Z^{l}(U) we see that r_{i}’s and ρ_{i}’s are rational functions of α_{kl}’s:

r_{1} = \frac{α_{04}}{α_{03}} ,  r_{2} = \frac{α_{03}α_{14} + α_{04}α_{13}}{α_{03}α_{12}} ,  etc.

Differential invariants we are going to construct are projective differential invariants of curves composing the considered PCB. Clearly, it is not possible to describe explicitly these curves. So, the problem is how to express these invariants in terms of the data at our disposal, i.e., X, Z and U. In what follows this problem is solved on the basis of a rather transparent analogy. For instance, the field Z restricted to one of these curves may be thought as the derivation with respect to a parameter along this curve, and so on. So, with similar interpretations in mind it is sufficient just to mimic a known construction of projective differential invariants for curves in order to obtain the desired result. In doing that we follows classical Wilczynski’s book [11]. By stressing the used analogy we pass to Wilczynski’s p_{i}’s and q_{j}’s instead of above r_{i}’s and ρ_{j}’s:

(18)  r_{1} = 4p_{1},  r_{2} = 6p_{2},  r_{3} = 4p_{3},  r_{4} = p_{4}

and

(19)  ρ_{1} = 4q_{1},  ρ_{2} = 6q_{2},  ρ_{3} = 4q_{3},  ρ_{4} = q_{4}.

In terms of these functions relations (14) and (17) read

(20)  Z^{4}(U) + 4p_{1}Z^{3}(U) + 6p_{2}Z^{2}(U) + 4p_{3}Z(U) + p_{4}U = 0,

(21) Z^{4}(X ∧ Z) + 4q_{1}Z^{3}(X ∧ Z) + 6q_{2}Z^{2}(X ∧ Z) + 4q_{3}Z(X ∧ Z) + q_{4}X ∧ Z = 0.

These relations are identical to Wilczynski’s formulas (see equation (1), page 238 of [11]). Z and U are unique up to a "gauge" transformation (Z, U) ⟼ (\overline{Z}, \overline{U})

(22)  Z = f\overline{Z}, U = h\overline{U}

with nowhere vanishing f, h ∈ C^{∞}(K). The corresponding transformation of coefficients {p_{i}} ⟼ {p̄_{i}} can be easily obtained from (20) by a direct computation:

(23)  \begin{matrix}p_{1} ⟼ p̄_{1} = \frac{Z(h)}{h} + \frac{p_{1}}{f} + \frac{3Z(f)}{2f}, \\ p_{2} ⟼ p̄_{2} = \frac{p_{2}}{f^{2}} + \frac{2p_{1} Z(f)}{f^{2}} + \frac{2p_{1} Z(h)}{3hf} + 3 \frac{Z(h)Z(f)}{hf} + \frac{7Z(f)^{2}}{6f^{2}} \\ +\frac{2Z^{2}(f)}{3f} + \frac{Z^{2}(h)}{h}, \\ p_{3} ⟼ p̄_{3} = \frac{p_{3}}{f^{3}} + \frac{3p_{2}Z(h)}{hf^{2}} + \frac{3p_{2}Z(f)}{2f^{3}} + \frac{p_{1} Z(f)^{2}}{f^{3}} + \frac{6p_{1}Z(h)Z(f)}{hf^{2}} \\ +\frac{3p_{1}Z^{2}(h)}{hf} + \frac{p_{1}Z^{2}(f)}{f^{2}} + \frac{Z^{3}(f)}{4f} + \frac{9Z(f)Z^{2}(h)}{2hf} + \frac{Z(f)^{3}}{4f^{3}} \\ +\frac{7Z(h)Z(f)^{2}}{2hf^{2}} + \frac{2Z(h)Z^{2}(f)}{hf} + \frac{Z^{3}(h)}{h} + \frac{Z^{2}(f)Z(f)}{f^{2}}, \\ p_{4} ⟼ p̄_{4} = \frac{p_{4}}{f^{4}} + \frac{4p_{3}Z(h)}{hf^{3}} + \frac{6p_{2}Z^{2}(f)}{hf^{2}} + \frac{6p_{2}Z(h)Z(f)}{hf^{3}} + \frac{4p_{1} Z^{3}(h)}{hf} \\ +\frac{4p_{1}Z(h)Z^{2}(f)}{hf^{2}} + \frac{4p_{1}Z(h)Z(f)^{2}}{hf^{3}} + \frac{12p_{1}Z(f)Z^{2}(h)}{hf^{2}} \\ +\frac{6Z(f)Z^{3}(h)}{hf} + \frac{Z^{4}(h)}{h} + \frac{7Z(f)^{2}Z^{2}(h)}{hf^{2}} + \frac{Z(h)Z^{3}(f)}{hf} \\ +\frac{4Z(h)Z^{2}(f)Z(f)}{hf^{2}} + \frac{Z(f)^{3}Z(h)}{hf^{3}} + \frac{4Z^{2}(h)Z^{2}(f)}{hf}. \end{matrix}

Now the problem is to combine p_{i}’s in a way to obtain expressions which are invariant with respect to transformations (23). To this end we first normalize (Z, U) by the condition p_{1} = 0. This can be easily done with f = 1 and a solution h of the equation

Z(h) + p_{1}h= 0.

After this normalization, equation (20) takes a simpler form

(24)  Z^{4}(\overline{U}) + 6P_{2}Z^{2}(\overline{U}) + 4P_{3}Z(\overline{U}) + P_{4}\overline{U} = 0

with

(25)  \begin{matrix}P_{2} := p_{2} − Z(p_{1}) − Z(p_{1})^{2}, \\ P_{3} := p_{3} − Z(Z(p_{1})) − 3p_{1}p_{2} + 2p_{1}^{3}, \\ P_{4} := p_{4} − 4p_{1}p_{3} − 3p_{1}^{4} − Z(Z(Z(p_{1}))) + 3Z(p_{1})^{2} \\ +6p_{1}^{2}Z(p_{1}) + 6p_{1}^{2}p_{2} − 6Z(p_{1})p_{2}. \end{matrix}

Proposition 12. Transformations (22) preserving the normalization p_{1} = 0 are subject to the condition

(26)  Z(h) + \frac{3}{2}Z(ln(f))h = 0.

Proof. A direct computation. □

Now the problem reduces to finding invariant combinations of P_{2}, P_{3} and P_{4} with respect to normalized, i.e., respecting condition (26), transformations (22). This can be done, for instance, by mimicking the construction of projective curvature and torsion in [11]. Namely, introduce first the functions

(27)  \begin{matrix}Θ_{3} = P_{3} − \frac{3}{2}Z(P_{2}), \\ Θ_{4} = P_{4} − 2Z(P_{3}) + \frac{6}{5}Z(Z(P_{2})) − \frac{81}{25}P_{2}^{2}, \\ Θ_{3·1} = 6Θ_{3}Z(Z(Θ_{3})) − 7Z(Θ_{3})^{2} − \frac{108}{5}P_{2}Θ_{3}^{2} \end{matrix}

They are semi-invariant with respect to normalized transformations (22), i.e., they are transformed according to formulas

(28)  \overline{Θ}_{3} = \frac{Θ_{3}}{f^{3}} ,  \overline{Θ}_{4} = \frac{Θ_{4}}{f^{4}} ,  \overline{Θ}_{3·1} = \frac{Θ_{3·1}}{f^{8}} .

Obviously, the following combinations of the Θ_{i}’s

(29)  κ_{1} = \frac{Θ_{4}}{Θ_{3}^{4/3}} ,  κ_{2} = \frac{Θ_{3·1}}{Θ_{3}^{8/3}}

are invariant with respect to normalized transformations (22). Thus we have

Proposition 13. κ_{1} and κ_{2} are scalar differential invariants of parabolic Monge-Ampere equations (10) with respect to contact transformations.

Explicit expressions of κ_{1} and κ_{2} in terms of coefficients of PMA (10) can be straightforwardly obtained from those of Z and U. However, they are not very instructive and too cumbersome to be reported here. Another invariant, which can be readily extracted from (28), is the invariant vector field

(30)  N_{1} = Θ_{3}^{−1/3}Z.

Another set of scalar differential invariants can be constructed in a similar manner by starting from equation (17). Indeed, the vector field Z and the bivector field X ∧ Z generating distributions R and D, respectively, are unique up to transformations

(31)  Z = f\overline{Z},  X ∧ Z = g\overline{X} ∧ \overline{Z}

with nowhere vanishing f, g ∈ C^{∞}(K). It is easy to check that coefficients q_{i}’s are transformed according to formulas (23) and one can repeat what was already done previously in the case of equation (14). In particular, equation (17) can be normalized as

(32)  Z^{4}(\overline{X} ∧ \overline{Z}) + 6Q_{2}Z^{2}(\overline{X} ∧ \overline{Z}) + 4Q_{3}Z(\overline{X} ∧ \overline{Z}) + Q_{4}\overline{X} ∧ \overline{Z} = 0

with

(33)  \begin{matrix}Q_{2} = q_{2} − Z(q_{1}) − Z(q_{1})^{2}, \\ Q_{3} = q_{3} − Z(Z(q_{1})) − 3q_{1}q_{2} + 2q_{1}^{3}, \\ Q_{4} = q_{4} − 4q_{1}q_{3} − 3q_{1}^{4} − Z(Z(Z(q_{1}))) + 3Z(q_{1})^{2} \\ +6q_{1}^{2}Z(q_{1}) + 6q_{1}^{2}q_{2} − 6Z(q_{1})q_{2}. \end{matrix}

This way we obtain the following scalar differential invariants of PMAs

(34)  τ_{1} := \frac{Λ_{4}}{Λ_{3}^{4/3}} ,  τ_{2} := \frac{Λ_{3·1}}{Λ_{3}^{8/3}}

with semi-invariants Λ_{i}’s defined by

(35)  \begin{matrix}Λ_{3} = Q_{3} − \frac{3}{2}Z(Q_{2}), \\ Λ_{4} = Q_{4} − 2Z(Q_{3}) + \frac{6}{5}Z(Z(Q_{2})) − \frac{81}{25}Q_{2}^{2}, \\ Λ_{3·1} = 6Λ_{3}Z(Z(Λ_{3})) − 7Z(Λ_{3})^{2} − \frac{108}{5}Q_{2}Λ_{3}^{2}. \end{matrix}

Similarly,

(36)  N_{2} = Λ_{3}^{−1/3}Z.

is an invariant vector field.

Remark 2. The observed parallelism in construction of two sets of differential invariants is explained by the fact that in both cases we compute the same invariants in two different PCB, namely, the first and the second ones, associated with the considered PMA.

Since vector fields N_{1} and N_{2} are invariant and N_{1} = λN_{2} the factor λ = Θ_{3}^{−1/3}Λ_{3}^{1/3} is a scalar differential invariant. So,

γ_{3} := λ^{3} = \frac{Λ_{3}}{Θ_{3}}

is a scalar differential invariant of a new “mixed” kind as well as ratios

γ_{4} := \frac{Λ_{4}}{Θ_{4}} , λ_{3·1} := \frac{Λ_{3·1}}{Θ_{3·1}}

By applying to already constructed scalar differential invariants

κ_{1}, κ_{2}, τ_{1}, τ_{2}, γ_{3}, γ_{4}, γ_{3·1}

various algebraic operations and arbitrary compositions of invariant vector fields N_{1} and N_{2} one can construct many other scalar differential invariants of PMAs. These does not exhaust all invariants. Nevertheless, various quintuples of (functionally) independent invariants can be composed from them, and this is the only one need in order to apply the “principle of n invariants”. For instance, we have Theorem 5. Quintuples (κ_{1}, κ_{2}, τ_{1}, τ_{2}, γ_{3}) and (γ_{3}, N_{1}(γ_{3}), N_{1}^{2}(γ_{3}), κ_{1}, κ_{2}) are composed of independent invariants. Proof. It is sufficient to exhibit an example for which invariants composing each of these two quintuples are independent. For instance, for both quintuples such is the equation determined by the directing distribution R generated by

Z = q∂_{x} + y∂_{y} + (qp + yq)∂_{z} + x∂_{p} − xz∂_{q}.

In this case the corresponding Lagrangian distribution D is generated by

X_{1} = a_{1}(∂_{x} + p∂_{u}) + a_{2}∂_{p} + a_{3}∂_{q}, X_{2} = a_{1}(∂_{y} + q∂_{u}) + a_{3}∂_{p} + a_{4}∂_{q}

with

\begin{matrix}a_{1} = xyu − (x − y)q,  a_{2} = −(x − y)x − (u + xp)y^{2}, \\ a_{3} = x^{2}u + (u + xp)yq,  a_{4} = −(q + xu)xu − (u + xp)q^{2}, \end{matrix}

and the corresponding PMA is

(37) a_{1}^{2}(rt − s^{2}) + a_{1}a_{4}r + 2a_{1}a_{3}s − a_{1}a_{2}t + xa_{1}(xyup − xu − xpq + yu^{2} − uq) = 0.

Explicit expressions of invariants (κ_{1}, κ_{2}, τ_{1}, τ_{2}, γ_{3}) and (γ_{3}, N_{1}(γ_{3}), N_{1}^{2}(γ_{3}), κ_{1}, κ_{2}) for equation (37) are too cumbersome to be reported here. A direct check shows that they are functionally independent in each of above two quintuples. □

Thus, according to the “principle of n-invariants” (see [1, 10]), the proven existence of five independent scalar differential invariants solves in principle the equivalence problem for generic PMA equations. It should be stressed, however, that a practical implementation of this result could meet some boring computational problems.

8. Concluding remarks

Representation of a PMA equation E by means of the associated PCB makes clearly visible the nature of its nonlinearities. For example, if all curves of this bundle are projectively nonequivalent each other, then E does not admit contact symmetries, etc. The instance of this can be detected by means of invariants constructed in the previous section. On the contrary, it may happen that all curves composing a PCB are projectively equivalent, i.e., nonlinearities of the corresponding PMA E are “homogeneous”. The above constructed invariants are not sufficient to distinguish one homogeneous in this sense PMA from another, and a need of new finer invariants arises. It is remarkable that in similar situations PCBs themselves give an idea of how such invariants can be constructed. For instance, in the above homogeneous case one can observe that the bundle P T^{∗}N → N is naturally supplied with a full parallelism structure which immediately furnishes the required new invariants. It is not difficult to imagine various intermediate situations, which demonstrate the diversity and complexity of the world of parabolic Monge-Ampere equations. In particular, the problem of describing all strata of the characteristic diffiety (see [10]) for parabolic Monge-Ampere equations is a task of a rather large scale. Further results in this direction will appear in a series of forthcoming publications.

References

[1] D. V. Alekseevsky, V. V.Lychagin, A. M. Vinogradov, Basic ideas and concepts of differential geometry, Encyclopedia of Math. Sciences, 28 (Springer-Verlag, Berlin, 1991).

[2] Su Buchin, The General Projective Theory of Curves (Science Press, Peking, 1973).

[3] I. S. Krasil’shchik, V. V. Lychagin and A. M. Vinogradov, Geometry of Jet spaces and Nonlinear Partial Differential Equations, Gordon and Breach, New York, 1986

[4] Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Ed. by I. S. Krasil’shchik and A. M. Vinogradov, “Translation of Mathematical Monographs Series ”182 (Amer.Math.Soc., Providence, RI, 1999).

[5] A. Kushner, V. Lychagin, V. Rubtsov, Contact Geometry and Nonlinear Differential Equations (Cambridge Univ. Press, Cambridge, 2007).

[6] T. Morimoto, in Symplectic Singularities and Geometry of Gauge Fields, Banach Center Publ. 39, 105-121 (1997).

[7] J. Nestruev, Smooth Manifolds and Observables (Springer-Verlag, Berlin - New-York, 2002).

[8] A. M. Vinogradov, On geometry of Second Order Parabolic Differential Equations in Two Indipendent Variables, Dokl.Math. 423(5), (2008).

[9] A. M. Vinogradov, Geometric Singularities of Solutions of Nonlinear Partial Differential Equations, in Proceedings of Conference "Differential Geometry and Applications", Brno, 1986.

[10] A. M. Vinogradov, in Mechanics, Analysis and Geometry: 200 Years after Lagrange, Ed. by M. Francaviglia (North Holland, North Holland, 1991).

[11] E. J. Wilczinsky, Projective Differential Geometry of Curves and Ruled Surfaces (Chealsea, 1961).

Dipartimento di Matematica, Università Statale di Milano, Via Cesare Saldini 50, 20133 Milano (MI), Italy

 E-mail address: catalano@mat.unimi.it

Dipartimento di Matematica e Informatica, Università di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA),Italy, Istituto Nazionale di Fisica Nucleare, sezione di Napoli-Salerno, and Diffiety Institute, Russia.

 E-mail address: vinograd@unisa.it