GEOMETRY OF THIRD{ORDER EQUATIONS OF MONGE{AMPERE …urbanski/Moreno.pdf · GEOMETRY OF THIRD{ORDER...

GEOMETRY OF THIRD–ORDER EQUATIONS OF MONGE–AMPERE TYPE

GIANNI MANNO AND GIOVANNI MORENO

Abstract. We study the class of scalar third–order PDEs in two independent variables which can be defined by means of

a differential two–form on the first prolongation of a five–dimensional contact manifold and represent the natural analoguesof the classical Monge–Ampere equations (MAEs) in the sense of Boillat–Lychagin. The peculiarity of such equations is

that their characteristic directions form a particularly simple geometric object, dubbed “characteristic cone”, which contains

all relevant information about the equation itself. In a sense specified in the paper, they are “reconstructable” from theircharacteristics. We observe that, in the case of a decomposable two–form, the corresponding equation, referred to as a

Goursat–type MAE, can be described as the set of Lagrangian planes which are non–transversal to a three–dimensional

sub–distribution of the second–order contact structure. We show that there can be up to three such sub–distributionsdetermining the same equation, and that these distributions are “orthogonal” each other in a generalized sense, namely

with respect to the Levi form of the contact distribution, called here “meta–symplectic structure”. We finally characterise

intermediate integrals of Goursat–type MAEs in terms of their characteristics.

Contents

1. Introduction 21.1. The characteristic cone and Monge-Ampere equations 21.2. Structure of the paper 3Conventions 32. Preliminaries on (prolongations of) contact manifolds, and (meta)symplectic structures 42.1. Contact manifolds, their prolongations and PDEs 42.2. The meta–symplectic structure on C1 73. Description of the main results 74. Vertical geometry of contact manifolds and of their prolongations 94.1. Vertical geometry of M (k) and three–fold orthogonality in M (1) 94.2. Rank–one lines, characteristic directions and characteristic hyperplanes 104.3. Canonical directions associated with orthogonal distributions 115. Characteristics of 3rd order PDEs 115.1. Characteristics of a 3rd order PDE and relationship with its symbol 125.2. The irreducible component VEI of the characteristic cone of a 3rd order PDE E 135.3. The characteristic variety as a covering of the characteristic cone 146. Proof of the main results 146.1. Reconstruction of PDEs by means of their characteristics 146.2. Proof of Theorem 2 166.3. Proof of Theorem 3 176.3.1. Proof of the sufficient part of Theorem 3 186.3.2. Proof of the necessary part of Theorem 3 206.4. Proof of Theorem 4 216.4.1. Proof of the statement 1 of Theorem 4 226.4.2. Proof of the statement 2 of Theorem 4 256.4.3. Proof of the statement 3 of Theorem 4 257. Intermediate integrals of Goursat–type 3rd order MAEs 268. Perspectives 27Inequivalence of rank–one vectors 28Multiplicity of distributions 28Rank–one lines in Dv 28Conclusions 28

Date: March 14, 2014.1991 Mathematics Subject Classification. 53D10, 35A30, 58A30, 14M15.Key words and phrases. Hypersurfaces in Lagrangian Grassmannians, Third–order PDEs, Prolongations of contact manifolds, Levi form of

a distribution, Characteristics of PDEs.The first author was partially supported by the grant “Finanziamento Giovani Studiosi”. The second author is thankful to the Grant Agency

of the Czech Republic (GA CR) for financial support under the project P201/12/G028.

1

2 GIANNI MANNO AND GIOVANNI MORENO

SOFFITTA 29References 29

1. Introduction

Classical Monge-Ampere equations (MAEs with one unknown function and two independent variables) constitute adistinguished class of scalar 2nd order (nonlinear) PDEs owing to two remarkable properties: first, they can be, in a sense,reconstructed from their characteristics [2] and, second, their so–called “characteristic cone”1 degenerates into the unionof zero, one or two 2D planes (according to the elliptic, parabolic or hyperbolic character of the equation). By contrast,the characteristic cone of multidimensional MAEs, which are defined by a linear combination of the minors of an Hessianmatrix equated to zero, is the union of n–dimensional linear subspaces only in special cases, dubbed “Goursat–type”[2, 13]. Thus, in the classical, i.e., two–dimensional, case the general MAEs and the MAEs of Goursat type form a singleclass.

The primary motivation of this paper was to determine whether, and to what extent, such a phenomenon occurs alsoin the context of 3rd order PDEs. Indeed, the above notion of “characteristic cone” is not exclusive to 2nd order PDEs(though it is far more exploited in this context), pertaining to PDEs of arbitrary order and number of (in)dependentvariables. This very introduction is the appropriate place to acquaint the reader with such a notion, which it is going tobe the main gadget for our analysis. Putting off an intrinsic geometric definition (see (9) and (78) later on), we deemedit convenient to introduce it here in a friendly coordinate way.

1.1. The characteristic cone and Monge-Ampere equations. Let

(1) E : F (x1, x2, u, p1, p2, . . . , pi1···il , . . . ) = 0 , l ≤ k,

be a scalar kth order PDE with one unknown function u = u(x1, x2) and two independent variables (x1, x2). As usual(see, e.g., [17, 18] and references therein) in the geometric theory of PDEs, the variables pi1···il correspond to the partial

derivatives ∂lu∂xi1 ···∂xil

, with the indices i1 ≤ i2 ≤ · · · ≤ il ranging in 1, 2.A Cauchy problem is obtained by complementing (1) with some initial data

(2) f(X1(t), X2(t)

)= U(t) ,

∂`f

∂z`(X1(t), X2(t)

)= Q`(t), ` ≤ k − 1,

where

(3) X1(t), X2(t), U(t), Q`(t)

are given functions, and ∂∂z is the derivative along the normal direction2 of the curve (X1(t), X2(t)). A solution of the

Cauchy problem (1)–(2) is a function u = f(x1, x2) which, together with its derivatives, satisfies both (1) and (2).In a more geometric perspective, the initial data (3) can be used to construct a particular curve

(4) Φ(t) = (X1(t), X2(t), U(t), P1(t), P2(t), . . . , Pi1···il(t), . . . ), l ≤ k − 1,

in the space with coordinates (x1, x2, u, . . . , pi1···il , . . . ), l ≤ k − 1, which we can then interpret as a Cauchy datum [23].If such curve is non–characteristic for equation (1), then, assuming F real analytic, Cauchy problem (1)–(2) admits a(locally) unique analytic solution. Hence, characteristic curves play a crucial role in the analysis of Cauchy problems.They can be defined as follows.

A curve (4) is characteristic for E at the point3 mk−1 ∈ E , i.e., a point

(5) mk−1 = (x1, x2, u, . . . , pi1···il , . . . ), l ≤ k,

whose coordinates satisfy (1), if there exists a tangent vector ν = Φ(t0) at the point

(6) mk−2 = Φ(t0) = (x1, x2, u, . . . , pi1···il , . . . ), l ≤ k − 1,

such that

(7)∑

`1+`2=k

(−1)`1∂F

∂p1 · · · 1︸︷︷︸`1

2 · · · 2︸︷︷︸`2

∣∣∣∣∣mk−1

(ν2)`1(ν1)`2 = 0,

1This definition is used here for the first time, though the notion behind it is rather old, and may have appeared in other guises someplace

else. For instance, it is called “fold–type singularity equation” in [27].2We adopted the same notation used in [28], where the reader may also find a gentle introduction to the theory of characteristics and

singularities of nonlinear PDEs.3The choice of notation “mk−2” and “mk−1” will be motivated later on.

GEOMETRY OF THIRD–ORDER EQUATIONS OF MONGE–AMPERE TYPE 3

where

(8) ν = ν1

∂x1 + p1∂u +∑

1≤i2≤···≤ih≤2

h≤k

p1i2···ih∂pi2···ih

+ ν2

∂x2 + p2∂u +∑

1≤i2≤···≤ih≤2

h≤k

p2i2···ih∂pi2···ih

.

From equation (7) it is clear that one can associate with any point (5) of E a number ≤ k of directions (8) in the spacewith coordinates (x1, x2, u, . . . , pi1···il , . . . ), l ≤ k − 1. So, if we keep the point (6) fixed and let the point (5) vary in E ,the aforementioned directions form, in general, a number ≤ k of cones (polynomial (7) might possess multiple and/orimaginary roots). We call such set of directions (see Fig. 1) the characteristic cone of E at mk−2 and denote it by

(9) VEmk−2 := Directions ν as in (8) | ∃mk−1 ∈ E such that (ν1, ν2) satisfies (7).Intuitive definition of the characteristic cone of E at mk−2.

In this paper we deal with the interesting issue of studying a PDE E through the above defined characteristic coneVE : such a matter has been given a relatively marginal attention and its potential is not yet fully understood. It shouldbe stressed, however, that E cannot be completely replaced by VE in all contexts, having a profoundly different nature.Nevertheless, the study of VE can be very convenient, for instance, in facing classification problems. For example, the only(bidimensional) 2nd order PDEs E whose VE degenerates into two 2D planes are the classical MAEs (see, e.g., [4, 3, 21]),so that the problem of classifying (classical) MAEs entails classifying 2D certain distributions on a 5D contact manifold[4]. A similar result holds true in the multidimensional case, but only for the class of MAEs introduced by Goursat [13].

The step ahead we are going to take here is the following. Basically, we study those PDEs that correspond to 3Ddistributions on the 8D first prolongation of a 5D contact manifold, borrowing the key ideas from the above works. Inspite on the evident methodological analogy with the 2nd order case, the 3rd order one will display a few unexpectedpeculiarities, mostly due to the richer structure of prolonged contact manifolds. It should be underlined that, as abyproduct of this study, we obtain an invariant definition of 3rd order bidimensional MAEs which, unlike 2nd order MAEs,intensively studied since the second half of XIX century by many mathematicians such as Darboux, Lie, Goursat and, morerecently, by Lychagin [19], Morimoto [24] and their school in the context of contact and symplectic geometry, have beendefined just recently by Boillat [7] as the only PDEs which satisfies the property of complete exceptionality in the sense ofLax [20]. It seems that a systematic analysis of such PDEs in a differential–geometric context is still lacking as, up to now,no serious effort has been made to extend the classical theory to the case of 3rd order MAEs, is spite of their importance[1, 10, 12, 26]. Much as in the geometric approach to classical MAEs it is conventient to exploit the contact/symplecticgeometry underlying 2nd order PDEs, to deal with 3rd order MAEs we shall make use of the prolongation of a contactmanifold, equipped with its Levi form (see, e.g., [25], Section 2), a structure known as meta–symplectic, quickly reviewedbelow.

Classical MAEs ↔ contact manifolds and symplectic geometry

3rd order MAEs ↔ prolongation of contact manifolds and meta–symplectic geometry

1.2. Structure of the paper. The exposition is pivoted around the central Section 6, where we prove the main results,Theorems 2, 3 and 4, whose statements are given earlier in Section 3. As their formulation requires simple but non–standard generalizations of some basic tools of contact/symplectic geometry, we added the preliminary Section 2 withthe idea of introducing the three theorems above in the most self–contained way. Moreover, between their statement andthe proof, we slotted two more sections to accommodate some preparatory results. More precisely, in Section 4, afterreviewing the standard notions of rank–one vectors and characteristics covectors, we introduce the key notion of “three–fold orthogonality” and study its main properties, while in the subsequent Section 5 we go deeper into the analysis of themain gadget of our study, the above defined characteristic cone (9), with a particular emphasis on its relationship withthe more widely known characteristics variety. In Section 7 we study the intermediate integrals of Goursat–type 3rd orderMAEs in terms of their characteristics, thus providing a tangible application of the main results. Their full applicativerange, however, goes beyond the scope of this paper, and we could only give it a quick try in the concluding Section 8.

Conventions. We make an extensive usage of the symbol “P” in order to avoid too many repetitions of the sentence“up to a conformal factor”. Nevertheless, if this latter is born in mind, all projective bundles can be replaced by theirlinear counterparts (taking care of increasing all dimensions by 1). One–dimensional linear object may be identified withtheir generators. As a rule, if P → X is a bundle, and x ∈ X, we denote by Px the fiber of it at x. Symbol D′ denotesthe derived distribution [D,D] of a distribution D. If there is any vertical distribution in the surrounding manifold, Dvdenotes the intersection of a distribution D with the vertical one. Differential forms on a manifold N (resp., distributionD) are denoted by Λ∗N (resp., Λ∗D∗) and f∗ is the tangent map of f : N → Q. If T is a tensor on N , or a distribution,we sometimes skip the index “n” in Tn, if it is clear from the context that T has been evaluated in n ∈ N . As a rule,we use the term “line” for contravariant quantities, and “direction” for covariant ones, and sometimes we gloss over the


E := F = 0

mk−11

mk−12

mk−13

mk−2

VEmk−2

ν

Lmk−11

Lmk−11

Lmk−12

Lmk−12

Lmk−13

L E

Figure 1. If coordinates (x1, x2, u, . . . , pi1···il , . . . ), with l ≤ k− 1, corresponding to the red linear space,are held fixed, then a PDE E reduces an hypersurface in the (. . . , pi1···il , . . .)–space, with l = k, depictedas a grey smooth object inside a blue linear container. A tangent direction ν to the red space can becoupled with the differential of F at a point mk−1 ∈ E via (7): those yielding zero form the cone VE

mk−2 ,the number and multiplicity of whose sheets are determined by the degree k polynomial appearing at theleft–hand side of (7).

distinction between “the symbol of E = F = 0” and “the symbol of E”. The symmetric tensor product is denoted by .The Einstein summation convention will be used, unless otherwise specified. Also, when a pair jk (resp., a triple ijk) runsin a summation, such summation is performed over 1 ≤ j ≤ k ≤ 2 (resp., 1 ≤ i ≤ j ≤ k ≤ 2), unless stated differently.See also Table 1. Throughout the paper, by M (0), C0, and m(0) we shall always mean M , C, and m, respectively.

2. Preliminaries on (prolongations of) contact manifolds, and (meta)symplectic structures

2.1. Contact manifolds, their prolongations and PDEs. We introduce here a minimal set of notions needed to laydown an invariant coordinate–free definition of the characteristic cone, superseding (9). Throughout this paper, (M, C)


(M, C) a 5D contact manifoldθ the contact form on M

L(Cm) the Lagrangian Grassmannian of CmM (1) π−→M the Lagrangian bundle

Ω the meta–symplectic form on M (1)

M (2) π2,1−→M (1) the meta–symplectic Lagrangian bundleW (1) 1st prolongation of W

Λ∗M (k) differential forms on M (k)

L −→M (k) tautological bundles

M (2) compactification of M (2)

x1, x2, u, pi, pij , pijk coordinates on M (2)

C = 〈D1, D2, ∂p1 , ∂p2〉 the contact distribution on Mξi basis for Lm2

dxi basis for L∗m2

ν = (ν1, ν2) = νiξi a tangent vector to M (1)

H a hyperplane (i.e., line) in Lmk

E −→M (1) a 3rd order PDEEm1 fiber of E

VM (k), V E vertical bundlesVM the relative vertical bundle of M

char RE characteristic variety of EVE the characteristic cone of EVEI an irreducible component of VEVEII the closure of of VE r VEIEω Boillat–type PDE determined by ωD a 3D sub–distribution of C1

Dv the vertical part of DD a 2D sub–distribution of C1

ED the Goursat–type PDE determined by DGr a Grassmannian manifold

Table 1: List of main symbols.

will be a 5D contact manifold. In particular, C is a completely non–integrable distribution of hyperplanes (i.e., 4Dtangent subspaces) on M locally described as C = ker θ, where the 1–form θ is determined up to a conformal factor andθ∧dθ∧dθ 6= 0. The restriction dθ|C defines on each hyperplane Cm, m ∈M , a conformal symplectic structure: Lagrangian(i.e., maximally dθ–isotropic) planes of Cm are tangent to maximal integral submanifolds of C and, as such, their dimensionis 2. We denote by L(Cm) the Grassmannian of Lagrangian planes of Cm and by

(10) π : M (1) :=∐m∈M

L(Cm)→M

the bundle of Lagrangian planes, also known as the 1st prolongation of M . The key property of the manifold M (1) is thatit is naturally endowed with a 5D distribution, defined by

(11) C1m1 := ν ∈ Tm1M (1) | π∗(ν) ∈ Lm1,

where Lm1 ≡ m1 is a point of M (1) considered as a Lagrangian plane in Cm.Let us denote by θ(1) the distribution of 1-forms on M (1) vanishing on C1. Then, by definition, a Lagrangian plane of M (1)

is a 2D subspace which is π–horizontal and such that both distributions of forms θ(1) and dθ(1) vanish on it. In analogywith (10), we define the 2nd prolongation M (2) of a contact manifold (M, C) as the first prolongation of M (1), that is

(12) M (2) = (M (1))(1) := Lagrangian planes of M (1) .

Observe that, even if the operation of prolongation (10) is formally identical to (12), it lacks the requirement of horizontality,due to the absence in M of a fibred structure. Nevertheless, projection (10) is the beginning of a tower of natural bundles

(13) M (2) π2,1−→M (1) π−→M,

which, for the present purposes, will be exploited only up to its 2nd term. It is well known that π2,1 is an affine bundle(see, e.g., [16]).


On the other hand, if horizontality is dropped in (12), one augments M (2) with the so–called “singular” integral elementsof C1 (which, in the case of PDEs, formalize the notion of “singular solutions”: see, e.g., [22, 27, 28]) thus obtaining a

projective bundle M (2), which can be thought of as the “compactification” of M (2). Such a compactification is defined bythe same formula (12), but without the horizontality condition on Lagrangian planes.Previous constructions lead naturally to consider a geometrical object which will play a prominent role in our analysis,namely the so–called tautological bundle over M (i)

(14) L→M (i),

where the fiber Lmi is mi itself, understood as a 2D subspace of Ci−1, with i = 1, 2. We keep the same symbol L for boththe tautological bundles over M (1) and M (2), since it will be always clear from the context which is which.

Now we focus on the local description of the geometrical objects just introduced. There exist coordinates (xi, u, pi) on M ,i = 1, 2 such that θ = du− pidxi, called contact coordinates, such that C is described by

(15) C = 〈D1, D2, ∂p1 , ∂p2〉 ,

where Di is the total derivative with respect to xi, truncated to the 0th order. A system of contact coordinates (xi, u, pi)induces coordinates

(16) (xi, u, pi, pij = pji, 1 ≤ i, j ≤ 2)

on M (1) as follows: a point m1 ≡ Lm1 ∈ M (1) has coordinates (16) iff m = π(m1) = (xi, u, pi) and the correspondingLagrangian plane Lm1 is given by:

(17) Lm1 = 〈Di + pij∂pj | i ∈ 1, 2〉 ⊂ Cm.

Similar reasonings lead to the following local descriptions

(18) C1 = 〈D1, D2, ∂p11 , ∂p12 , ∂p22〉 ,

where now Di stand for total derivatives truncated to the 1st order and a point m2 ∈M (2) has coordinates (xi, u, pi, pij =pji, pijk = pikj = pjik = pjki = pkij = pkji) if the corresponding Lagrangian plane is given by

(19) Lm2 =⟨Di + pijk∂pjk | i ∈ 1, 2

⟩⊆ C1

m1 ,

where the vector fields Di and ∂pij are tacitly assumed to be evaluated at m1.

Remark 1. We always use the symbol Di for the truncated total derivative with respect to xi, i = 1, 2, the order oftruncation depending on the context. For instance, the order of truncation is 0 in (15) and it is 1 in (18) and (19). It isconvenient to set

(20) ξi := Di|mk , k = 1, 2,

where the total derivatives appearing in (20) are truncated to the (k − 1)st order. Indeed, both (17) and (19) simplify as

(21) Lmk = 〈ξ1, ξ2〉 , k = 1, 2,

and their dual as L∗mk =⟨dx1, dx2

⟩, respectively.

A generic point of M (resp., M (1), M (2)) is denoted by m (resp., m1, m2). As a rule, when both m1 and m2 (resp. mand m1) appear in the same context, the former is always the π2,1–image (resp., the π–image) of the latter.In compliance with the geometric understanding of PDEs in the context of jet spaces (see (22) below), by a kth orderPDE we always mean a sub–bundle E ⊆ M (k−1) of codimension one whose fiber Emk−2 at mk−2 is henceforth assumed,

without loss of generality, to be connected and with compact closure in M (k−1). We retain the same symbol L for thetautological bundle L|E −→ E restricted to E .

We conclude this section by observing that, locally, M (k) is the k+ 1st jet–extension of the trivial bundle R2×R→ R2,so that the reader more at ease with jet formalism may perform the substitution

(22) M (k) ←→ Jk+1(2, 1)

which does not affect the logic of this paper, with i ≥ 0.

By prolongation of a subspace W ⊆ TmkMk we mean the subset W (1) ⊆ M(k+1)

mk made of the Lagrangian planes

which are tangent to W . The prolongation of a submanifold W ⊆ Mk is the sub–bundle of W(1) ⊆ M (k+1) defined by

W(1)

mk := (TmkW)(1), mk ∈M (k). The prolongation to M (k) of a local contactomorphism ψ of M is denoted by ψ(k).


2.2. The meta–symplectic structure on C1. We point out that the canonical bundle epimorphism θglob : TM −→ TMC

can be regarded as a “global” analog of the local form θ defining C, in the sense that C = ker θglob everywhere. However,

even though TMC is a rank–one bundle, it is, in general, nontrivial, so that, strictly speaking, θglob cannot be considered

as a 1–form. Similarly, the (local) conformal symplectic structure ω = dθ|C admits a “global” analog, namely

(23) ωglob : C ∧ C −→ C′

C.

Indeed, using the local coordinates (15), one immediately sees that the derived distribution C′ is spanned by C and ∂u, so

that the quotient [C,C]C = 〈∂u〉 is again rank–one, and ωglob identifies with ω.

One of the main gadgets of our analysis is the co–restriction to (C1)′ of the Levi form of C1, firstly investigated by V.Lychagin [18] as a “twisted” analog of the symplectic form (23) for the prolonged contact distribution C1 and called, forthis reason, meta–symplectic:

(24) Ω : C1 ∧ C1 −→ (C1)′

C1.

Meta–symplectic form on C1.

Notice that, unlike (23), the form (24) takes its values into a rank–two bundle so that, even locally, it cannot be regardedas a 2–form in the standard sense. Nevertheless, it can be used in place of dθ(1) for defining Lagrangian subspaces: indeed,a 2D π2,1–horizontal subspace of C1

m1 is Lagrangian if it is Ω–isotropic.

Using the local coordinates (18), it is easy to realize that (C1)′ is spanned by C1 and 〈∂p1 , ∂p2〉, so that the quotient (C1)′

C1identifies with the latter, and (24) reads

(25) Ω = dpij ∧ dxi ⊗ ∂pj .

As a vector–valued differential 2–form, Ω can be identified with the pair Ω ≡ (−dθ1,−dθ2) where θi = dpi − pijdxj are

the local 1–forms defining C1, i.e., the generators of θ(1) ⊂ Λ1M (1). An intuitive definition of Ω in the context of kth orderjet spaces can be found in Section 3 of [5].

3. Description of the main results

In the case of a 3rd order PDE E , polynomial (7) always admits a linear factor, and this fact reflects on the correspondingcharacteristic cone VE , which always contains an irreducible component henceforth denoted by VEI , as opposed to theclassical MAEs, where such a component exists only for non–elliptic equations. In [8] Boillat proposed a 3rd ordergeneralization E = F = 0 of a MAE by taking F of the form

(26) F = det

p111 p112 p122

p112 p122 p222

A

+ B · (p111, p112, p122, p222)T + C,

where A (resp., B, C) is an R3–(resp., R4–, R–)valued smooth function on M (1). The right–hand side of (26) is nothingbut a linear combination of the minors of the so-called Hankel matrix(

p111 p112 p122

p112 p122 p222

).

Let us stress that (26) locally describes the intrinsically defined hypersurface of M (2)

(27) Eω := m2 ∈M (2) | ω|Lm2 = 0Geometric definition of Boillat–type 3rd order MAEs.

associated to a 2–form ω ∈ Λ2M (1).The first result of this paper is concerned with the characteristic cone VE of the equations Eω. Namely, we show

that VE can be used to reconstruct the equation itself, since there is an obvious way to “invert” the construction of thecharacteristic cone (9) out of a PDE E : take any sub–bundle V ⊆ PC1 and associate with it the following subset

(28) EV := m2 ∈M (2) | ∃H ∈ V : Lm2 ⊃ H ⊆M (2) .

Now, if V is regular enough, the corresponding EV turns out to be a genuine 3rd order PDE, and examples of “regularenough” sub–bundles are provided by characteristic cones of 3rd order PDEs themselves. In particular, it always holdsthe inclusion EVE ⊇ E and we call reconstructable (from its characteristics) an equation E such that

(29) E = EVE .

Theorem 2. Any 3rd order MAE E of Boillat–type is reconstructable from its characteristics, i.e., E = EVE .


The simplest examples of equations (28) are obtained when V = PD, where D ⊆ C1 is a 3D sub–distribution. Evenif Goursat himself never spoke explicitly of 3rd order MAEs, we dub after him these equations to honor his idea tocharacterize multi–dimensional 2nd order MAE through linear objects involving only 1st order derivatives. In our case,the role of such a linear object is played by D, and (28) reads

(30) ED := EPD = m2 ∈M (2) | Lm2 ∩ D 6= 0,Geometric definition of Goursat–type 3rd order MAEs.

which, locally, means that E can be brought either to a quasi–linear equation, or to the form E = F = 0 with

(31) F = det

p111 − f111 p112 − f112 p122 − f122

p112 − f112 p122 − f122 p222 − f222

A

,

where A is as above and fijk ∈ C∞(M (1)). Furthermore, equations (30) form the sub–class of the equations (27) whichare determined by 2–forms which are decomposable modulo the differential ideal generated by contact forms (see thebeginning of Section 6.3).

The second result of this paper allows to characterizes Goursat–type 3rd order MAEs in terms of their characteristiccone.

Theorem 3. Let E be a 3rd order PDE. Then E is locally a 3rd order MAE of Goursat–type (i.e., E is either quasi–linearor of the form E = F = 0, with F given by (31)), if and only if its characteristic cone VE contains an irriduciblecomponent VEI which is a 2D linear projective sub–bundle.

Theorem 3 implies that E ⊆M (2) is a Goursat–type 3rd order MAE if and only if its characteristic cone decomposes as

(32) VE = PD1 ∪ VEII,

where D1 ⊆ C1 is a 3D sub–distribution and VEII encompasses all the remaining irreducible components of VE . Surprisinglyenough, if (32) holds, then all the information about E is encapsulated in D1. For instance, if VEII is in its turn reducible,then its components PD2 and PD3 are linear as well and can be unambiguously characterized by PD1 through the formula

(33) Ω(h1,Dvi ) = Ω(hi,Dv1) = 1D space,

where hi is a nonzero horizontal vector in Di, i = 1, 2, 3.The third result of this paper finalizes our characterization of Goursat–type 3rd order MAEs through their character-

istics.

Theorem 4. Let E = EVE be a 3rd order Goursat–type MAE, i.e., such that (32) is satisfied. Then E = ED1, with

dimDv1 = 1, 2, and

(1) E is quasi–linear if and only if dimDv1 = 2: in this case, VEII is either empty or equal to PD2 ∪ PD3, where Di isunambiguously defined by (33) and EDi

= ED1, i = 2, 3;

(2) E is fully nonlinear if and only if dimDv1 = 1: in this case, if the projective sub–bundle VEII is not empty, then itcannot contain any linear irreducible component and E = EVEII ;

(3) if E = ED2, then either D1 = D2 or D1 and D2 are “orthogonal” in the sense of (33).

As a main byproduct of Theorem 4 we shall obtain a method for finding intermediate integrals, discussed in Section 7,while more consequences, mainly concerning classification issues, will be collected in the final Section 8. Even without athorough grasp of the above results, the reader may appreciate Example 5 below, where a hands–on computation revealsthe distributions behind some very simple equations, which will be recalled in the key moments of this paper.

Example 5. Let E be as in (1), with F given by p111 − p112 − 2p122 (resp., p122 and p111). Then equation (7) reads

(34) (ν2)3 + (ν2)2ν1 − 2ν2(ν1)2 = 0 (resp., ν2(ν1)2 = 0 and (ν2)3 = 0).

Applying definition (9), one easily obtains that VE = ∪3i=1PDi, with

D1 = 〈D1 , ∂p11 + ∂p12 , 2∂p11 + ∂p22〉 ,(35)

D2 = 〈D1 +D2 , 2∂p11 + ∂p12 , ∂p22〉 ,(36)

D3 = 〈D1 − 2D2 , ∂p11 − ∂p12 , ∂p22〉(37)

(resp.,

(38) D1 = 〈D1 , ∂p11 , ∂p12〉 , D2 = D3 = 〈D2 , ∂p11 , ∂p22〉

and

D1 = D2 = D3 = 〈D1 , ∂p12 , ∂p22〉).


Observe that, in the first case, equation (34) splits as

(39) ν2(ν2 − ν1)(ν2 + 2ν1) = 0,

and that the “horizontal components” hi (i.e., the first generators) of (35), (36) and (37) are precisely the three distinctroots of (39). Notice also that, in general, the distributions associated with the same quasi–linear 3rd order MAE need notto be contactomorphic. Here it follows an easy counterexample. For instance, consider the equation p122 = 0 above: itscharacteristic cone consists of the two distributions D1 and D2 = D3 (see (38)), which are not contactomorphic. Indeed,the derived distribution D′1 := D1 + [D1,D1] of D1 is 5-dimensional, whereas dim(D′2) = 4.

4. Vertical geometry of contact manifolds and of their prolongations

The departing point of our analysis of 3rd order PDEs is to identify them with sub–bundles of the 1st prolongationM (2) = (M (1))(1) of M (1). Hence, a key role will be played by their vertical geometry, i.e., the 1st order approximationof their bundle structure, and, in particular, by the so–called rank–one vectors, which are in turn linked to the notion ofcharacteristics. In order to introduce these concept, we being with the vertical geometry of the surrounding bunlde, i.e.,M (2) itself.

4.1. Vertical geometry of M (k) and three–fold orthogonality in M (1). For k ∈ 1, 2, we define the vertical bundleover M (k) as follows:

VM (k) :=∐

mk∈M(k)

TmkM(k)

mk−1 .

As regard to the contact manifold M , i.e. the case k = 0, even if it does not possess a naturally defined “vertical bundle”,it is still equipped with the relative vertical bundle

(40) VM :=∐

m1∈M(1)

Cπ(m1)

Lm1

.

Note that the bundle VM is, in fact, a bundle over M (1) with 2D fibers Vm1M .

Lemma 6. It holds the following canonical isomorphism:

(41) VM (k) ' Sk+1L∗, k ≥ 0.

Proof. A “fibered”, i.e., jet–theoretic version of the proof is provided, e.g., by Theorem 3.2 of [6].

Directly from the definition (40) of VM and the isomorphism (41) for k = 0 one obtains

(42)Cπ(m1)

Lm1

∼= L∗m1 .

Example 7. Of particular importance will be the vertical vectors on M (2) which correspond to perfect cubes of covectorson L via the fundamental isomorphism (41). For instance, if

(43) α := αidxi ∈ L∗m2

(see Remark 1), then the corresponding vertical vector is

(44) S3L∗m2 3 α⊗3 ←→∑i+j=3

αi1αj2

∂

∂p1 · · · 1︸︷︷︸i

2 · · · 2︸︷︷︸j

∣∣∣∣∣m2

∈ Vm2M (2).

Even if the sections of VM are not, strictly speaking, vector fields, and a such they lack an immediate geometricinterpretation, the bundle VM itself has important relationships with the contact bundle C1. Namely, on one hand, it iscanonically embedded into the module of 1–forms on C1 (see (48) later on) and, on the other hand, it is identified with

the quotient distribution (C1)′

C1 (Lemma 8 below).

Lemma 8. There is a (conformal4) natural isomorphism

(45)(C1)′

C1∼= L∗.

Proof. It follows from a natural isomorphsim between the left–hand sides of (42) and (45). Indeed, the map

(46)(C1)′m1

C1m1

−→Cπ(m1)

Lm1

induced from π∗ is well–defined and linear, for all m1 ∈ M (1). In other words, (46) is a bundle morphism, and a directcoordinate approach shows that it is in fact an isomorphism.

4In the sense that (45) is canonical only up to a conformal factor.


We stress that there is no canonical way to project the bundle C1 over the tautological bundle L, if both are understoodas bundles over M (2). Nevertheless, if C1 and L are regarded as bundles over M (1), then π∗ turns out to be a bundleepimorphism from C1 to L, i.e.,

(47) π∗ : C1 −→ L = π∗(C1).

Dually, epimorphism (47) leads to the bundle embedding L∗ → C1 ∗ which can be combined with the identificationL∗ ∼= VM . The result is a (conformal) embedding of bundles over M (1),

(48) VM → C1 ∗ ,

which will be useful in the sequel. In local coordinates, (48) reads ∂pi |m 7−→ dm1xi, i = 1, 2.Now we are in position to define the concept of orthogonality in the meta–symplectic context (that has apparently neverbeen observed before), which generalizes the “symplectic orthogonality” within the contact distribution C of M . Indeed,

an immediate consequence of Lemma 8 is that the meta–symplectic form Ω is L∗–valued, i.e., C1 ∧C1 Ω−→ L∗ is a trilinearform

(49) C1 ∧ C1 ⊗M(1) LΩ−→ C∞(M (1)).

In turn, thanks to the canonical projection (47) of C1 over L, the form (49) descends to a trilinear form C1 ∧ C1 ⊗ C1 Ω−→C∞(M (1)).

Consider for a moment a 4D linear symplectic space (V, ω). Two vectors v1, v2 ∈ V are orthogonal if ω(v1, v2) = 0 andthe orthogonal complement L⊥ of a given 2D subspace L ⊆ V is defined by the system of two equations ω(L, · ) = 0, Lbeing Lagrangian if L = L⊥. Such notions (orthogonal vectors, orthogonal complement, Lagrangian subspaces) can befound also in a 5D meta–symplectic space, but with more ramifications. For example, there can be up to two distinct, soto speak, “orthogonal complements” to a given subspace.

Definition 9 (Three–fold orthogonality). Elements X1, X2, X3 ∈ C1 are orthogonal if and only if Ω(X1, X2, X3) = 0.

Remark 10. In this perspective, condition (33) expresses precisely the fact that the pair D2,D3 is the orthogonalcomplement of the distribution D1, in the sense that

(50) Ω(Di,Dj ,Dk) = 0, i, j, k = 1, 2, 3.Observe that the notion of a Lagrangian sub–distribution of C1 is richer with respect to the case of sub–distributions ofC: first, the pair D2,D3 needs not to exist, then there is the case when D2,D3 degenerates to a singleton, and theextreme case when it is the whole triple D1,D2,D3 which degenerates.

4.2. Rank–one lines, characteristic directions and characteristic hyperplanes. Isomorphism (41) shows that

there is a canonical distinguished subset of tangent directions to M(k)

mk−1 , k ∈ 1, 2, namely those sitting in the image

of the Veronese embedding PL∗ → PSk+1L∗, i.e., the “kth powers” of sections of L∗. More geometrically, these are thetangent directions at m2 = m2(0) to the curves mk(t) such that the corresponding family Lm2(t) of Lagrangian subspaces

in M (k−1) “rotates” around a common hyperplane (which, in our case, is a line).

Definition 11. A line ` ∈ PVmkM (k) is called a rank–one line if ` =⟨α⊗(k+1)

⟩, for some 〈α〉 ∈ PL∗mk , in which case we

call α a characteristic covector and 〈α〉 a characteristic (direction) in the point mk. The subspace Hα := kerα ≤ Lmk

is called the characteristic hyperplane associated to the rank–one line `. Furthermore, if ω ∈ Ck ∗ is a vertical form, ahyperplane Hα is called characteristic for ω if ω|` = 0.

Denomination rank–one refers to the rank of the multi–linear symmetric form on L involved in the definition (see, e.g.,[10]). Observe that, in our contest, dimHα = 1, so that we shall speak of a characteristic line and call characteristicvector a generator of Hα. The geometric relationship between Hα and ` =

⟨α⊗3

⟩is well–known (see, e.g., [28]) and it can

be rendered by

(51) ` = PTmk

(H(1)α

),

where the 1st prolongation H(1)α = mk ∈M (k)

mk−1 | Lmk ⊇ Hα of Hα has dimension one.

Remark 12 (Lines are hyperplanes). Take the local expression (43) for α and use the basis (20) of Lm2 . Then Hα =ker(α1dx

1 + α2dx2) = 〈α2ξ1,−α1ξ2〉, i.e., one can safely identify α↔ Hα via a counterclockwise π

2 rotation:

(52) α ≡ (α1, α2)↔ (α2,−α1) ≡ Hα.

From now on, characteristic directions (which are covariant objects in M (1)) and characteristic lines (which are contravari-ant objects in M (1)) will be taken as synonyms, thus breaking the separation proposed in Table 2. It should be stressedthat in the multidimensional case5 it is no longer possible to regard the characteristic directions as lines lying in L, sincethe latter are not hyperplanes.

5As those treated in [2].


contravariant object covariant object

M (2) ` = Tm2H(1)α

rank–one line

M (1) Hα = kerα ∈ PLm2

characteristic hyperplane≡line〈α〉 ∈ PL∗m2 ,

characteristic direction

α ∈ L∗m2 r 0characteristic covector

Table 2: To the intuitive idea of a “characteristic line”, which formalizes an infinitesimal Cauchy datum giving rise to anill–posed Cauchy problem (see the discussion of Section 1.1), there corresponds a multitude of heterogeneous yet equivalentobjects, but such a redundancy is necessary to fit the same idea into multiple contexts.

4.3. Canonical directions associated with orthogonal distributions. It is easy to see that there are two charac-teristic lines H1 and H2 which are characteristics for a vertical form ω ∈ C1 ∗ which is decomposable by means of Lemma6. The meta–symplectic form (27) links these two lines with the vertical distribution V := kerω determined by ω.

Lemma 13. It holds the following identification

V yΩHi' Hj ⊗

L∗

Hi, i, j = 1, 2,

where “'” is meant via (48). In particular, dim V yΩHi

= 1, i = 1, 2.

Proof. We shall assume that the coefficient of dp11 is not zero: other cases can be dealt with likewise. So, being ωdecomposable, it can be brought, up to a scaling, to the form ω = dp11 − (k1 + k2)dp12 + k1k2dp22 ∈ C∗. Accordingly,Hi = ∂p2 + ki∂p1 and a straightforward computation shows that V = 〈X1, X2〉, with

(53) X1 = −k1k2∂p11 + ∂p22 , X2 = (k1 + k2)∂p11 + ∂p12 .

Then

(54) V yΩ = 〈X1yΩ, X2yΩ〉 =⟨k1k2dx

1 ⊗ ∂p1 − dx2 ⊗ ∂p2 ,−(k1 + k2)dx1 ⊗ ∂p1 − dx1 ⊗ ∂p2 − dx2 ⊗ ∂p1⟩.

If the basis Hi, ∂p1 for VM ' L∗ (see (41)) is chosen, then the factor of (54) with respect to Hi is the line

(55)⟨(kjdx

1 + dx2)⊗ ∂p1⟩, i 6= j.

Recall now that in the case of a classical, i.e., 2nd order MAE ED determined by D = 〈X1, X2〉 ⊂ C, the annihilator ofD⊥ is described by

(56) θ = 0, X1ydθ = 0, X2ydθ = 0 ,

and, moreover, the 2–form ω = X1(θ) ∧X2(θ) is such that Eω = ED (see [2]).Proposition 32 later on will generalize (56) to the case of 3rd order MAEs, where there can be up to three mutually

“orthogonal” distributions, in the sense of Definition 9. Lemma 13 is the key to achieve such a generalization since itimplies that, associated with a quasi–linear MAE, whose symbol is completely decomposable, there are three canonicallines in M .

Example 14. Let E be the first equation of Example 5. The following three lines

1st line: Ω(D1,D2) = 〈2∂p1 + ∂p2〉 ,(57)

2nd line: Ω(D1,D3) = 〈∂p1 − ∂p2〉 ,(58)

3rd line: Ω(D2,D3) = 〈∂p2〉 ,(59)

are canonically associated with the triple (D1,D2,D3), i.e., with the equation E . It is worth observing that the verticalpart Dv1 is an integrable vertical distribution on M (1), so that, in this case, lines (58) and (59) are the characteristic linesof the family of equations 2p22−p11 +p12 = k(x1, x2, u). Nevertheless, integrability Dv1 is not indispensable for associatingdirections (58) and (59) with the distribution D1, since they occur as the characteristic lines of the (not necessarily closed)vertical covector 2dp22 − dp11 + dp12, which annihilates Dv1 .

5. Characteristics of 3rd order PDEs

Now we turn our attention to the hypersurfaces in M (2) that play the role of 3rd order PDEs in our analysis.

Definition 15. A characteristic covector α ∈ L∗m2 , with m2 ∈ Em1 , is characteristic for E in m2 if the correspondingrank–one line ` is tangent to Em1 in m2, in which case 〈α〉 is a characteristic (direction) for E in m2 and the subspace Hα

is a characteristic hyperplane for E.


Formula (51) says precisely that the rank–one line ` which corresponds to⟨α⊗3

⟩via the isomorphism (41) is precisely

the (one–dimensional) tangent space to the prologation H(1)α . In this perspective, 〈α〉 is a characteristic for E at m2 if and

only if H(1)α is tangent to Em1 at m2 but, in general, H

(1)α does not need to touch Em1 in any other point.

Definition 16. If H(1)α is entirely contained into Em1 , then 〈α〉 is called a strong characteristic (direction) and Hα a

strongly characteristic line (in m2).

Example 17. Function (1) determines, for k = 2, the 3rd order PDE E = F = 0. Now equation (7) can be correctlyinterpreted as follows: it is satisfied if and only if the vector ν = νiξi given by (8) spans a characteristic line of E . Thevery same equation (7) tells also when the covector α = ν2dx1 − ν1dx2 spans a characteristic direction of E . In the lastperspective, equation (7) is nothing but the right–hand side of (44) applied to F and equated to zero.

5.1. Characteristics of a 3rd order PDE and relationship with its symbol. For any m2 ∈ Em1 we define thevertical tangent space to E at m2 as the subspace

(60) Vm2E := Tm2Em1 ≤ Vm2M (2),

called the symbol of E by many authors (herewith we prefer to use the term “symbol” only for the function determiningE). Obviously, vertical tangent spaces can be naturally assembled into a linear bundle

(61) V E :=∐m2∈E

Vm2E ,

called the vertical bundle of E . Directly from (60) it follows the bundle embedding V E ⊆ VM (2)∣∣E , where fibers of the

former are hyperplanes in the fibers of the latter. Thanks to Lemma 6, Vm2E can also be regarded as a subspace of S3L∗m2 ,and, being the identification (41) manifestly conformal, such an inclusion descends to the corresponding projective spaces,i.e.,

(62) PVm2E ⊆ PS3L∗m2 .

The dual of canonical inclusion (62) reads

(63) Ann (Vm2E) ∈ PS3Lm2 ,

where Ann (Vm2E) is the (one–dimensional) subspace(Vm2M (2)

)∗made of covectors vanishing on the hyperplane Vm2E ,

viz.,

(64) Ann (Vm2E) ≤(Vm2M (2)

)∗= (S3L∗m2)∗ = S3Lm2 .

From a global perspective, (63) is nothing but the definition of a section

(65) PS3L // E .Ann (V E)rr

Remark 18 (Symbol of a function). Let E = F = 0, and use the same coordinates (21) of Remak 1. Then

(66) Ann (Vm2E) = 〈Smblm2F 〉 ,

where Smblm2F is the symbol of F at m2, i.e.,

(67) Smblm2F =∂F

∂pijk

∣∣∣∣m2

ξiξjξk,

where ξi has been defined in (20). Observe that Ann (Vm2E) is independent of the choice of F in the ideal determined byE , so that Ann (V E) can be replaced with SmblF .

In view of (64), a direction 〈α〉 ∈ PL∗m2 is a characteristic one for E at m2 if and only if

(68) 〈Smblm2F, α⊗3〉 = 0,

where 〈 · , · 〉 is the canonical pairing on S3Lm2 . Needless to say, (68) is independent of the choice of α (resp., F )representing 〈α〉 (resp., E). Similarly, H = 〈ν〉, with ν = νiξi, is a characteristic line if and only if∑

i+j=3

(−1)i∂F

∂p1 · · · 1︸︷︷︸i

2 · · · 2︸︷︷︸j

∣∣∣∣∣m2

(ν1)i(ν2)j = 0,

which eventually clarifies formula (7).


Remark 19 (Coordinates of V E). If E = F = 0 is a 3rd order PDE according to the definition given at the end of Section2.1, then dF |Em1 is nowhere zero, for all m1 ∈M (1). So,

(69) Vm2E = ker∂F

∂pijk

∣∣∣∣m2

dpijk

is an hyperplane in the 4D space Vm2M (2).

Remark 20 (Types of 3rd order PDEs). Observe that (67) is a 3rd order homogeneous polynomial in two variables, so thatwe can introduce its discriminiant ∆ = 18abcd− 4b3d+ b2c2 − 4ac3 − 27a2d2, where a, b, c, d ∈ C∞(M (2)) are defined by

a(m2) =∂F

∂p111

∣∣∣∣m2

, b(m2) =∂F

∂p112

∣∣∣∣m2

, c(m2) =∂F

∂p122

∣∣∣∣m2

, d(m2) =∂F

∂p222

∣∣∣∣m2

.

Thus, any point m2 ∈ E can be of three different types, according to the signature of ∆, namely,

“hyperbolic”: ∆(m2) > 0 ⇔ (67) decomposes into three distinct linear factors;“parabolic”: ∆(m2) = 0 ⇔ (67) contains the square of a linear factor;

“elliptic”: ∆(m2) < 0 ⇔ (67) contains an irreducible quadratic factor.

Remark 21. Equation E = F = 0 may be called “fully parabolic” in m2 ∈ E if (67) reduces to the cube of linear factorin the point m2. We shall prove (see Proposition 31 later on) that if E is of Boillat type and fully parabolic in every point,then it is of Goursat type, i.e., E = ED, where D is unique. In this case, F = p111 + 3kp112 + 3k2p122 + k3p222, and D =⟨Dx + kDy,−2k∂p11 + ∂p12 ,−3k2∂p11 + k∂p12 + ∂p22

⟩. It is easy to realise that such equations are not all contactomorphic

each other. Indeed, direct computations show that if k = k(x, y, u) (resp., k ∈ C∞(M), k ∈ C∞(M (1))), then the derivedflag (D,D′,D′′) has generically index (3, 4, 5) (resp., (3, 4, 6), (3, 6, 8)).

5.2. The irreducible component VEI of the characteristic cone of a 3rd order PDE E. Regard L as a sub–bundleof C1 bearing in mind the diagram

(70) L //

C1

E // M (1),

Lm2 //

C1m1

E 3 m2

π2,1 // m1 ∈M (1).

Let E = F = 0 and identify Ann (V E) with SmblF as in Remark 18. As a homogeneous cubic tensor on L (see (65)),SmblF possesses a linear factor, i.e., a section

(71) PL // ESmbl IFtt

such that

(72) SmblF = Smbl IF Smbl IIF,

where Smbl IIF is a section of PS2L→ E .Plugging (72) into (68) shows that, in order to have

(73) 〈Smbl I,m2F Smbl II,m2F, α α2〉 = 〈Smbl I,m2F, α〉〈Smbl II,m2F, α2〉 = 0

it suffices that

(74) 〈Smbl I,m2F, α〉 = 0.

In other words, if (74) is satisfied, then 〈α〉 ∈ PL∗m2 is a characteristic direction (incidentally revealing that 3rd orderPDEs always possess a lot of them).

Moreover, in view of the identification of characteristic directions with characteristic lines (Remark 12), (74) shows alsothat the line Smbl I,m2F ∈ PLm2 is always a characteristic line for E , inasmuch as there always is an α such that (74) issatisfied.

It is clear now how diagram (70) can be made use of and how to define, much as we did in Section 4.2, a subset VEI ⊆ VEfitting into the commutative diagram

(75) Smbl IF //

VEI

E // M (1),

parallel to (79). This gives a solid background to the statement of Theorem 3, where VEI was mentioned without exhaustiveexplanations.


5.3. The characteristic variety as a covering of the characteristic cone. Now we can come back to the directionν defined by (8) and notice that it lies in Lm2 , and also provide an intrinsic way to check whether ν is characteristicor not. Indeed, in view of the fundamental isomorphism (41), one can regard the cube ν⊗3 of ν as a tangent vector to

the fiber M(2)

m2 , and equation (7) tells precisely when ν⊗3 belongs to the sub–space Vm2E = Tm2Em1 (see (60)), up to aline–hyperplane duality (see also Remark 12). In other words, the set of characteristic lines

(76) char Rm2E := H ∈ PL∗m2 | H is a characteristic hyperplane of E ⊆ PL∗m2

is the projective sub–variety in PL∗m2 cut out by (7). Fig. 1 displays in green the characteristic lines, inscribed into thecorresponding fibers of the tautological bundle.

Since we plan to carry out an analysis of certain PDEs via their characteristics, it seems natural to consider allcharacteristic lines at once, i.e., as a unique geometric object. Traditionally, one way to accomplish this is to take thedisjoint union

(77) char RE :=∐m2∈E

char Rm2E ,

known as the characteristic variety of E (see, e.g., [9]). As a bundle over E , the family of the fibers of char RE coincides

with the equation E itself. So, the bundle char RE , in spite of its importance for the study of a given equation E , cannotbe used to define E as an object pertaining to M (1). Still this can be arranged: it suffices “to project everything one stepdown”, so to speak. The result is precisely the characteristic cone VE :

(78) Projective sub–bundle of PL→M (2)

Characteristic variety char RE

(π2,1)∗−→ Projective sub–bundle of PC1 →M (1).Characteristic cone VE

As revealed by the choreography of Fig. 1, some characteristic lines, which are distinct entities in char RE , may collapseinto VE (e.g., when the corresponding fibers of the tautological bundle have nonzero intersection): hence, VE it is not themost appropriate environment for the study of characteristics. On the other hand, VE is a bundle over M (1), where thereis no trace of the original equation E : as such, it may effectively replace the equation itself and serves as a source of itsinvariants.

Now we can give a rigorous definition of our main tool, the characteristic cone of E . Indeed, thanks to Remark 12, thecharacteristic variety char RE can be regarded as a sub–bundle of PL −→ E . In turn, this makes it possible to use the

commutative diagram (70) to map char RE to PC1 and define the sub–bundle VE ⊆ PC1 as the image of such a mapping.

Definition 22 (Main). Above defined bundle VE −→M (1) is the characteristic cone of E.

By its definition, VE fits into the commutative diagram

(79) char RE // //

VE

E // M (1),

revealing that, in a sense, VE is covered by char RE (see again Fig. 1).It is worth observing that, from a local perspective, (9), (78), and Definition 22 all define the same object. For

instance, by Definition 22, a line H = 〈ν〉 ∈ PC1m1 belongs to VEm1 if and only if there is a point m2 ∈ Em1 such that H

is a characteristic line for E at the point m2. But Example 17 shows that this is the case if and only if the generatorν = (ν1, ν2) of H satisfies equation (7).

6. Proof of the main results

Now we are in position to prove the main results, Theorem 2, concerned with the structure of Boillat–type 3rd orderMAEs, Theorem 3 and Theorem 4, concerned with the structure of Goursat–type 3rd order MAEs. Since the former arespecial instances of the latter, we prefer to clarify their relationship beforehand.

6.1. Reconstruction of PDEs by means of their characteristics. In spite of its early introduction (9), the mainobject of our interest has been defined again in a less direct way, which passes through the characteristic variety (see

above Definition 22). The reason behind this choice is revealed by diagram (79): the characteristic variety char RE playsthe role of a “minimal covering object” for both the equation E itself and its characteristic cone VE .

This situation is common to various area of modern Mathematics and goes under different names, depending on thecontext (Backlund transformation, double fibration transform, Penrose transform, etc.6), and it often boils down to the

6See also [14] on this concern.


introduction of a suitable incidence correspondence. In the geometric theory of PDEs it is easy to guess7 that such acorrespondence must be defined in terms of Ω–isotropic flags on M (1), in the sense of Section 2.2, i.e. by means of thecommutative diagram

(80) Fl iso2,1(C1)

%1

$$

%2

zzM (2)

%% %%

PC(1)

yyyyM (1),

where Fl iso2,1(C1) := (m2, H) ∈M (2) ×M(1) PC1 | Lm2 ⊃ H is the flag bundle of Ω–isotropic elements of C1.

The “double fibration transform” associated to the diagram (80) allows to pass from a sub–bundle of M (2) (e.g., a3rd order PDE E) to a sub–bundle of PC1 (e.g., its characteristic cone VE), and vice–versa. Lemma 23 indicates how toproceed in one direction.

Lemma 23. Let E be a PDE.

(1) Take the pre–image %−12 (E) of E and select the points of tangency with the %1–fibers: the result is char RE.

(2) Project char RE onto PC1 via %1: the result is VE .

Proof. Item 1 follows from the fact that the %1–fibers are the prolongations H(1) of hyperplanes H ∈ PC(1), and thetangency condition means that they are determined by characteristics (see (51)). Item 2 is just a paraphrase of Definition22.

Coming back is easier. Indeed, given a sub–bundle V ⊆ PC1, the same definition (28) of EV given earlier can be recastas

(81) EV = %2(%−11 (V)).

In spite of the name “double fibration transforms”, performing (81) first, and then applying Lemma 23 to the resultingequation EV , does not return, as a rule, the original sub-bundle V.

Corollary 24. Let V ⊆ PC1 be a sub–bundle. Then

(82) V ⊆ VEV

Proof. Directly from Lemma 23 and formula (81).

We stress that, for any odd–order PDE E , it holds the inclusion

(83) E ⊆ EVE .This paper begins to tackle the problem of determining those PDEs which are reconstructable from their characteristicsin the sense that (29) is valid. Lemma 25 below provides a simple evidence that, as a matter of fact, not all PDEs arereconstructable.

Lemma 25. V is made of strongly characteristic lines for EV .

Proof. Let H ∈ V be a line belonging to V. Then %2(%−11 (H)) is nothing but H(1) (see Definition 16) and (81) tells precisely

that H(1) is entirely contained into EV . Hence, H is a strongly characteristic line for EV in any point of %2(%−11 (H)).

The meaning of Theorem 3 is that 3rd order MAEs of Goursat–type are, in a sense, those PDEs whose characteristiccone takes the simplest form, namely that of a linear projective sub–bundle, and, moreover, they are also reconstructablefrom it. Observe that (30) is a particular case of (81), so that Goursat–type 3rd order MAEs constitute a remarkableexample of equations determined by a projective sub–bundles of PC1. Then Lemma 25 reveals that all the lines lying inD are strongly characteristic lines for ED, in strict analogy with the classical case [2].

Example 26. In this example we shall use a different notation from the rest of the paper. Let L = 〈x, y〉 be a 2D realvector space, L∗ = 〈ξ, η〉 its dual, and ω := ξ ∧ x + η ∧ y the canonical symplectic form on V := L ⊕ L∗. Regard S2L∗

as the open and dense subset of the Lagrangian Grassmannian L(V, ω) which is made of planes transversal to L∗, i.e.,the generic fiber of the bundle J2(2, 1)→ J1(2, 1). In this perspective, a quasi–linear 2nd order PDE in two independentvariables corresponds to a hyperplane

(84) E = q = 0 ⊆ S2L∗,

7In spite of its plainness, there is no trace of such a structure in the classical literature. Apparently, it is the second author who firstintroduced it explicitly [23].


where q ∈ (S2L∗)∗ = S2L is a symmetric tensor on L, understood as a linear map on S2L∗. The “linear” analog ofdiagram (80) reads

(85) L2,1(V, ω)

%1

$$

%2

yyL(V, ω) PV ,

where L2,1(V, ω) is the manifold of ω–isotropic flags of type (2, 1). It is possible to prove that the equation E given by (84)is the double–fibration transform of PQ ⊆ PV , where Q := Q = 0 is a quadric hypersurface in V , canonically associatedwith q, i.e.,

(86) E = EPQ,

where EPQ is the “linear” analog of (81). We sketch (see also [15]) the procedure to obtain Q ∈ S2V ∗ just by using q andω.

First, regard q as a homomorphism L∗q−→ L, and take its adjoint L

q∗−→ L∗. Second, observe that q + q∗ can be seenboth as an endomorphism of V and as an endomorphism of V ∗, so that the diagram

(87) Vq+q∗ //

ω

V

ω

V ∗

q+q∗ // V ∗

makes sense. It is easy to verify that the commutator Q := [q + q∗, ω] provides us with the sought–for quadratic form onV . In particular, to realize that Q is symmetric, just write down

(88) q ≡(a b

2b2 c

)in the above basis, and observe that

(89) Q ≡

0 0 c− a −b0 0 −b a− c

c− a −b 0 0−b a− c 0 0

.

6.2. Proof of Theorem 2. We called “Boillat–type” 3rd order MAEs the equations (27) defined by means of a 2–form,disregarding the fact that, actually, Boillat introduced them by imposing the vanishing of a certain 6 × 6 matrix ofsymmetric second derivatives of F . This condition describes the complete exceptionality of a 3rd order PDE E = F = 0and it allowed Boillat to write down the local form (26). For the sake of completeness, we propose here a coordinate–freeformulation of the object involved in Boillat’s original definition.

Remark 27. Fix m1 and observe that all members of the family Lm2m2∈M(2)

m1are canonically identified each other, being

all of them horizontal: hence, the restriction SmblF |M

(2)

m1of the section SmblF (see (65) and Remark 18) can be seen as

a map

(90) SmblF |M

(2)

m1: M

(2)m1 −→ S3Lm2 ,

for any choice of Lm2 . Take now the tangential map of (90),

(91)

(SmblF |

M(2)

m1

)∗

: Tm2M

(2)m2 −→ TSmbl m2FS

3Lm1 .

After some obvious identifications, (91) becomes a linear map S3L∗m2 −→ S3Lm2 , i.e., an element of the tensor square(S3Lm2)⊗2 which can be projected, via symmetrization, over S6Lm2 . Hence, letting m2 vary, we obtain a section Ann ′V E :M (2) −→ P(S6L) representing a sort of “vertical covariant derivative” of (65). By the same reasons explained in Remark18, we can safely identify Ann ′V E with

(92) Smbl ′F :=∂2F

∂p(ijk∂pi′j′k′)ξiξi′ξjξj′ξkξk′ .

The section Smbl ′F conformally defined by (92) above can be called the (symmetric) derivative of the symbol of E , anda direct computation (see [7, 8]) shows that E is locally of Boillat–type if and only if Smbl ′F = 0.


Fix now a point m1 ∈ M (1) and recall (see Lemma 6) that the fiber M(2)m1 is an affine space modeled over S3L∗m1 . Let

ω ∈ Λ2C1 ∗ and regard the corresponding 2–form ωm1 on C1m1 as a linear map Λ2C1

m1

φ−→ R. The linear projective subspace

(93) H := P kerφ ⊂ PΛ2C1m1

is the so–called hyperplane section determined by ωm1 . Now we show that H corresponds precisely to the fiber Eω,m1 of

the Boillat–type 3rd order MAE determined by ω, via the Plucker embedding.Indeed, S3L∗m1 is embedded into the space L∗m1⊗S2L∗m1 of all, i.e., not necessarily Lagrangian, 2D horizontal subspaces,

via the Spencer operator. In turn, L∗m1 ⊗S2L∗m1 is an affine neighborhood of Lm1 in the Grassmannian Gr (C1m1 , 2), which

is sent to PΛ2C1m1 by the Plucker embedding. Hence, the subspace Eω,m1 of M

(2)m1 can be regarded as a subspace of PΛ2C1

m1 ,and (27) tells precisely that such a subspace coincides with H defined by (93).

We are now in position to generalize a result about classical MAEs ([2], Theorem 3.7) to the context of 3rd order MAEs.

Proposition 28. A characteristic direction for Eω is also strongly characteristic.

Proof. Let H ∈ PC1m1

be a characteristic direction for Eω at the point m2. This means that H ⊂ Lm2 and that the

prolongation H(1) determined by H is tangent to Eω,m1 at m2. We need to prove that the whole H(1) is contained intoEω,m1 .

To this end, recall that H(1) ⊂M (2)m1 is a 1D affine subspace modeled over S3AnnH, passing through Lm2 (see, e.g., the

proof of Theorem 1 in [5]). By the above arguments, H(1) can be embedded into PΛ2C1m1 as well. Now we can compare

H(1) and H: they are both linear, they pass through the same point Lm2 , where they are also tangent each other. Hence,H(1) ⊂ H.

Now Proposition 28 allows us to prove Theorem 2.

Proof of Theorem 2. Inclusion (82) is valid for any 3rd order PDE. Conversely, if m2 ∈ EVE , then there is a line H ∈ VE ,such that Lm2 ⊃ H. But H is a strong characteristic line for E thanks to Proposition 28, so that all Lagrangian planespassing through H and, in particular, Lm2 ≡ m2 itself, belong to E .

6.3. Proof of Theorem 3. As it was outlined in Section 3, Goursat–type MAEs are the Boillat–type MAEs whichcorrespond to decomposable forms, modulo a certain ideal. Before proving Theorem 3, we make rigorous this statement.To this end, we shall need the submodule of contact 2–forms

(94) Θ = ω ∈ Λ2M (1) | ω|Lm2 = 0 ∀m2 ∈M (2) ⊂ Λ2M (1),

and the corresponding projection

(95) Λ2M (1) −→ Λ2M (1)

Θ.

Observe that the quotient bundle Λ2M(1)

Θ is canonically isomorphic to the rank–one bundle Λ2L∗ over M (1). Hence, (95)

can be thought of as (Λ2L∗)–valued.

Remark 29. Two 2–forms have the same projection (95) if and only if they differ by an element of Θ.

Proposition 30. It holdsED = Eω ⇔ ω = ρ1 ∧ ρ2 mod Θ.

Moreover, the 3D sub–distribution D ⊆ C1 is given by

(96) D = ker ρ1|C1 ∩ ker ρ2|C1 .

Proof. If AnnD = 〈ρ1, ρ2〉 ⊂ C1 ∗ is the annihilator of D in C1, then ED can be written as Eρ1∧ρ2 , where ρ1, ρ2 ∈ Λ1M (1)

are extensions of ρ1, ρ2, respectively. The result follows from Remark 29.Conversely, in light of (27) and (94), ω and ρ1 ∧ ρ2 + Θ give rise to the same equation, i.e.,

(97) Eω = Eρ1∧ρ2 .It remains to be proved that the right–hand side of (97) is of the form ED. To this end, it suffices to define D as in (96),

(98) ED = Eρ1|C1∧ρ2|C1

,

and observe that the right–hand sides of (97) and (98) coincide since ρi|C1 − ρi ∈ Ann C1 ⊂ Θ, i = 1, 2.

Next Proposition 31 provides an interesting link between the (complete) decomposability of the symbol of a Boillat–typeMAE E and the decomposability of the corresponding two–form. Even if its proof relies on Theorem 4, which will beproved later one, this is the appropriate moment to present it.

Proposition 31. It E = F = 0 is a 3rd order MAE of Boillat–type whose symbol SmblF is completely decomposable,then E is quasi–linear of Goursat–type.


Proof. Let V be the characteristic cone of E , and VI an its irreducible component. Then, by Theorem 2, E = EV . Observealso that if m2 ∈ EV , then Lm2 contains, in particular, a line H ∈ VI, since each irreducible component of V correspondsto one of the factors of SmblF . It follows that E = EVI as well (this fact will be proved for all Goursat–type 3rd orderMAEs in Corollary 36).

If we prove that V contains (at least) two irreducible components VI which are linear, then, by Theorem 4, E must beof the form ED and, by the same Theorem 4, also quasi–linear. To this end, notice that, for any point m1, the restrictionSmblF |

M(2)

m1is polynomial in the pijk’s of degree at most one. Hence, at least two out of it its three factors must be

constant, and they correspond precisely to the sought–for linear components of V.

Observe also that equation (120) may be thought of as the condition for the function G corresponding to a localBoillat–type 3rd order MAE (26) to give a Goursat–type MAE.

Before starting the proof of Theorem 3, we provide a meta–symplectic analog of the well–known formula AnnD⊥ = Dyω.

Proposition 32. Let D1 ⊆ C1 be a 3D sub–distribution with dimDv1 = 2 and E = ED1 the corresponding quasi–linear 3rd

order MAE. Then, if VEII = PD2 ∪ PD3,

(99) AnnDi =D1yΩHi

⊆ C1 ∗, i = 2, 3,

where H2 and H3 are the characteristic lines of Dv1 .

Proof. Let D1 = 〈aD1 + bD2, X1, X2〉, with X1 and X2 as in (53). Then, the quasi–linear 3rd order MAE determined byD1 is

ED1 = ap111 + (b− a(k1 + k2))p112 + (ak1k2 − b(k1 + k2))p122 + bk1k2p222.

In order to obtain the right–hand side of (99), compute first aD1 + bD2yΩ = (adp11 + bdp12)⊗∂p1 + (adp12 + bdp22)⊗∂p2 ,and factor it by Hi,

(100) aD1 + bD2yΩ = (adp11 + (b− kia)dp12 − kibdp22)⊗ ∂p1 mod Hi.

Combining (100) above with (55), yields

(101)D1yΩHi

'⟨kjdx

1 + dx2, adp11 + (b− kia)dp12 − kibdp22

⟩Finally, the wedge product of the two 1–forms spanning the module (101) above, is the 2–form

ω = akjdp11 ∧ dx1 + (b− kia)kjdp12 ∧ dx1 − bkikjdp22 ∧ dx1 + adp11 ∧ dx2 + (b− kia)dp12 ∧ dx2 − kibdp22 ∧ dx2,

and direct computations show that Eω = ED1. The result follows from Proposition 30.

Now we turn back to Theorem 3, and deal separately with its two implications.

6.3.1. Proof of the sufficient part of Theorem 3. The sufficient part of Theorem 3 will be proved through Lemma 33 below,which is purely local, that is, all objects involved are defined on the domain of a local chart.

Lemma 33. Let E = F = 0, where F is given either by (31), or

(102) F = ap111 + bp112 + cp122 + dp222 + e,

where a, b, c, d, e ∈ C∞(M (1)). Then there exists a 3D sub–distribution D ⊆ C(1) such that VE = VEI ∪ VEII, with VE = PD.Moreover, E = ED, i.e., according to (30), it is a 3rd order MAE of Goursat–type.

Proof. Let us begin with the fully nonlinear case.In order to verify that E = ED, it suffices to put

(103) D := 〈D1 + f111∂p11 + f112∂p12 + f122∂p22 , D2 + f112∂p11 + f122∂p12 + f222∂p22 , R∂p11 + S∂p12 + T∂p22〉 ,

where (R,S, T ) = A is the same appearing in (31), and observe that (see also (19))

Lm2 = 〈D1 + p111∂p11 + p112∂p12 + p122∂p22 , D2 + p112∂p11 + p122∂p12 + p222∂p22〉

belongs to ED (see (30)) if and only if the 5× 5 determinant

(104) det

∣∣∣∣∣∣∣∣∣∣1 0 p111 p112 p122

0 1 p112 p122 p222

1 0 f111 f112 f122

0 1 f112 f122 f222

0 0 R S T

∣∣∣∣∣∣∣∣∣∣= det

∣∣∣∣∣∣∣∣∣∣0 0 p111 − f111 p112 − f112 p122 − f122

0 0 p112 − f112 p122 − f1221 p222 − f222

1 0 f111 f112 f122

0 1 f112 f122 f222

0 0 R S T

∣∣∣∣∣∣∣∣∣∣= −F

is zero.


We now prove that VE = PD. First of all, solve equation (31) with respect to p111 and take the remaining coordinatesp112, p122, p222 as local coordinates on E . Thus SmblF (see (67)) is, up to a factor, equal to

(105)

((S(p222 − f222)− T (p122 − f122)

)ξ1 +

(T (p112 − Tf112)− S(p122 − f122)

)ξ2

)·

·((T (p122 − f122)− S(p222 − f222)

)ξ21 +

(T (p112 − f112) +R(p222 − f222)

)ξ1ξ2 +

(S(p112 − f112)−R(p122 − f122)

)ξ22

)so that (see also Section 5.2)

(106) VEI =

((S(p222 − f222)− T (p122 − f122)

)D1 +

(T (p112 − f112)− S(p122 − f122)

)D2

)m2

| m2 ∈ E.

A direct computation shows that

(107) VEI =(S(p222 − f222)− T (p122 − f122)

)(D1 + f111∂p11 + f112∂p12 + f122∂p22)

+(T (p112 − f112)− S(p122 − f122)

)(D2 + f112∂p11 + f122∂p12 + f222∂p22)

+ (p112p222 − p2122 + f112f222 − f2

122 − p112f222 − f112p222 + 2p122f122)(R∂p11 + S∂p12 + T∂p22) |p112, p122, p222 ∈ R

so that VEI turns out to be the 3D linear space (103).

Let us now pass to the quasi–linear case.One of the coefficients of the third derivatives of (102) must be nonzero. Assume that a 6= 0; the remaining cases can

be treated similarly. Again, we solve equation (102) with respect to p111, so that p112, p122, p222 become local coordinateson it, and

(108) SmblF = aξ31 + bξ2

1ξ2 + cξ1ξ22 + dξ3

2 = (kξ1 + hξ2)q(ξ1, ξ2),

where q(ξ1, ξ2) is a homogeneous quadratic function in ξ1 and ξ2. Thus, we have that (see again Section 5.2).

(109) VEI =⟨(kD1 + hD2)m2 | m2 ∈ E

⟩.

Let us assume d = 0. In order to verify that E = ED, it is enough to put

(110) D :=

⟨D1 −

e

a∂p11 ,−

b

a∂p11 + ∂p12 ,−

c

a∂p11 + ∂p22

⟩.

To prove that VEI = PD, observe that the symbol (108) contains the linear factor ξ1, and

VEI = 〈D1|m2〉m2∈E =

D1 −

e

a∂p11 + p112

(− ba∂p11 + ∂p12

)+ p122

(− ca∂p11 + ∂p22

)| p112, p122 ∈ R

is the projectivization of (110).Finally assume d 6= 0.Observe that the lines in VEI , which, in view of (109), are generated by

kD1 + hD2 −ke

a∂p11 + p112

((−kba

+ h

)∂p11 + k∂p12

)+ p122

(−kca∂p11 + h∂p12 + k∂p22

)+ p222

(−kda∂p11 + h∂p22

),

p112 , p122 , p222 ∈ R, fill a 3D linear space since ξ1ξ2

= −hk is a solution to (108), i.e.,

(111) − ah3

k3+ b

h2

k2− ch

k+ d = 0.

So, if we put

(112) D :=

⟨D1 +

h

kD2 −

e

a∂p11 ,

(− ba

+h

k

)∂p11 + ∂p12 ,

k

h

d

a∂p11 − ∂p22

⟩with h, k 6= 0 such that (111) is satisfied, we obtain E = ED and VEI = PD.


6.3.2. Proof of the necessary part of Theorem 3. Being Em1 a closed submanifold of codimension 1, in the neighborhoodof any point m2 ∈ Em1 , we can always present E in the form E = F = 0, with F := pijk −G, with G not depending onpijk, for some (i, j, k) which we shall assume equal to (1, 1, 1), since the other cases are formally analog. Then the symbolof F at m1 is

(113) Smblm1F = ξ31 −Gp112ξ2

1ξ2 −Gp122ξ1ξ22 −Gp222ξ3

2 .

The right–hand side of (113) is a 3rd order homogeneous polynomial with unit leading coefficient: hence, there exist uniqueβ,A,B such that Smblm1F = (ξ1 + βξ2)(ξ2

1 + Aξ1ξ2 + Bξ22). Following the general procedure (see also formula (78) and

Definition 22), to construct the characteristic cone VE , one easily sees that VE contains the following 3–parametric familyof lines:

VEI := D1 + βD2 + (G+ βp112)∂p11 + (p112 + βp122)∂p12 + (p122 + βp222)∂p22 | p112, p122, p222 ∈ R.

Observe that, with the same notation as (8), the direction v = (v1, v2) belongs to VEI if and only if there exists m2 ∈ Em1

such v1 + β(m2)v2 = 0 (see also Remark 12).Suppose now that there exists a 3D sub–distribution D ⊂ C1 such that

(114) VEI = PD.

Since D has codimension 2 in C1, condition (114) can be dualized as follows: there are two independent forms ρ1, ρ2 ∈ C1 ∗,

ρ1 = k1dx1 + k2dx

2 + k11dp11 + k12dp12 + k22dp22,(115)

ρ2 = h1dx1 + h2dx

2 + h11dp11 + h12dp12 + h22dp22,(116)

such that AnnD = 〈ρ1, ρ2〉, i.e., D = ker ρ1 ∩ ker ρ2. In other words, (114) holds true if and only if ρ1(v) = ρ2(v) = 0 forall v ∈ VEI , i.e.,

k1 + k12p112 + k22p122 + (k2 + k11p112 + k12p122 + k22p222)β + k11G = 0,(117)

h1 + h12p112 + h22p122 + (h2 + h11p112 + h12p122 + h22p222)β + h11G = 0,(118)

identically in p112, p122, p222, where the 2× 5 matrix

(119)

∣∣∣∣ k1 k2 k11 k12 k22

h1 h2 h11 h12 h22

∣∣∣∣of functions on M (1) has rank two everywhere.

Now (117)–(118) must be regarded as linear system of two equations in the unknowns β and G. Its discriminant iseasily computed, ∆ := (h11k22− k11h22)p222 + (h11k12− k11h12)p122 + h11k2− k11h2, and ∆ is a polynomial in p122, p222.

Suppose ∆(m2) = 0, and interpret (117)–(118) as a linear system in the variables β and G. The condition ∆(m2) = 0,together with the compatibility conditions of such a system, implies that the matrix (119) is of rank 1 in m1. Thiscontradicts the hypothesis that dimDm1 = 3, so that ∆ must be everywhere nonzero.

Then, in the points with ∆(p122, p222) 6= 0 one can find G as a polynomial expression of p112, p122, p222, whose coefficientsturn out to be minors of the matrix (119). In particular, p111 − G can be singled out from the so–obtained expression,namely

−∆(p111 −G) = −h1k2 + k1h2 − (h11k2 − k11h2)p111 − (k11h1 − h11k1 + k2h12 − k12h2)p112(120)

− (k2h22 + k12h1 − k22h2 − k1h

12)p122 − (k22h1 − k1h22)p222 + (k12h22 − h12k22)(p112p222 − p2

122)

+ (k22h11 − k11h22)(−p111p222 + p112p122) + (k11h12 − h11k12)(p111p122 − p2112).

We need to show that (120) is satisfied if and only if there is a nowhere zero factor λ such that

(121) F = λ(p111 −G),

where F is given by (31), i.e., equation (121) needs to be solved with respect to fijk, R, S, T . By equating the coefficientsof p111, one obtains

(122) λ = T (p122 − f122)− S(p222 − f222),

and replacing (122) into (121) yields

(123) F − (T (p122 − f122)− S(p222 − f222))(p111 −G) = 0.

Direct computations show that the left–hand side of (123) is a rational function whose numerator is a 2nd order polynomialin p112, p122, p222. From the vanishing of this polynomial, one can express fijk, R, S, T as functions of the entries of the


matrix (119). In particular,8 the vector (R,S, T ) is conformally defined by the equation

det

∣∣∣∣∣∣k11 k12 k22

h11 h12 h22

R S T

∣∣∣∣∣∣ = 0.

6.4. Proof of Theorem 4. It will be accomplished in steps. As a preparatory result, we show that a 3D sub–distributionof C1 with 2D vertical part determines a quasi–linear 3rd order MAE (Lemma 34 below). The converse statement, i.e., thatthe characteristic cone of a generic quasi–linear 3rd order PDE possesses a linear sheet determined by a 3D sub–distributionD ⊆ C1 with 2D vertical part, has been proved by Lemma 33 above. Then we pass to generic 3D sub–distributions of C1,and derive the expression of the corresponding equation ED (Lemma 35). As a consequence of these results (Corollary36), we prove the initial statement of Theorem 4.

The next three sections 6.4.1, 6.4.2, 6.4.3 are devoted to the specific proofs of the items 1, 2, 3 of Theorem 4, respectively.Besides the proof of Theorem 4 itself, a few interesting byproducts will be pointed out. For instance, Corollary 36 means

that for quasi–linear 3rd order MAEs, all characteristic lines are strong, generalizing an analogous result for classical MAEs(see [2], Theorem 3.7). Also Corollary 39 is a nontrivial and unexpected generalization of a phenomenon firstly observedin the classical case (see [2], Theorem 1.1).

Lemma 34. Let D1 = 〈h1,Dv1〉, with

h1 = aD1 + bD2 + f11∂p11 + f12∂p12 + f22∂p22 ,(124)

Dv1 = 〈Ri∂p11 + Si∂p12 + Ti∂p22 | i = 1, 2〉 .(125)

Then ED1 is of the form (26) with

A = 0,(126)

C = det

∣∣∣∣∣∣f11 f12 f22

R1 S1 T1

R2 S2 T2

∣∣∣∣∣∣ ,(127)

B =

(−adet

∣∣∣∣ S1 T1

S2 T2

∣∣∣∣ ,−bdet

∣∣∣∣ S1 T1

S2 T2

∣∣∣∣+ a det

∣∣∣∣ R1 T1

R2 T2

∣∣∣∣ ,bdet

∣∣∣∣ R1 T1

R2 T2

∣∣∣∣− a det

∣∣∣∣ R1 S1

R2 S2

∣∣∣∣ ,−bdet

∣∣∣∣ R1 S1

R2 S2

∣∣∣∣) .(128)

Proof. Just observe that Lm2 = 〈D1 + p111∂p11 + p112∂p12 + p122∂p22 , D2 + p112∂p11 + p122∂p12 + p222∂p22〉 belongs to ED1

if and only if the determinant of the 5× 5 matrix

(129)

∣∣∣∣∣∣∣∣∣∣a b f11 f12 f22

0 0 R1 S1 T1

0 0 R2 S2 T2

1 0 p111 p112 p122

0 1 p112 p122 p222

∣∣∣∣∣∣∣∣∣∣is zero (see also (19) and (30)). Standard matrix manipulations reveal that the determinant of (129) coincides in turnwith (26) under conditions (126), (127) and (128).

Observe that dim Gr (C1, 3) = 6, i.e., it takes 6 parameters to identify a 3D sub-distribution D ⊂ C1 and, hence, thecorresponding equation ED.

Lemma 35. The local form of a generic ED with D ∈ Gr (C1, 3), in the case dimDv = 1 and up to a change of variables,is given by (31).

Proof. We can always find a coordinate system such that the (conformally unique) generator υ of Dv is given by

(130) υ = R∂p11 + S∂p12 + T∂p22

with RT − S2 6= 0. Indeed, if RT − S2 = 0 in some coordinates (16), then a linear change of independent variables(x1, x2)→ (x1, x2) = (ax1 + bx2, cx1 + dx2) (which is an obvious contact transformation) is enough to obtain, in the new

coordinates, RT − S2 6= 0.

So, without loss of generality, let us suppose that the generator of Dv is given by (130) with R,S, T ∈ C∞(M (1)) suchthat RT − S2 6= 0. Accordingly, distribution D can be always put in the form D = 〈D1 + b111∂p11 + b112∂p12 + b122∂p22 ,

D2 + b211∂p11 + b212∂p12 + b222∂p22 , υ〉, with bijk ∈ C∞(M (1)), or, equivalently,

(131) D = 〈D1 + b111∂p11 + b112∂p12 + b122∂p22 + αυ , D2 + b211∂p11 + b212∂p12 + b222∂p22 + βυ , υ〉,

8The expression of fijk, easily obtainable with a software of symbolic computations, is too lengthily to be laid down here.


for any α, β ∈ C∞(M (1)). Moreover, α and β can be chosen in such a way that

(132) b112 + αS = b211 + βR , b122 + αT = b212 + βS,

since system (132), in view of the condition RT − S2 6= 0, is always compatible for any R,S, T, b112, b211, b122, b212. Toconclude the proof, set

f111 = b111 , f112 = b112 + αS , f122 = b122 + αT , f222 = b222,

substitute them into (131), and compute the corresponding ED by means of the analog of matrix (129). The result isprecisely (31).

Corollary 36. Let E be an equation with VEI = PD1. Then E = ED1.

Proof. Observe that any Lm2 , with m2 ∈ E , always contains a characteristic line H corresponding to a fixed linear factorof the symbol, i.e., an element H belonging to VEI (see Section 5.2). In other words, inclusion (83) can be made moreprecise:

(133) EVEI ⊇ E .

It remains to prove that the inverse of inclusion (133) is valid when VEI = PD1, and to this end it is convenient to regard

(133) as an inclusion of subsets in the compactification M (2) (see Section 5.1).Indeed, Lemma 34 and Lemma 35 together guarantee that the left–hand side of (133) is a closed submanifold of

codimension one and, hence, fiber–wise compact, so being M (2). So, the right–hand side of (133) is a closed submanifoldof codimension zero, i.e., an open and closed subset of the left–hand side, which is also connected. Hence, the two of themmust coincide as well.

6.4.1. Proof of the statement 1 of Theorem 4. The proof of its first claim is contained into the next corollary.

Corollary 37. A 3rd order MAE ED is quasi–linear if and only dimDv = 2.

Proof. A direct consequence of Lemmas 33, 34 and 35.

The remainder of statement 1 is concerned with the “other component” VEII of VE , i.e., the one associated with thequadratic factor of AnnV E (see Section 5.2). Recall that VEII might be empty, in which case there is nothing to prove. Atthe far end, there is the case when VEII is, in its turn, decomposable, i.e., it consists of two linear sheets, which is dealtwith by Lemma 38 below.

Lemma 38. Let E = F = 0 be a 3rd order quasi–linear PDE such that SmblF is completely decomposable, i.e.,

(134) SmblF = (ξ + k1η) (ξ + k2η) (ξ + k2η).

If

(135) Di :=⟨D1|m2 + ki D2|m2 | m2 ∈ Em1

⟩is the 3D sub–distribution of C1 corresponding9 to the ith linear factor of (134) and

(136) D =⟨h,Dv

⟩⊆ C1

is a generic 3D sub–distribution of C1 with dimDv = 2, then D satisfies condition (33) if and only if D = Dj with j 6= i.

Proof. To begin with, (134) dictates some restrictions on F , which must be of the form

(137) F = p111 + (k1 + k2 + k3)p112 + (k1k2 + k1k3 + k2k3)p122 + k1k2k3p222 + c.

We begin with the “homogeneous” case, i.e., we assume c = 0 since, as we shall see at the end of the proof, the generalcase can be easily brought back to this one.

Equating (137) to zero allows to express p111 as a linear combination of p112, p122, and p222, i.e., to identify Em1 withR3 ≡ (p112, p122, p222). In turn, this makes it possible to parametrize the space of vertical elements of (135) by threereal parameters, viz.,(138)Dvi = 〈−((k1+k2+k3−ki)p112+(k1k2+k1k3+k2k3)p122+k1k2k3p222)∂p11 + (p112+kip122)∂p12 + (p122+kip222)∂p22 | p112, p122, p222 ∈ R〉

It is worth observing that, in compliance with Lemma 33, the dimension of Dvi , which equals the rank of the 3× 3 matrix

(139) Mi =

∣∣∣∣∣∣−k1 − k2 − k3 + ki −k1k2 − k1k3 − k2k3 −k1k2k3

1 ki 00 1 ki

∣∣∣∣∣∣ ,is 2: indeed, detMi = −(k1 − ki)(k2 − ki)(k3 − ki) vanishes for i = 1, 2, 3.

9See also, on this concern, the construction of VEI in Section 5.2.


In order to find a basis for (138), regard the matrix (139) as a (rank–two) homomorphism Mi : R3 −→ V =〈∂p11 , ∂p12 , ∂p22〉 and compute its kernel:

(140) kerMi =⟨k2i ,−ki, 1

⟩.

Then, independently on i (and on the value of ki as well), R3 = 〈(1, 0, 0), (0, 1, 0)〉 ⊕ kerMi, andDvi = 〈Mi · (1, 0, 0), Mi· (0, 1, 0)〉 is the sought–for basis. In other words, instead of (135), we shall work with the handier

description Di = 〈hi, Dvi 〉, where

hi = D1 + kiD2,

(141)

Dvi = 〈−(k1 + k2 + k3 − ki)∂p11 + ∂p12 ,−(k1k2 + k1k3 + k2k3)∂p11 + ki∂p12 + ∂p22〉 .(142)

Concerning (136), introduce similar descriptions of its horizontal and vertical part:

h = aD1 + bD2,(143)

Dv = 〈r1∂p11 + s1∂p12 + t1∂p22 , r2∂p11 + s2∂p12 + t2∂p22〉 .(144)

This concludes the preliminary part of the proof. Now impose condition (33):

(145) Ω(hi,Dv) = Ω(h,Dvi ) = 1D subspace.

Observe that (145) consists, in fact, of two requirements, which are going to be dealt with separately.The first one corresponds to the one–dimensionality of the subspace

Ω(hi,Dv) = 〈r1∂p1 + s1∂p2 + ki(s1∂p1 + t1∂p2), r2∂p1 + s2∂p2 + ki(s2∂p1 + t2∂p2)〉

= 〈(r1 + kis1)∂p1 + (s1 + kit1)∂p2 , (r2 + kis2)∂p1 + (s2 + kit2)∂p2〉 ,(146)

i.e., to the equation

(147) det

∣∣∣∣ r1 + kis1 s1 + kit1r2 + kis2 s2 + kit2

∣∣∣∣ = k2iA− kiB + C = 0,

with

A = det

∣∣∣∣ s1 t1s2 t2

∣∣∣∣ ,(148)

B = −det

∣∣∣∣ r1 t1r2 t2

∣∣∣∣ ,(149)

C = det

∣∣∣∣ r1 s1

r2 s2

∣∣∣∣ .(150)

Interestingly enough, above coefficients (148), (149) and (150) characterize the dual direction of Dv, i.e., a nonzero co–vector ω′ = Adp11 +Bdp12 + Cdp22 ∈ V ∗, defined up to a nonzero constant, such that

(151) Dv = kerω′.

This dual perspective on Dv allows to rewrite (147) as

(152) ω′(k2i ∂p11 − ki∂p12 + ∂p22) = 0.

In turn, (152) dictates the form of ω′:

(153) ω′ = Adp11 +Bdp12 + (Bki −Ak2i )dp22, (A,B) 6= (0, 0).

Together, (153) and (151) produce a simplified expression

(154) Dv =⟨(Ak2

i −Bki)∂p11 +A∂p22 ,−B∂p11 +A∂p12⟩,

which, in comparison with (144), depending on 6 parameters, needs only 2 of them, or even 1, if a nonzero constant isneglected. Now, thanks to (154), it is easier to see that the space (146) is 1D. Indeed, (146) reads

Ω(hi,Dv) =

⟨(Ak2

i −Bki)∂p1 +Aki∂p2 ,−B∂p1 +A∂p2 +Aki∂p1⟩,

where the first vector equals the second multiplied by ki, so that it can be further simplified:

(155) Ω(hi,Dv) = 〈(Aki −B)∂p1 +A∂p2〉 .


We can pass to the other condition dictated by (145). In particular, the subspace

Ω(h,Dv) =〈a(−(k1 + k2 + k3 − ki)∂p1 + ∂p2) + b∂p1 ,

a(−(k1k2 + k1k3 + k2k3)∂p1 + ki∂p2) + b(ki∂p1 + ∂p2)〉=〈(b− a(k1 + k2 + k3 − ki))∂p1 + a∂p2 ,(156)

(bki − a(k1k2 + k1k3 + k2k3))∂p1 + (aki + b)∂p2〉must be 1D, i.e.,

(157) det

∣∣∣∣ b− a(k1 + k2 + k3 − ki) abki − a(k1k2 + k1k3 + k2k3) aki + b

∣∣∣∣ =∏j 6=i

(b− akj) = 0.

Above equation (157), makes it evident that, for any i = 1, 2, 3, the space (155) is 1D if and only if

(158) b = akj , j ∈ 1, 2, 3r i (a 6= 0),

which corresponds to

(159) h = D1 + kjD2, j ∈ 1, 2, 3r i.Plugging (158) into (156), we find a unique vector

(160) (ki + kj − k1 − k2 − k3)∂p1 + ∂p2

generating Ω(h,Dv). Introducing the complement c(i, j) of i, j in 1, 2, 3, (160) reads

(161) Ω(h,Dv) =⟨−kc(i,j)∂p1 + ∂p2

⟩, j ∈ 1, 2, 3r i.

To conclude the proof, recall that, besides their one–dimensionality, (145) also requires the equality of the subspaces (154)

and (155), i.e., Ω(hi,Dv) =

⟨(Aki −B)∂p1 +A∂p2 ,−kc(i,j)∂p1 + ∂p2

⟩= Ω(h,Dv) or, alternatively,

(162) det

∣∣∣∣ Aki −B A−kc(i,j) 1

∣∣∣∣ = 0.

Thanks to (162),

(163) B = A(ki + kc(i,j)), j ∈ 1, 2, 3r i,and (163) allows to eliminate B from (154):

(164) Dv =⟨(Ak2

i −A(ki + kc(i,j))ki)∂p11 +A∂p22 ,−A(ki + kc(i,j))∂p11 +A∂p12⟩.

Being a nonzero constant,10 A can also be removed from (164), which becomes:

(165) Dv =⟨−kc(i,j)ki∂p11 + ∂p22 , (kc(i,j) + ki)∂p11 + ∂p12

⟩.

To enlighten the conclusions, it is useful to rewrite together (159) and (165) above: there are exactly 2 distributions Dwhich are “compatible” (in the sense of (145)) with the distribution Di given by (135). More precisely, their horizontaland vertical parts are:

h = D1 + kjD2,(166)

Dv =⟨−kc(i,j)ki∂p11 + ∂p22 , (kc(i,j) + ki)∂p11 − ∂p12

⟩,(167)

respectively, for the only two possible values of j ∈ 1, 2, 3r i. One only needs to realize that (167) is one of the Dvi ’sfrom (142). To this end, rewrite (142) replacing i with j: Dvj = 〈−(k1 +k2 +k3−kj)∂p11 +∂p12 ,−(k1k2 +k1k3 +k2k3)∂p11+kj∂p12 − ∂p22〉, and subtract from the second vector the first one multiplied by kj :

(168) Dvj = 〈−(k1 + k2 + k3 − kj)∂p11 + ∂p12 , (kj(k1 + k2 + k3 − kj)− (k1k2 + k1k3 + k2k3))∂p11 + ∂p22〉 .Formulas

kj(k1 + k2 + k3 − kj) = kj∑l 6=j

kl,(169)

k1k2 + k1k3 + k2k3 = kj∑l 6=j

kl +∏l 6=j

kl,(170)

show that the coefficient of ∂p11 in the second vector of (168) reduces to −∏l 6=j kl. Hence, (168) reads

(171) Dvj =

⟨−

∏l 6=j

kl

∂p11 + ∂p22 ,

∑l 6=j

kl

∂p11 − ∂p12

⟩.

10If A = 0, then from (154) follows dimDv

= 1


Comparing now (167) with (171) it is evident that Dv must equal Dvj , with j 6= i, and the proof of the “homogeneous”case is complete.

To deal with the general case, denote by E the equation determined by F as in (137), with c = 0, and by Ec a “inhomo-geneous” equation, i.e., one with c 6= 0. Observe that there is a natural identification ic : 〈D1, D2〉 −→ 〈D1 − c∂p11 , D2〉between horizontal planes, giving rise to an automorphism ϕc := ic ⊕ idVM(1) of C1. Easy computations show thatϕc(Di), i = 1, 2, 3, are precisely the three distributions associated with the factors of the symbol of Ec, and plainlyΩ(h,Dv) = Ω(ϕc(h), ϕc(Dv)). So, the “inhomogeneous” case reduces to the “homogeneous” one, which has been estab-lished above.

Corollary 39 (Proof of statement 1 of Theorem 4). Let E = F = 0 be a quasi–linear 3rd order PDE, and U ⊆M (1) the

open locus where the bundle VEII is not empty. Then VE =⋃3i=1 PDi on U , where Di ⊆ C|U are orthogonal (see Definition

9) with respect to the meta–symplectic structure on M (1) and E = EDion U , for all i = 1, 2, 3.

If E = F = 0 is a quasi–linear 3rd order PDE, and m1 ∈ M (1) is such that(VEII)m16= ∅, then it means that the

function ∆ (see Remark 20) is positive on the whole fibre M(1)m1 and, hence, on the neighbouring fibres. So, U is indeed

open.

Proof of Corollary 39. Letm1 ∈ U . Then VEII 6= ∅, i.e., there is a pointm2 ∈ Em1 such that Smblm2F is fully decomposable.But, in view of the quasi–linearity of F , this means that SmblF is fully decomposable on the whole fiber Em1 , i.e., Efullfills the hypotheses of Lemma 38 on U .

6.4.2. Proof of the statement 2 of Theorem 4. The first part of this statement merely rephrases the fist part of statement1.

Proving its second part consists in completing the proof of Lemma 33 above with the computation of VEII, whichcorresponds to the quadratic factor (see Section 5.2) of (105). Formally, the procedure to get a description of VEII out ofthe quadratic factor of (105) is the same as that for obtaining (106) from its linear factor, but it is more involved whenit comes to computations, and this is the reason why we kept them separate.

So, suppose that VEII 6= ∅. This means that there are points where the the quadratic factor of (105), rewritten below,admits real roots.

(172)(T (p122 − f122)− S(p222 − f222)

)ξ21 +

(T (p112 − f112) +R(p222 − f222)

)ξ1ξ2 +

(S(p112 − f112)−R(p122 − f122)

)ξ22

If ∆ is the discriminant of (172), then one factor of (172) is given by

(173) ξ1 −Tp112 − Tf112 +Rp222 −Rf222 +

√∆

2(−Tp122 + Tf122 + Sp222 − Sf222)ξ2,

and the corresponding line H1 is obtained by replacing11 ξi with Di|E in (173). Now it is possible to assign certain values(namely (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)) to the internal parameters of E (which are p112, p122, p222), in such a way thatthe corresponding lines H1 do not lie all in the same 3D subspace of C1. This shows that VEII cannot contain any linearirreducible component.

In order to prove that

(174) ED1= EVEII ,

observe that the “⊇” inclusion follows from the fact that, by definition, VEII is made of characteristic lines of ED1which

are also strongly characteristic by Proposition 28, so that all Lagrangian planes containing a line in VEII belong also to ED1

(see also Definition 16).Inclusion “⊆” can be obtained by a straightforward computation: indeed the 3 × 5 matrix whose first line is formed

by the components of H1 and the remaining ones by the components of the vectors spanning a generic Lagrangian planebelonging to ED1 , is of rank 2. The same result is attained by considering the line H2 obtained by changing the sign of√

∆ in (173).

6.4.3. Proof of the statement 3 of Theorem 4. Given any 3D sub–distribution D ⊆ C1, and the corresponding Goursat–typeequation ED (see (30)), we are now in position to clarify how many and which sort of elements can be found in the

(175) D | D is a 3D sub–distribution of C1 such that ED = ED.More precisely,

(1) in the points of M (1) where dimDv = 3, we have ED = M (2), so that the latter cannot be even considered as a 3rd

order PDE according to our understanding of PDEs (see Section 2.1), and this case must be excluded a priori ;12

(2) in the points of M (1) where dimDv = 1, the equation ED is fully nonlinear and the set (175) contains only D;

11This is a circumstance where the ξi’s and the Di’s are vector fields, not vectors (still, this notation is consistent with the conventions

established in Remark 1).12Still, the description of the set (175) is very easy: it contains only D, which in turn coincides with VM(1).


(3) in the points of M (1) where dimDv = 2, the equation ED is quasi–linear and the set (175) contains three elementsDi, i = 1, 2, 3, possibly repeated and comprising D itself, which are orthogonal each other (see Definition 9).

Proof of 2. and 3. Let D be an element of the set (175). Then, in particular,

(176) VED = VED .

In view of Corollary 24, equality (176) implies

PD ⊆ VED ,(177)

PD ⊆ VED .(178)

In other words, condition (32) is fulfilled by both equations ED and ED, so that the statements 1 and 2 of Theorem 4 (seeabove Section 6.4.1 and Section 6.4.2) can be made use of. To this end, rewrite (177) and (178) as

VED = PD ∪ VEDII ,(179)

VED = PD ∪ VEDII ,(180)

respectively. Because of statement 1 (reps., 2) of Theorem 4, from (179) it follows that ED is quasi–linear (resp., fullynonlinear) if and only if dimDv = 2 (resp., 1) and from (180) it follows that ED is quasi–linear (resp., fully nonlinear) if

and only if dim Dv = 2 (resp., 1). Summing up, dimDv = dim Dv = 1, 2, and the two cases can be treated separately.Directly from (179) and (180) we get

PD ∪ VEDII = PD ∪ VEDII ,(181)

PD ∪ VEDII = PD ∪ VEDII .(182)

Prove now 2. If dimDv = 1, then also dim Dv = 1 and the statement 2 of Theorem 4 (see Section 6.4.2) guarantees that

VEDII (resp., VEDII ) cannot contain PD (resp., PD). It follows from (181) and (182) that PD = PD.

Finally prove 3. If dimDv = 2, then also dim Dv = 2 and, in view of the statement 1 of Theorem 4 (see Section 6.4.1),

VEDII (resp., VEDII ) is either empty, in which case PD = PD, or consists of two, possibly repeated, distributions “orthogonal”

to D (resp., D).

The result presented in this section mirrors the analogous result for classical multi–dimensional MAEs (see [2],Theorem1), but displays some new and unexpected features: the threefold multiplicity of the notion of orthogonality and thedistinction of the cases according to the dimension of the vertical part.

7. Intermediate integrals of Goursat–type 3rd order MAEs

Theorem 4 established a powerful link between 3rd order MAEs of Goursat–type, i.e., nonlinear PDEs of order three, and3D sub–distributions of C1, i.e., linear objects involving (at most) second–order partial derivatives. Besides its aestheticvalue, such a perspective also allows to formulate concrete results concerning the existence of solutions, as the first integralsof D can be made use of in order to find intermediate integrals of ED, along the same lines of the classical case (see [2],Section 6.3). Proposition 43 is the main result of this last section, showing that, for a Goursat–type 3rd order MAE E ,the notions of an intermediate integral of E and of a first integral of any distribution D such that ED = E are actually thesame. To facilitate its proof, we deemed it convenient to introduce equations of the form E = EV where V = PD, with, forthe first time in this paper, dim D = 2. By analogy with (30), we still write ED instead of EPD, but we warn the readerthat, unlike all the cases considered so far, ED is actually a system of two independent equations (see Lemma 41 below).

Recall that a function f ∈ C∞(M (1)) determines, in the neighborhood of an its nonsingular point, the hyperplanedistribution ker df on M (1), each of whose leaves is identified by a value c ∈ R. Following the same notation as [2], we set(

M (1))f−c

:= m1 ∈M (1) | f(m1) = c.

Definition 40. A function f ∈ C∞(M (1)) is an intermediate integral of a PDE E ⊆M (2) if

(183)(M (1)

)(1)

f−c⊆ E

for all c ∈ R.

Inclusion (183) can be interpreted as follows: if f = c defines a 2nd order PDE (in the sense given in Section 2.1), thenany solution of such an equation is also a solution of the 3rd order equation E .

Lemma 41. Let D ⊂ C1 be a 2D non–vertical sub distribution. Then ED −→M (1) is a nonlinear bundle of rank 2.


Proof. To simplify the notations, let E := ED. Then, fix m1 ∈M (1) and observe that Em1 = m2 ∈M (2) | Lm2 ∩D 6= 0 is

a 2D manifold. Indeed, up to a zero–measure subset, Em1 is a rank–one bundle over the 1D manifold PD, its fibers beingthe prolongations (or “rays”) of the lines lying in D (see [11], Proposition 2.5).

To prove nonlinearity of E , write down D in local coordinates as D = 〈aiDi + biDi +Ri∂p11 + Si∂p12 + Ti∂p22 | i = 1, 2〉and observe that E is given by the vanishing of the five 4× 4 minors of the matrix∣∣∣∣∣∣∣∣

a1 b1 R1 S1 T1

a2 b2 R2 S2 T2

1 0 p111 p112 p122

0 1 p112 p122 p222

∣∣∣∣∣∣∣∣ .Tedious computations show that the only cases when such minors are quasi–linear, is that either when D fails to betwo–dimensional or when D is vertical, which are forbidden by the hypotheses.

Corollary 42. Let D ⊂ C1 be a 2D sub distribution contained into ker df . Then for any m1 ∈M (1) there is a m2 ∈M (2)m1

such that Lm2 ⊂ ker dm1f and Lm2 ∩ Dm1 = 0.

Proof. Towards an absurd, suppose that

(184) m2 ∈M (2) | Lm2 ⊂ ker dm1f ⊆ m2 ∈M (2) | Lm2 ∩ Dm1 6= 0and set c := f(m1). Then (184) means that, over the point m1,

(185)(M (1)

)(1)

f−c⊆ ED

which is impossible, since the left–hand side of (185) is linear while its right–hand side is either empty (in the case whenD ⊆ VM (1)) or nonlinear, thanks to above Lemma 41.

Proposition 43. Let E be a Goursat–type 3rd order MAE and f ∈ C∞(M (1)). Then f is a first integral of D, where Dis a distribution such that ED = E, if and only if f is an intermediate integral of E.

Proof. If f is a first integral of D, then

(186) D ⊆ C1 ∩ ker df ⊆ C1

is a (3, 4, 5)–type flag of distributions (C1 does not posses first integrals). By definition,

(187)(M (1)

)(1)

f−c= m2 ∈M (2) | f(π2,1(m2)) = c, df |Lm2 = 0

and condition df |Lm2 = 0 means precisely that the 2D subspace Lm2 of C1 is also contained into ker df , i.e., Lm2 lies in

the 4D subspace C1 ∩ ker df of C1. Because of (186), the 3D subspace D is also contained into C1 ∩ ker df and, as such, itcannot fail to nontrivially intersect Lm2 . This means that m2 ∈ ED, and (183) follows from the arbitrariness of m2.

Suppose, conversely, that f is an intermediate integral of E . In view of (187) and inclusion (183), one has that

(188) Lm2 ⊆ ker df ⇒ Lm2 ∩ D 6= 0

for any distribution D such that ED = E . We wish to show that, there is at least one among these distributions suchthat f is an its first integral, i.e., that the first inclusion of (186) is satisfied. Towards a contradiction, suppose that nodistribution D is contained into ker df , i.e., that Df := D∩ker df is a 2D sub–distribution of the 4D distribution C1∩ker df ,for all distributions D such that ED = E . Then, by Corollary 42, it is possible to find a (2D subspace) Lm2 which iscontained into C1 ∩ ker df and also trivially intersects one Df , thus violating (188).

Corollary 44. If the derived flag of D never reaches TM (1), then ED admits an intermediate integral.

8. Perspectives

The correspondence between 3D sub–distributions of C1 and Goursat–type 3rd order MAEs established in Section 6above is a convenient departing point for a classification of such equations, which is a problem beyond the scope of thispaper. We stress that in all the key moments of our reasonings we never made use of coordinates, and we relied themonly for clarifying the results and laying down examples. In other words, the framework we proposed is well–defined upto contactomorphisms. In particular, for Boillat–type MAEs, this means that

(189) ψ(2)(Eω) = Eψ(1) ∗(ω),

where ψ is a (local) contactomorphism of M and ψ(i) : M (i) −→ M (i) is the prolongation of i to M (i), with i = 1, 2. Inthe special case of Goursat–type MAEs, (189) reduces to

(190) ψ(2)(ED) = Eψ

(1)∗ (D)

.


So, a sufficient condition for Goursat–type 3rd order MAEs to be contactomorphic is that so are the corresponding 3Dsub–distributions of C1, but, as showed by Example 5, the condition needs not to be necessary too. Nevertheless, formula(190) allows one to find some convenient normal forms for a class of (generally) nonlinear PDEs through the analysis ofcertain objects canonically associated with them, which display two evident advantages: they are linear and also involvelower–order derivatives.

In view of the three–fold orthogonality, some precautions must be taken in facing equivalence problems. Indeed, if

a contact–invariant property P is possessed by a distribution D determining E = ED, then any equation E which is

contact–equivalent to E must be determined by a distribution D, having the same property P.

Inequivalence of rank–one vectors. 3D sub–distributions of C1 possess some evident invariants, like, e.g., the dimension ofthe vertical part and, in the case of 1D vertical parts, their rank. This reflects on the corresponding PDEs.

For instance, within the class of fully nonlinear Goursat–type 3rd order MAEs, equations of the form E = ED, withD = 〈H, v〉, are distinguished according to the rank of v, which can be 1 or 2. Furthermore, the meta–symplectic formΩ provides a tool to distinguish between two equations whose distributions have both rank–one vertical part. Namely, arank–one vector v is not always conctactomorphic to ∂p11 . Indeed, independently on the horizontal 2D component H ofD,

(191) dim Ω(H, ∂p11) = 1,

whereas the rank–one vector a2∂p11 + 2ab∂p12 + b2∂p22 , with a, b 6= 0, is such that above dimension (191) is 2.

Multiplicity of distributions. Within the class of quasi–linear Goursat–type 3rd order MAEs E , the fact that there existsD such that E = ED and the dimension of the space Ω(h,Dv) is one or two, with D = 〈h,Dv〉, is a contact invariant ofthe equation. More precisely,

(192) dim Ω(h,Dv) = 1 (resp., 2)⇔ D has multiplicity ≥ 2 (resp., 1).

In order to prove (192), let h and Dv be as in (124) and (125), respectively. Then direct computations show that the symbolof the determinant of the matrix (129) contains the square of a linear term if and only if Ω(h,Dv) is one–dimensional.

Rank–one lines in Dv. One can further distinguish quasi–linear equations according to the integrability of Dv, and also bythe number of rank–one lines contained into Dv, which is a contact invariant. For instance, the two integrable distributions〈∂p11 , ∂p22〉, 〈∂p11 , ∂p12〉 cannot be contactomorphic. Indeed, a∂p11 + b∂p22 is a perfect square if and only if a = 0 or b = 0,whereas a∂p11 + b∂p12 ' aξ2

1 + bξ1ξ2 is a perfect square if and only if b = 0. We stress that from the non–equivalence ofdistributions does not follow the non–equivalence of the corresponding equations, as shown by Example 5.

Conclusions. Summing up, any PDE E appeared so far had two remarkable properties:

(i) E is of 3rd order;(ii) characteristics of E are also strongly characteristics.

Above properties guaranteed, respectively, that

(i) each Lagrangian plane in E possesses a characteristic line (see Corollary 36), i.e., it belongs to EVE ;(ii) the set of Lagrangian planes containing a line which belongs to VE is entirely contained into E (see Lemma 25),

i.e., inclusions E ⊆ EVE and EVE ⊆ E , respectively. In other words, we dealt with equations which can be reconstructedfrom their characteristics (see Section 3).

Boillat–type and Goursat–type 3rd order MAEs provided us with an interesting class of such equations, whose charac-teristic cone took a particularly simple form, especially in the case Goursat–type equations, where an its single irreduciblecomponent (which turned out to be linear) carried all the relevant information (see the statement 1 of Theorem 4).

Even if our analysis of Goursat–type 3rd order MAEs was quite comprehensive, no clues have been found concerningthe structure of the characteristic cone of Boillat–type 3rd order MAEs, which is something worth trying. Indeed, themain Theorems 3 and 4 went deep into the structure of the characteristic cone of Goursat–type MAEs, whereas a similarinvestigation of Boillat–type MAEs was rather superficial and limited to Theorem 2, which just established that also thesekinds of equations can be reconstructed from their characteristics.

An intrinsic formula analogous to that discussed in Example 26 is still missing in the meta–symplectic context, and itcould help in facing this problem, even beyond the class of MAEs, which played the role of a sort of testing ground forthe theory. It should be stressed that, the fact that we dealt with 3rd order equations added a remarkable simplification,due to the presence of a real root for the symbol, and investigating, e.g., 4th order PDEs may turn out to be much morecomplicated.

Our analysis of intermediate integrals (Section 7) was limited to 2nd order intermediate integrals, and it should beextended in order to encompass 1st order intermediate integrals as well.

We concluded this paper with some ideas and preliminary results concerning the classification of Goursat–type 3rd orderMAEs through the properties of the corresponding characteristic cone which, in this case, reduced to a 3D distribution.Finding normal forms of such distributions along the same lines as [4] will require deeper insights on the meta–symplecticstructure and it will be carried out in a forthcoming publication.


SOFFITTA. CARATTERIZZARE I CASI IN CUI LA COPPIA ORTOGONALE ESISTE DAVVERO?[CONTROLLARE I TIPI DIFFERENTI DI INTEGRALI PRIMI][verificare che i the coefficienti sono VERAMENTE liberi!]controllare E = ED localmente[controllare bene che vale il se e solo se in (192); pare che Dv contiene 2 direzioni di rango uno; potrebbe

essere che la molteplicita di D e’ legata al numero di direzioni di rango uno][controllare quella storia di ∆ 6= 0 in 6.3.2]

References

[1] S. I. Agafonov and E. V. Ferapontov. Systems of conservation laws in the setting of the projective theory of congru-ences: reducible and linearly degenerate systems. Differential Geom. Appl., 17(2-3):153–173, 2002. ISSN 0926-2245.doi: 10.1016/S0926-2245(02)00105-5. URL http://dx.doi.org/10.1016/S0926-2245(02)00105-5. 8th Interna-tional Conference on Differential Geometry and its Applications (Opava, 2001).

[2] Dmitri V. Alekseevsky, Ricardo Alonso-Blanco, Gianni Manno, and Fabrizio Pugliese. Contact geometry of multidi-mensional Monge-Ampere equations: characteristics, intermediate integrals and solutions. Ann. Inst. Fourier (Greno-ble), 62(2):497–524, 2012. ISSN 0373-0956. doi: 10.5802/aif.2686. URL http://dx.doi.org/10.5802/aif.2686.

[3] R. Alonso Blanco, Gianni Manno, and Fabrizio Pugliese. Contact relative differential invariants for non genericparabolic Monge-Ampere equations. Acta Appl. Math., 101(1-3):5–19, 2008. ISSN 0167-8019. doi: 10.1007/s10440-008-9204-8. URL http://dx.doi.org/10.1007/s10440-008-9204-8.

[4] R. Alonso-Blanco, G. Manno, and F. Pugliese. Normal forms for Lagrangian distributions on 5-dimensional contactmanifolds. Differential Geom. Appl., 27(2):212–229, 2009. ISSN 0926-2245. doi: 10.1016/j.difgeo.2008.06.019. URLhttp://dx.doi.org/10.1016/j.difgeo.2008.06.019.

[5] M. Bachtold and G. Moreno. Remarks on non–maximal integral elements of the cartan plane in jet spaces. Journalof Geometry and Physics, page to appear, 2014.

[6] M. J. Bachtold. Fold–type solution singularities and charachteristic varieties of non–linear pdes. PhD thesis, Universityof Zurich, 2009. URL http://dx.doi.org/10.5167/uzh-42719.

[7] Guy Boillat. Sur l’equation generale de Monge-Ampere a plusieurs variables. C. R. Acad. Sci. Paris Ser. I Math.,313(11):805–808, 1991. ISSN 0764-4442.

[8] Guy Boillat. Sur l’equation generale de Monge-Ampere d’ordre superieur. C. R. Acad. Sci. Paris Ser. I Math., 315(11):1211–1214, 1992. ISSN 0764-4442.

[9] R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths. Exterior differential systems,volume 18 of Mathematical Sciences Research Institute Publications. Springer-Verlag, New York, 1991. ISBN 0-387-97411-3.

[10] Pietro De Poi and Emilia Mezzetti. Congruences of lines in P5, quadratic normality, and completely exceptionalMonge-Ampere equations. Geom. Dedicata, 131:213–230, 2008. ISSN 0046-5755. doi: 10.1007/s10711-007-9228-7.URL http://dx.doi.org/10.1007/s10711-007-9228-7.

[11] Bocharov et al. Symmetries and conservation laws for differential equations of mathematical physics, volume 182of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1999. ISBN 0-8218-0958-X.

[12] E. V. Ferapontov. Decomposition of higher-order equations of Monge-Ampere type. Lett. Math. Phys., 62(3):193–198,2002. ISSN 0377-9017. doi: 10.1023/A:1022236709346. URL http://dx.doi.org/10.1023/A:1022236709346.

[13] E. Goursat. Sur les equations du second ordre a n variables analogues a l’equation de Monge-Ampere. Bull. Soc.Math. France, 27:1–34, 1899. ISSN 0037-9484. URL http://www.numdam.org/item?id=BSMF_1899__27__1_0.

[14] G infinity (http://mathoverflow.net/users/22606/g infinity). Submanifolds in the grassmannian of n-dimensionalsubspaces determined by a submanifold in the grassmannian of l-dimensional subspaces. MathOverflow, . URLhttp://mathoverflow.net/q/138544. URL:http://mathoverflow.net/q/138544 (version: 2013-08-06).

[15] G infinity (http://mathoverflow.net/users/22606/g infinity). When a hyperplane of symmetric forms is de-termined by a quadric hypersurface? MathOverflow, . URL http://mathoverflow.net/q/141113.URL:http://mathoverflow.net/q/141113 (version: 2013-09-03).

[16] Ivan Kolar, Peter W. Michor, and Jan Slovak. Natural operations in differential geometry. Springer-Verlag, Berlin,1993. ISBN 3-540-56235-4.

[17] Joseph Krasil’shchik and Alexander Verbovetsky. Geometry of jet spaces and integrable systems. J. Geom. Phys., 61(9):1633–1674, 2011. ISSN 0393-0440. doi: 10.1016/j.geomphys.2010.10.012. URL http://dx.doi.org/10.1016/j.

geomphys.2010.10.012.[18] Boris Kruglikov and Valentin Lychagin. Geometry of differential equations. In Handbook of global analysis, pages

725–771, 1214. Elsevier Sci. B. V., Amsterdam, 2008.[19] Alexei Kushner, Valentin Lychagin, and Vladimir Rubtsov. Contact geometry and non-linear differential equations,

volume 101 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2007.ISBN 978-0-521-82476-7; 0-521-82476-1.

http://dx.doi.org/10.1016/S0926-2245(02)00105-5

http://dx.doi.org/10.5802/aif.2686

http://dx.doi.org/10.1007/s10440-008-9204-8

http://dx.doi.org/10.1016/j.difgeo.2008.06.019

http://dx.doi.org/10.5167/uzh-42719

http://dx.doi.org/10.1007/s10711-007-9228-7

http://dx.doi.org/10.1023/A:1022236709346

http://www.numdam.org/item?id=BSMF_1899__27__1_0

http://mathoverflow.net/q/138544

http://mathoverflow.net/q/141113

http://dx.doi.org/10.1016/j.geomphys.2010.10.012

http://dx.doi.org/10.1016/j.geomphys.2010.10.012


[20] P. D. Lax and A. N. Milgram. Parabolic equations. In Contributions to the theory of partial differential equations,Annals of Mathematics Studies, no. 33, pages 167–190. Princeton University Press, Princeton, N. J., 1954.

[21] V. V. Lychagin. Singularities of multivalued solutions of nonlinear differential equations, and nonlinear phenomena.Acta Appl. Math., 3(2):135–173, 1985. ISSN 0167-8019.

[22] V. V. Lychagin. Geometric theory of singularities of solutions of nonlinear differential equations. In Problems ingeometry, Vol. 20 (Russian), Itogi Nauki i Tekhniki, pages 207–247. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. iTekhn. Inform., Moscow, 1988. Translated in J. Soviet Math. 51 (1990), no. 6, 2735–2757.

[23] Giovanni Moreno. The geometry of the space of Cauchy data of nonlinear PDEs. Cent. Eur. J. Math., 11(11):1960–1981, 2013. ISSN 1895-1074. doi: 10.2478/s11533-013-0292-y. URL http://dx.doi.org/10.2478/

s11533-013-0292-y.[24] Tohru Morimoto. Monge-Ampere equations viewed from contact geometry. In Symplectic singularities and geometry

of gauge fields (Warsaw, 1995), volume 39 of Banach Center Publ., pages 105–121. Polish Acad. Sci., Warsaw, 1997.[25] Paolo Piccione and Daniel V. Tausk. The single-leaf Frobenius theorem with applications. Resenhas, 6(4):337–381,

2005. ISSN 0104-3854.[26] I. A. B. Strachan. On the integrability of a third-order Monge-Ampere type equation. Phys. Lett. A, 210(4-5):267–

272, 1996. ISSN 0375-9601. doi: 10.1016/0375-9601(95)00944-2. URL http://dx.doi.org/10.1016/0375-9601(95)

00944-2.[27] A. M. Vinogradov. Geometric singularities of solutions of nonlinear partial differential equations. In Differential

geometry and its applications (Brno, 1986), volume 27 of Math. Appl. (East European Ser.), pages 359–379. Reidel,Dordrecht, 1987.

[28] Luca Vitagliano. Characteristics, bicharacteristics and geometric singularities of solutions of pdes. InternationalJournal of Geometric Methods in Modern Physics, to appear. URL http://arxiv.org/abs/1311.3477.

(G. Manno) Department of Mathematics, University of Padova, Via Trieste 63, 35121 Padova, Italy

E-mail address, G. Manno: [email protected]

(G. Moreno) Mathematical Institute in Opava, Silesian University in Opava, Na Rybnicku 626/1, 746 01 Opava, Czech RepublicE-mail address, G. Moreno: [email protected]

http://dx.doi.org/10.2478/s11533-013-0292-y

http://dx.doi.org/10.2478/s11533-013-0292-y

http://dx.doi.org/10.1016/0375-9601(95)00944-2

http://dx.doi.org/10.1016/0375-9601(95)00944-2

http://arxiv.org/abs/1311.3477

GEOMETRY OF THIRD{ORDER EQUATIONS OF MONGE{AMPERE …urbanski/Moreno.pdf · GEOMETRY OF THIRD{ORDER...

Documents

Transcript of GEOMETRY OF THIRD{ORDER EQUATIONS OF MONGE{AMPERE …urbanski/Moreno.pdf · GEOMETRY OF THIRD{ORDER...