Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics

This paper is a natural companion of [Alekseevsky D.V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497-524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order (2D) Monge-Ampère equations, by using the so-called ''...

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Дата:	2016
Автори:	Manno, G., Moreno, G.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2016
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics / G. Manno, G. Moreno // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 29 назв. — англ.

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spelling	irk-123456789-1477302019-02-16T01:25:06Z Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics Manno, G. Moreno, G. This paper is a natural companion of [Alekseevsky D.V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497-524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order (2D) Monge-Ampère equations, by using the so-called ''meta-symplectic structure'' associated with the 8D prolongation M⁽¹⁾ of a 5D contact manifold M. We write down a geometric definition of a third-order Monge-Ampère equation in terms of a (class of) differential two-form on M⁽¹⁾. In particular, the equations corresponding to decomposable forms admit a simple description in terms of certain three-dimensional distributions, which are made from the characteristics of the original equations. We conclude the paper with a study of the intermediate integrals of these special Monge-Ampère equations, herewith called of Goursat type. 2016 Article Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics / G. Manno, G. Moreno // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 29 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D10; 35A30; 58A30; 14M15 DOI:10.3842/SIGMA.2016.032 http://dspace.nbuv.gov.ua/handle/123456789/147730 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	This paper is a natural companion of [Alekseevsky D.V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497-524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order (2D) Monge-Ampère equations, by using the so-called ''meta-symplectic structure'' associated with the 8D prolongation M⁽¹⁾ of a 5D contact manifold M. We write down a geometric definition of a third-order Monge-Ampère equation in terms of a (class of) differential two-form on M⁽¹⁾. In particular, the equations corresponding to decomposable forms admit a simple description in terms of certain three-dimensional distributions, which are made from the characteristics of the original equations. We conclude the paper with a study of the intermediate integrals of these special Monge-Ampère equations, herewith called of Goursat type.
format	Article
author	Manno, G. Moreno, G.
spellingShingle	Manno, G. Moreno, G. Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Manno, G. Moreno, G.
author_sort	Manno, G.
title	Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics
title_short	Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics
title_full	Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics
title_fullStr	Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics
title_full_unstemmed	Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics
title_sort	meta-symplectic geometry of 3rd order monge-ampère equations and their characteristics
publisher	Інститут математики НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/147730
citation_txt	Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics / G. Manno, G. Moreno // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 29 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT mannog metasymplecticgeometryof3rdordermongeampereequationsandtheircharacteristics AT morenog metasymplecticgeometryof3rdordermongeampereequationsandtheircharacteristics
first_indexed	2025-07-11T02:43:49Z
last_indexed	2025-07-11T02:43:49Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 032, 35 pages Meta-Symplectic Geometry of 3rd Order Monge–Ampère Equations and their Characteristics? Gianni MANNO † and Giovanni MORENO ‡ † Dipartimento di Scienze Matematiche “G.L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy E-mail: giovanni.manno@polito.it ‡ Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland E-mail: gmoreno@impan.pl URL: https://www.impan.pl/en/sites/gmoreno/home Received October 29, 2015, in final form March 16, 2016; Published online March 26, 2016 http://dx.doi.org/10.3842/SIGMA.2016.032 Abstract. This paper is a natural companion of [Alekseevsky D.V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497–524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order (2D) Monge–Ampère equations, by using the so-called “meta-symplectic structure” associated with the 8D prolon- gation M (1) of a 5D contact manifold M . We write down a geometric definition of a third- order Monge–Ampère equation in terms of a (class of) differential two-form on M (1). In particular, the equations corresponding to decomposable forms admit a simple description in terms of certain three-dimensional distributions, which are made from the characteristics of the original equations. We conclude the paper with a study of the intermediate integrals of these special Monge–Ampère equations, herewith called of Goursat type. Key words: Monge–Ampère equations; prolongations of contact manifolds; characteristics of PDEs; distributions on manifolds; third-order PDEs 2010 Mathematics Subject Classification: 53D10; 35A30; 58A30; 14M15 1 Introduction Classical Monge–Ampère equations (MAEs with one unknown function and two independent variables) constitute a distinguished class of scalar 2nd order (non-linear) PDEs owing to a re- markable property: the totality of their characteristics, herewith called characteristic cone1, degenerates into the union of zero, one or two 2D planes (according to the elliptic, parabolic or hyperbolic character of the equation). The primary motivation of this paper was to determine whether, and to what extent, such a phenomenon occurs also in the context of 3rd order PDEs, where there can be found physically interesting analogues of the classical MAEs [1, 13, 14, 15, 28]. To the authors’ best knowledge, such PDEs were defined by Boillat by using Lax’s complete exceptionality [9, 10, 21]. We introduce now, in a friendly coordinate way, the notion of characteristic cone, whose intrinsic definition will be given later on (see (1.9) and (5.9)). ?This paper is a contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti. The full collection is available at http://www.emis.de/journals/SIGMA/Benenti.html 1The notion behind it is rather old, and may have appeared in other guises someplace else. mailto:giovanni.manno@polito.it gmoreno@impan.pl https://www.impan.pl/en/sites/gmoreno/home http://dx.doi.org/10.3842/SIGMA.2016.032 http://www.emis.de/journals/SIGMA/Benenti.html 2 G. Manno and G. Moreno The main idea: studying PDEs using the geometry of their characteristics Let E : F ( x1, x2, u, p1, p2, . . . , pi1···il , . . . ) = 0, l ≤ k, (1.1) be a scalar kth order PDE with one unknown function u = u(x1, x2) and two independent variables (x1, x2). As usual (see [16] and, more recently, e.g., [18, 19] 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 (1.1) with some initial conditions f ( X1(t), X2(t) ) = U(t), ∂`f ∂z` ( X1(t), X2(t) ) = Q`(t), ` ≤ k − 1, (1.2) where X1(t), X2(t), U(t), Q`(t) (1.3) 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 (1.1), (1.2) is a function u = f(x1, x2) which, together with its derivatives, satisfies both (1.1) and (1.2). Initial data (1.3) can be used to construct the curve Φ(t) = ( X1(t), X2(t), U(t), P1(t), P2(t), . . . , Pi1···il(t), . . . ) , l ≤ k − 1, (1.4) in the space with coordinates (x1, x2, u, . . . , pi1···il , . . . ), l ≤ k − 1, which we can then interpret as a Cauchy datum (see, e.g., [25] for a jet-theoretic treatment of the space of Cauchy data). If such curve is non-characteristic for equation (1.1), then, assuming F real analytic, Cauchy problem (1.1), (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 (1.4) is characteristic for E at the point3 mk−1 ∈ E , i.e., a point mk−1 = ( x1, x2, u, . . . , pi1···il , . . . ) , l ≤ k, (1.5) whose coordinates satisfy (1.1), if the tangent vector ν = Φ̇(t0) at the point mk−2 = Φ(t0) = ( x1, x2, u, . . . , pi1···il , . . . ) , l ≤ k − 1, (1.6) such that ∑ `1+`2=k (−1)`1 ∂F ∂p1 · · · 1︸︷︷︸ `1 2 · · · 2︸︷︷︸ `2 ∣∣∣∣∣ mk−1 ( ν2 )`1(ν1 )`2 = 0, (1.7) where 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 ) . (1.8) 2We adopted the same notation used in [29], where the reader may also find a gentle introduction to the theory of characteristics and singularities of non-linear PDEs. 3The choice of notation “mk−2” and “mk−1” will be motivated later on. 3rd Order Monge–Ampère Equations 3 From (1.7) one can associate with any point (1.5) of E a number ≤ k of directions (1.8) (the polynomial (1.7) might possess multiple and/or imaginary roots) in the space with coordinates (x1, x2, u, . . . , pi1···il , . . . ), l ≤ k − 1. So, if we keep the point (1.6) fixed and let the point (1.5) vary in E , the aforementioned directions form, in general, a cone. The set VE mk−2 := {directions ν as in (1.8) \| ∃mk−1 ∈ E such that (ν1, ν2) satisfies (1.7)} is what we call the characteristic cone of E at mk−2, and the union VE := ⋃ mk−2 VE mk−2 (1.9) is a geometric object naturally associated to E (see Section 5.2 later on). In this paper, we basically study those PDEs that correspond to 3D distributions on the 8D first prolongation of a 5D contact manifold. Since the correspondence equation↔distribution is essentially the same as in the case of 2nd order MAEs, the PDEs we are interested in are precisely the 3rd order analogues of the classical (i.e., second-order) bidimensional MAEs. Our main motivation was the apparent lack of a systematic geometric analysis of 3rd order MAEs, carried out in the same spirit as for classical MAEs. Much as in the geometric approach to classical MAEs it is convenient to exploit the contact/symplectic geometry underlying 2nd order PDEs [4, 5, 12, 20, 22, 26], to deal with 3rd order MAEs we shall make use of the prolongation of a contact manifold, equipped with its Levi form (see, e.g., [27, Section 2]), a structure known as meta-symplectic, quickly reviewed below. It is worth stressing that our main gadget, i.e., the correspondence between E and VE , is just an adaptation of similar techniques traditionally found in other areas of modern mathematics4. In consequence, in this paper there coexist differential and algebraic geometric constructions, and the authors are aware that this may give a certain feeling of incompatibility. Section 4.5 below should help in this concern. Structure of the paper Section 2 reviews the classical notions needed for the formulation of the main Theorems 3.3, 3.4 and 3.5, which is given in Section 3, and the Sections 4, 5 introduce new ideas, namely the three- fold orthogonality and its relationship with the characteristics variety, necessary to the proof of the main results, worked out in Section 6. In Section 7 we study the intermediate integrals of Goursat-type 3rd order MAEs in terms of their characteristics. Notation and conventions All objects and morphisms herewith considered are assumed of class C∞, up to mild exceptions (see Section 4.5 below). We use the symbol “P” in order to avoid too many repetitions of the sentence “up to a confor- mal factor”. One-dimensional linear object may be identified with their generators. As a rule, if P → X is a bundle, and x ∈ X, we denote by Px the fibre of it at x. Symbol D′ denotes the derived distribution D + [D,D] of a distribution D. Differential forms on a manifold N (resp., distribution D) are denoted by Λ∗N (resp., Λ∗D∗) and f∗ denotes the tangent map of f . 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 . The symmetric tensor product is de- noted by �. The Einstein summation convention will be used, unless otherwise specified. Also, when a pair jk (resp., a triple ijk) runs in a summation, such summation is performed over 1 ≤ j ≤ k ≤ 2 (resp., 1 ≤ i ≤ j ≤ k ≤ 2), unless stated differently. Throughout the paper, by M (0), C0, and m(0) we shall always mean M , C, and m, respectively. 4Bäcklund transformation, double fibration transform, Penrose transform, etc. (see also [24] on this concern). 4 G. Manno and G. Moreno Remark 1.1. If there is any bundle structure, or a vertical distribution, in the surrounding manifold, the symbol Dv denotes the vertical part of the distribution D. 2 Preliminaries on (prolongations of) contact manifolds, and (meta)symplectic structures 2.1 Contact manifolds, their prolongations and PDEs Throughout this paper, (M, C) will be a 5D contact manifold, i.e., C is a completely non- integrable distribution of hyperplanes on M locally described as C = ker θ, where the 1-form θ is determined up to a conformal factor and θ ∧ dθ ∧ dθ 6= 0. The restriction dθ\|C defines on each 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 dimension is 2. We denote by L(Cm) the Grassmannian of Lagrangian planes of Cm and by π : M (1) := ∐ m∈M L(Cm)→M (2.1) the bundle of Lagrangian planes, also known as the 1st prolongation of M . The key property of the manifold M (1) is that it is naturally endowed with a 5D distribution, defined by C1 m1 := { 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 set 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-horizontal5 and such that all the forms belonging to the differential ideal generated by θ(1), vanish on it. In analogy with (2.1), we define the 2nd prolongation M (2) of a contact manifold (M, C) as the first prolongation of M (1), that is M (2) = ( M (1) )(1) := { Lagrangian planes of M (1) } . (2.2) Projection (2.1) is the beginning of a tower of natural bundles 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., [17]). The tautological bundle L → M (i) is defined by requiring that the fibre Lmi is mi itself, understood as a 2D subspace of Ci−1, with i = 1, 2. We keep the same symbol L for both the tautological bundles over M (1) and M (2), since it will be clear from the context which is which. 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. m and m1) appear in the same context, the former is always the π2,1-image (resp., the π-image) of the latter. By a kth order PDE we always mean a sub-bundle E ⊆ M (k−1) of codimension one whose fibre Emk−2 at mk−2 is henceforth assumed, without loss of generality, to be connected (see (4.26) below). Recalling that π2,1 is an affine bundle, we say that an equation E is quasi-linear at a point m1 ∈M (1) if the fibre Em1 is an affine subspace of π−1 2,1(m1), otherwise we say that it is non-linear at m1. We retain the same symbol L for the tautological bundle L\|E −→ E restricted to E . 5If horizontality is dropped in (2.2), one augments M (2) with the so-called “singular” integral elements of C1 which, in the case of PDEs, formalise the notion of “singular solutions”, originally introduced in [23] (see also the review paper [29] and references therein). 3rd Order Monge–Ampère Equations 5 2.2 The meta-symplectic structure on C1 The (local) conformal symplectic structure $ = dθ\|C admits a “global” analog, namely $glob : C ∧ C −→ C ′ C = TM C . (2.3) Indeed, since TM C is rank-one, $glob locally identifies with $. 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 [19] as a “twisted” analog of the symplectic form (2.3) for the prolonged contact distribution C1 and called, for this reason, meta-symplectic: Ω: C1 ∧ C1 −→ (C1)′ C1 . (2.4) Notice that, unlike (2.3), the form (2.4) takes its values into a rank-two bundle so that, even locally, it cannot be regarded as a 2-form in the standard sense. Nevertheless, it can be used for defining Lagrangian subspaces: indeed, a π2,1-horizontal 2D subspace of C1 m1 is Lagrangian if it is Ω-isotropic. 3 Description of the main results In the case of a 3rd order PDE E , polynomial (1.7) always admits a linear factor, so that the corresponding characteristic cone VE , always contains a linear irreducible component VEI (we stress that in the fully decomposable case, all the three irreducible components are linear, and it suffices to choose one). Definition 3.1. If ω ∈ Λ2M (1) is a 2-form, then the hypersurface Eω := { m2 ∈M (2) \|ω\|Lm2 = 0 } (3.1) in M (2) is called the Boillat-type 3rd order MAE (associated to ω). The first result of this paper is concerned with the characteristic cone VEω . Namely, we show that VEω can be used to recover the equation itself, since there is an obvious way to “invert” the construction of the characteristic cone (1.9) out of a PDE E : take any sub-bundle V ⊆ PC1 and associate with it the following subset EV := { m2 ∈M (2) \| ∃H ∈ V : Lm2 ⊃ H} ⊆M (2). (3.2) Now, if V is regular enough, the corresponding EV turns out to be a genuine 3rd order PDE, and examples of “regular enough” sub-bundles are provided by characteristic cones of 3rd order PDEs themselves. In particular, it always holds the inclusion EVE ⊇ E . Definition 3.2. We say that the equation E is recoverable (from its characteristics) if E = EVE . (3.3) Theorem 3.3. Any 3rd order MAE is recoverable from its characteristics. The simplest examples of equations (3.2) are obtained when V = PD, where D ⊂ C1 is a 3D sub-distribution: ED := EPD = { m2 ∈M (2) \|Lm2 ∩ Dm1 6= 0 } , (3.4) 6 G. Manno and G. Moreno with m2 projecting onto m1. Equations (3.4) are herewith dubbed Goursat-type 3rd order MAEs. Observe that the equations (3.4) form the sub-class of the equations (3.1) which are determined by 2-forms which are decomposable modulo the differential ideal generated by contact forms (see the beginning of Section 6.3). The second result of this paper allows to locally characterizes Goursat-type 3rd order MAEs in terms of their characteristic cone. Theorem 3.4. Let E be a 3rd order PDE. Then E is locally of the form (3.4), if and only if its characteristic cone VE contains an irreducible component VEI which is a 2D linear projective sub-bundle. Theorem 3.4 implies that E ⊆ M (2) is a Goursat-type 3rd order MAE if and only if its characteristic cone decomposes as VE = PD1 ∪ VEII, (3.5) where D1 ⊆ C1 is a 3D sub-distribution and VEII encompasses all the remaining irreducible components of VE . Surprisingly enough, if (3.5) 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 Ω ( h1,Dvi ) = Ω ( hi,Dv1 ) = 1D space, (3.6) where hi is a non-zero horizontal vector in Di, i = 1, 2, 3 (for the definition of Dv see Remark 1.1). Observe that, if a Goursat-type MAE is non-linear in a point m1 ∈M (1), then it is non-linear in a neighbourhood of m1; on the contrary, if it is linear in m1, then a neighbourhood of m1 where it is quasi-linear may not exist (see (4.25)). On account of this, we give the following Theorem 3.5, that finalises our characterisation of Goursat-type 3rd order MAEs through their characteristics. Theorem 3.5. Let E = EPD1, where D1 ⊂ C is a 3-dimensional distribution, be a Goursat-type MAE. Let m1 ∈M (1). Then E = EPD1∪VEII and 1) E is quasi-linear at m1 if and only if dim(Dv1)m1 = 2: in this case, (VEII)m1 is either empty or equal to P(D2)m1 ∪ P(D3)m1, where (Di)m1 is unambiguously defined by (3.6) and E(Di)m1 = E(D1)m1 , i = 2, 3; 2) E is non-linear at m1 if and only if dim(Dv1)m1 = 1: in this case, if (VEII)m1 is not empty, then it cannot contain any linear irreducible component and Em1 = E(VEII)m1 ; 3) if Em1 = E(D2)m1 for some 3D subspace (D2)m1 ⊂ C1 m1, then either (D1)m1 = (D2)m1 or (D1)m1 and (D2)m1 are “orthogonal” in the sense of (3.6). As a main byproduct of Theorem 3.5 we shall obtain a method for finding intermediate integrals, discussed in Section 7. We conclude this section with a local description of the main objects introduced so far. 3.1 Local coordinate description of the main objects Let (xi, u, pi) be contact coordinates on M , i = 1, 2. Then θ = du− pidxi, and C = 〈D1, D2, ∂p1 , ∂p2〉 , (3.7) where Di is the total derivative with respect to xi, truncated to the 0th order. The above- introduced system (xi, u, pi) induces coordinates( xi, u, pi, pij = pji, 1 ≤ i, j ≤ 2 ) (3.8) 3rd Order Monge–Ampère Equations 7 on M (1) as follows: a point m1 ≡ Lm1 ∈ M (1) has coordinates (3.8) iff m = π(m1) = (xi, u, pi) and the corresponding Lagrangian plane Lm1 is given by Lm1 = 〈Di + pij∂pj \| i ∈ {1, 2}〉 ⊂ Cm. (3.9) Similarly, C1 = 〈D1, D2, ∂p11 , ∂p12 , ∂p22〉 , (3.10) where now the Di’s stand for the 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 Lm2 = 〈 Di + pijk∂pjk \| i ∈ {1, 2} 〉 ⊆ C1 m1 , (3.11) where the vector fields Di and ∂pij are tacitly assumed to be evaluated at m1. Remark 3.6. We always use the symbol Di for the truncated total derivative with respect to xi, i = 1, 2, the order of truncation depending on the context. For instance, the order of truncation is 0 in (3.7) and it is 1 in (3.10) and (3.11). It is convenient to set ξi := Di\|mk , k = 1, 2, (3.12) where the total derivatives appearing in (3.12) are truncated to the (k − 1)st order. Indeed, both (3.9) and (3.11) simplify as Lmk = 〈ξ1, ξ2〉 , k = 1, 2, (3.13) and their dual as L∗ mk = 〈 dx1, dx2 〉 , respectively. Using the local coordinates (3.10), it is easy to realize that (C1)′ is spanned by C1 and 〈∂p1 , ∂p2〉, so that the quotient (C1)′ C1 identifies with the latter, and (2.4) reads Ω = dpij ∧ dxi ⊗ ∂pj . (3.14) As a vector-valued differential 2-form, Ω can be identified with the pair Ω ≡ (−dθ1,−dθ2) where θi = dpi − pijdxj . 3.1.1 Boillat and Goursat 3rd order MAEs We conclude with a local coordinate description of Boillat (3.1) and Goursat (3.4) equations E = {F = 0}. Locally, the former is given by F = det p111 p112 p122 p112 p122 p222 A + B · (p111, p112, p122, p222)T + C, (3.15) where A (resp., B, C) is an R3-(resp., R4-, R-)valued smooth function on M (1), and the latter is either given by F = det p111 − f111 p112 − f112 p122 − f122 p112 − f211 p122 − f212 p222 − f222 A  , (3.16) where A is as in (3.15) and fijk ∈ C∞(M (1)), or it is quasi-linear. 8 G. Manno and G. Moreno 4 Vertical geometry of (prolongations of) contact manifolds and their characteristics The departing point of our analysis of 3rd order PDEs is to identify them with sub-bundles of the 1st prolongation M (2) = (M (1))(1) of M (1). Hence, a key role will be played by their vertical geometry, i.e., the 1st order approximation of their bundle structure, and, in particular, by the so-called rank-one vectors, which are in turn linked to the notion of characteristics. In order to introduce these concepts, we begin with the vertical geometry of the surrounding bundle, 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 bundle over 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, it does not possess a naturally defined vertical bundle: we replace it by the following bundle on M (1): VM := ∐ m1∈M(1) Cπ(m1) Lm1 . (4.1) Lemma 4.1. It holds the following canonical isomorphism: VM (k) ' Sk+1L∗, k ≥ 0. (4.2) Proof. It is a generalisation of the proof of the jet-theoretic version of the statement (see, e.g., [6, Theorem 3.2]). � Directly from the definition (4.1) of VM and the isomorphism (4.2) for k = 0 one obtains Cπ(m1) Lm1 ∼= L∗m1 . (4.3) Example 4.2. Of particular importance will be the vertical vectors on M (2) which correspond to the perfect cubes of covectors on L via the fundamental isomorphism (4.2). For instance, if α := αidx i ∈ L∗m2 (4.4) (see Remark 3.6), then the corresponding vertical vector is S3L∗m2 3 α⊗3 ←→ ∑ i+j=3 αi1α j 2 ∂ ∂p1 · · · 1︸︷︷︸ i 2 · · · 2︸︷︷︸ j ∣∣∣∣∣ m2 ∈ Vm2M (2). (4.5) Even if the sections of VM are not, strictly speaking, vector fields, and a such they lack an immediate geometric interpretation, the bundle VM itself has important relationships with the contact bundle C1. Namely, on one hand, it is canonically embedded into the module of 1-forms on C1 (see (4.9) later on) and, on the other hand, it is identified with the quotient distribution (C1)′ C1 (Lemma 4.3 below). 3rd Order Monge–Ampère Equations 9 Lemma 4.3. There is a (conformal)6 natural isomorphism (C1)′ C1 ∼= L∗. (4.6) Proof. It follows from a natural isomorphsim between the left-hand sides of (4.3) and (4.6). Indeed, the map (C1)′m1 C1 m1 −→ Cπ(m1) Lm1 (4.7) induced from π∗ is well-defined and linear, for all m1 ∈M (1). In other words, (4.7) is a bundle morphism, is well-defined and linear, and surjective. Since the ranks of the bundles are the same, it is an isomorphism. � We stress that there is no canonical way to project the bundle C1 over the tautological bundle L, if both are understood as bundles over M (2). Nevertheless, if C1 and L are regarded as bundles over M (1), then π∗ turns out to be a bundle epimorphism from C1 to L, i.e., π∗ : C1 −→ L = π∗ ( C1 ) . (4.8) Dually, epimorphism (4.8) leads to the bundle embedding L∗ ↪→ C1 ∗ which can be combined with the identification L∗ ∼= VM . The result is a (conformal) embedding of bundles over M (1), VM ↪→ C1 ∗, (4.9) which will be useful in the sequel. In local coordinates, (4.9) 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 never been observed before), which generalizes the symplectic orthogonality within the contact distribution C of M . Indeed, an immediate consequence of Lemma 4.3 is that the meta-symplectic form Ω is L∗-valued, i.e., C1 ∧ C1 Ω−→ L∗ is a trilinear form C1 ∧ C1 ⊗M(1) L Ω−→ C∞ ( M (1) ) . (4.10) In turn, thanks to the canonical projection (4.8) of C1 over L, the form (4.10) descends to a trilinear form C1 ∧ C1 ⊗ C1 Ω̃−→ C∞(M (1)). Such notions (orthogonal vectors, orthogonal complement, Lagrangian subspaces) can be found also in a 5D meta-symplectic space, but with more subtleties. For example, there can be up to two distinct, so to speak, “orthogonal complements” to a given subspace. Definition 4.4 (three-fold orthogonality). Elements X1, X2, X3 ∈ C1 are orthogonal if and only if Ω̃(X1, X2, X3) = 0. Remark 4.5. In this perspective, condition (3.6) expresses precisely the fact that the pair {D2,D3} is the orthogonal complement of the distribution D1, in the sense that Ω̃(Di,Dj ,Dk) = 0, {i, j, k} = {1, 2, 3}. 6In the sense that (4.6) is canonical only up to a conformal factor. 10 G. Manno and G. Moreno 4.2 Rank-one lines, characteristic directions and characteristic hyperplanes Isomorphism (4.2) 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∗. Geometrically, these are the tangent directions at mk = mk(0) to the curves mk(t) such that the corresponding family Lmk(t) of Lagrangian subspaces in M (k−1) “rotates” around a common hyperplane (which, in our case, is a line). Local sections of the dual bundle Ck∗ will be called vertical forms, as they are the image of the local sections of V ∗M (k) under the natural projection7. Because of the dual of the fundamental isomorphism (see Lemma 4.1), Ck∗ is the epimorphic image of the symmetric power Sk+1L of the tautological bundle, and we can speak of decomposable vertical forms, if they come from exact powers. Definition 4.6. 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, a hyperplane 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., [13]). Observe that, in our contest, dimHα = 1, so that we shall speak of a characteristic line and call characteristic vector a generator of Hα. The geometric relationship between Hα and ` = 〈 α⊗3 〉 is well-known (see, e.g., [29]) and it can be rendered by ` = PTmk ( H(1) α ) , (4.11) where the 1st prolongation H (1) α = {mk ∈M (k) mk−1 \|Lmk ⊇ Hα} of Hα has dimension one. By prolongation of a subspace W ⊆ TmkM (k) we mean the subset W (1) ⊆M (k+1) mk made of the Lagrangian planes which are contained in W . The prolongation of a submanifold W ⊆ M (k) 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). Remark 4.7. Let α and Lm2 be locally described by (4.4) and (3.12). Then Hα = ker(α1dx 1 + α2dx 2) = 〈α2ξ1,−α1ξ2〉, i.e., one can identify α↔ Hα via a counterclockwise π 2 rotation: α ≡ (α1, α2)↔ (α2,−α1) ≡ Hα. (4.12) From now on, characteristic directions and characteristic lines will be taken as synonyms, thus breaking the separation proposed below: 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\{0} characteristic covector It should be stressed that in the multidimensional cases (see, e.g., [2]) it is no longer possible to regard the characteristic directions as lines lying in L, since the latter are not hyperplanes. 7We insist that “vertical forms” are not, globally speaking, differential forms on M (k), but rather equivalence classes of them. Locally, of course, one can find a representative in each class. 3rd Order Monge–Ampère Equations 11 4.3 Canonical directions associated with orthogonal distributions Suppose that the vertical form ω ∈ C1 ∗ is decomposable in the sense of Lemma 4.1. Then it is easy to see that there are two characteristic lines H1 and H2. The meta-symplectic form (3.1) links these two lines with the vertical distribution V := kerω determined by ω. Recall now that Ω is VM ' L∗-valued (see (4.2) and (4.10)), so that Hi can be considered as a subspace of VM (in view of Remark 4.7), and there are the induced maps Ω(X, · ) : C1 −→ VM Hi , X ∈ C1. Letting X vary in V , one gets the rank≤ 2 sub-bundle V yΩ Hi := 〈Ω(X, · ) +Hi \|X ∈ V 〉 ⊂ C1∗ ⊗ VM Hi . Finally, Hj ⊗ L∗ Hi can be regarded as a one-dimensional sub-bundle of C1∗ ⊗ VM Hi , via (4.9). Taking this into account, we have the following result. Lemma 4.8. It holds the following identification V yΩ Hi ' Hj ⊗ L∗ Hi , {i, j} = {1, 2}. In particular, dim V yΩ Hi = 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 ω decomposable, it can be brought, up to a scaling, to the form ω = dp11− (k1 + k2)dp12 + k1k2dp22 ∈ C1∗. Accordingly, Hi = ∂p2 + ki∂p1 and a straightforward computation shows that V = 〈X1, X2〉, with X1 = −k1k2∂p11 + ∂p22 , X2 = (k1 + k2)∂p11 + ∂p12 . (4.13) Then V yΩ = 〈X1yΩ, X2yΩ〉 = 〈 k1k2dx 1 ⊗ ∂p1 − dx2 ⊗ ∂p2 ,−(k1 + k2)dx1 ⊗ ∂p1 − dx1 ⊗ ∂p2 − dx2 ⊗ ∂p1 〉 . (4.14) If the basis {Hi, ∂p1} for VM ' L∗ (see (4.2)) is chosen, then the factor of (4.14) with respect to Hi is the line〈( kjdx 1 + dx2 ) ⊗ ∂p1 〉 , i 6= j, (4.15) and the lemma is proved. � Lemma 4.8 will be used later on in the proof of Proposition 6.7, establishing that, associated with a quasi-linear MAE with completely decomposable symbol, there are three canonical lines in M . 4.4 Some examples Let E be as in (1.1), with F given by p111−p112−2p122 (resp., p122 and p111). Then equation (1.7) reads ( ν2 )3 + ( ν2 )2 ν1 − 2ν2 ( ν1 )2 = 0 ( resp., ν2 ( ν1 )2 = 0 and ( ν2 )3 = 0 ) . (4.16) 12 G. Manno and G. Moreno Applying definition (1.9), one obtains that VE = ∪3 i=1PDi, with D1 = 〈D1, ∂p11 + ∂p12 , 2∂p11 + ∂p22〉 , (4.17) D2 = 〈D1 +D2, 2∂p11 + ∂p12 , ∂p22〉 , (4.18) D3 = 〈D1 − 2D2, ∂p11 − ∂p12 , ∂p22〉 (4.19) (resp., D1 = 〈D1, ∂p11 , ∂p12〉 , D2 = D3 = 〈D2, ∂p11 , ∂p22〉 (4.20) and D1 = D2 = D3 = 〈D1, ∂p12 , ∂p22〉). Observe that, in the first case, equation (4.16) factors as ν2 ( ν2 − ν1 )( ν2 + 2ν1 ) = 0, (4.21) and that the “horizontal components” hi (i.e., the first generators) of (4.17), (4.18) and (4.19) are precisely the three distinct roots of (4.21). Notice also that, in general, the distributions associated with the same quasi-linear 3rd order MAE need not to be contactomorphic. Here it follows an easy counterexample. For instance, consider the equation p122 = 0 above: its characteristic cone consists of the two distributions D1 and D2 = D3 (see (4.20)), which are not contactomorphic. Indeed, the derived distribution D′1 := D1 + [D1,D1] of D1 is 5-dimensional, whereas dim(D′2) = 4. Let E be the first equation from above, i.e., with F = p111−p112−2p122. The following three lines 1st line: Ω ( D1,D2) = 〈2∂p1 + ∂p2〉 , (4.22) 2nd line: Ω ( D1,D3) = 〈∂p1 − ∂p2〉 , (4.23) 3rd line: Ω ( D2,D3) = 〈∂p2〉 , (4.24) are canonically associated with the triple (D1,D2,D3), i.e., with the equation E . It is worth observing that the vertical part Dv1 is an integrable vertical distribution on M (1), so that, in this case, lines (4.23) and (4.24) are the characteristic lines of the family of equations 2p22−p11+p12 = k(x1, x2, u). Nevertheless, integrability Dv1 is not indispensable for associating directions (4.23) and (4.24) with the distribution D1, since they occur as the characteristic lines of the (not necessarily closed) vertical covector 2dp22 − dp11 + dp12, which annihilates Dv1 . Now we give another example that clarifies the type-changing phenomenon: p11(p112p122 − p111p222)− p122 = 0. (4.25) In fact, equation (4.25) is non-linear at the points with p11 6= 0, whereas it is linear at the points with p11 = 0. Equation (4.25) is of Goursat type and the corresponding 3-dimensional distribution D = 〈D1, p11D2 + ∂p11 , ∂p12〉 behaves accordingly with Theorem 3.5, i.e., the dimension of Dv is 1 if p11 6= 0 and it is 2 if p11 = 0. Moreover, at the points with p11 = 0 we have three planes, whose two of them coincide, defining the fibre of equation (4.25) (see (4.20)). 4.5 Smoothness and singularities issues For generic PDEs, we just assume that E −→ M (1) is the zero locus of a smooth function, without requiring the non-vanishing of its differential. In all the statements and reasonings, we tacitly restrict ourselves to the open and dense subset Ereg of smooth points. 3rd Order Monge–Ampère Equations 13 According to our definition of a PDE E as a sub-bundle of M (1), the natural projection E −→ M (1) is surjective, and that each fibre Em1 is a smooth submanifold of M (2) m1 , possibly with singularities. For example, the singularities which can occur for 3rd Monge–Ampère equations are of “al- gebraic type”, i.e., E −→ M (1) is a bundle of projective varieties. Indeed, each fibre M (2) m1 is naturally understood as the Grassmannian variety of isotropic elements with respect to the meta- symplectic structure which, in analogy with the standard Lagrangian Grassmannian, embeds into a suitable projective space via the Plücker embedding8. More precisely, a straightforward computation shows that the subset of the singular points of ED is made by the m2’s such that Lm2 ⊂ Dm1 . Such a possibility occurs only when dim(Lm2 ∩ Dm1) = 2 for m2 ∈ M (2) (in the case when Lm2 and Dm1 are not transversal, generically the dimension is 1, otherwise is generically 0). Methodologically, this paper is at the crossroad between algebraic and differential geometry. The geometric approach to non-linear PDEs is traditionally differential, while recent develop- ments revealed that many features of PDEs are genuinely algebraic-geometric. Even though a rigorous algebraic-geometric approach to the present topics is feasible, the authors have opted for the traditional and – in a sense, easier – methods, based on elementary differential geometry. Summing up, all the objects are herewith assumed to be smooth, and all maps to be of class C∞, except for: • the fibres of E , which are the zero loci of smooth functions (and, as such, may display singularities where the differential vanishes); • the sub-bundles V of the projectivised contact bundle PC1, which are smooth families of projective sub-varieties in each fibre, i.e., locally isomorphic to the product of an open subset of V by a projective subvariety in P4. If VI and VII are sub-bundles of V, in the aforementioned sense, we say that VI and VII are irreducible components of V if, for any point m1 ∈M (1), the projective variety Vm1 is reducible, and (VI)m1 and (VII)m1 are its irreducible components. We conclude this preliminary part of the paper 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 M (k) ←→ Jk+1(2, 1), k ≥ 0, (4.26) and easily adapt the main results to the new setting. 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 5.1. A characteristic covector α ∈ L∗m2\{0}, with m2 ∈ Em1 , is characteristic for E in m2 if the corresponding rank-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 (4.11) says precisely that the rank-one line ` which corresponds to 〈 α⊗3 〉 via the isomorphism (4.2) is precisely the (one-dimensional) tangent space to the prolongation 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. 8The general theory of this would lead us beyond the scope of the present paper. 14 G. Manno and G. Moreno Definition 5.2. If H (1) α is entirely contained in Em1 , then 〈α〉 is called a strong characteristic (direction) and Hα a strongly characteristic line (in m2). Example 5.3. Function (1.1) determines, for k = 2, the 3rd order PDE E = {F = 0}. Now equation (1.7) can be correctly interpreted as follows: it is satisfied if and only if the vector ν = νiξi given by (1.8) spans a characteristic line of E . The very same equation (1.7) tells also when the covector α = ν2dx1 − ν1dx2 spans a characteristic direction of E . In the last perspective, equation (1.7) is nothing but the right-hand side of (4.5) 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 the vertical tangent space to E at m2 as the subspace Vm2E := Tm2Em1 ≤ Vm2M (2), (5.1) called the symbol of E by many authors9. Obviously, vertical tangent spaces can be naturally assembled into a linear bundle V E := ∐ m2∈E Vm2E , called the vertical bundle of E . Directly from (5.1) it follows the bundle embedding V E ⊆ VM (2) ∣∣ E , where fibres of the former are hyperplanes in the fibres of the latter. Thanks to Lemma 4.1, Vm2E can also be regarded as a subspace of S3L∗m2 , and, being the identification (4.2) manifestly conformal, such an inclusion descends to the corresponding projective spaces, i.e., PVm2E ⊆ PS3L∗m2 . (5.2) The dual of canonical inclusion (5.2) reads Ann(Vm2E) ∈ PS3Lm2 , (5.3) where Ann(Vm2E) is the (one-dimensional) subspace ( Vm2M (2) )∗ made of covectors vanishing on the hyperplane Vm2E , viz. Ann(Vm2E) ≤ ( Vm2M (2) )∗ = (S3L∗m2)∗ = S3Lm2 . (5.4) From a global perspective, (5.3) is nothing but the definition of a section PS3L // E . Ann(V E) rr (5.5) Remark 5.4 (symbol of a function). Let E = {F = 0}, and use the same coordinates (3.13) of Remak 3.6. Then Ann(Vm2E) = 〈Smblm2 F 〉 , where Smblm2 F is the symbol of F at m2, i.e., Smblm2 F = ∂F ∂pijk ∣∣∣∣ m2 ξiξjξk, (5.6) where ξi has been defined in (3.12). Observe that Ann(Vm2E) is independent of the choice of F in the ideal determined by E , so that Ann(V E) can be replaced with SmblF . 9Herewith we prefer to use the term “symbol” only for the function F determining E , and not for E itself. The two things, however, are the same, if one just regards F as a the (3rd order) differential operator defining the equation E . 3rd Order Monge–Ampère Equations 15 In view of (5.4), a direction 〈α〉 ∈ PL∗m2 is a characteristic one for E at m2 if and only if〈 Smblm2 F, α⊗3 〉 = 0, (5.7) where 〈 · , · 〉 is the canonical pairing on S3Lm2 . Needless to say, (5.7) 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 (1.7). Remark 5.5 (coordinates of V E). If E = {F = 0} is a 3rd order PDE according to the definition given at the end of Section 2.1, then dF \|Em1 is nowhere zero, for all m1 ∈M (1) (recall Section 4.5). So, Vm2E = ker ∂F ∂pijk ∣∣∣∣ m2 dpijk is an hyperplane in the 4D space Vm2M (2). Remark 5.6 (types of 3rd order PDEs). Observe that (5.6) is a 3rd order homogeneous poly- nomial in two variables, so that we 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, • ∆(m2) > 0 ⇔ (5.6) decomposes into three distinct linear factors; • ∆(m2) = 0 ⇔ (5.6) contains the square of a linear factor; • ∆(m2) < 0 ⇔ (5.6) contains an irreducible quadratic factor. Equation E = {F = 0} may be called “fully parabolic” in m2 ∈ E if (5.6) reduces to the cube of linear factor in the point m2. 5.2 The characteristic variety as a covering of the characteristic cone Let E = {F = 0}, and regard L as a sub-bundle of the pull-back of C1 to the equation E , bearing in mind the diagram L � � // �� C1 �� E //M (1) Lm2 � � // �� C1 m1 �� E 3 m2 π2,1 // m1 ∈M (1) (5.8) Now we can come back to the direction ν defined by (1.8) and notice that it lies in Lm2 , and also provide an intrinsic way to check whether ν is characteristic or not. Indeed, in view of the fundamental isomorphism (4.2), one can regard the cube ν⊗3 of ν as a tangent vector to the fibre M (2) m2 , and equation (1.7) tells precisely when ν⊗3 belongs to the sub-space Vm2E = Tm2Em1 16 G. Manno and G. Moreno (see (5.1)), up to a line-hyperplane duality (see also Remark 4.7). In other words, the set of characteristic lines charRm2 E := {H ∈ PL∗m2 \|H is a characteristic hyperplane of E} ⊆ PL∗m2 is the projective sub-variety in PL∗m2 cut out by (1.7). Since we plan to carry out an analysis of certain PDEs via their characteristics, it seems na- tural to consider all characteristic lines at once, i.e., as a unique geometric object. Traditionally, one way to accomplish this is to take the disjoint union charR E := ∐ m2∈E charRm2 E , known as the characteristic variety of E (see, e.g., [11]). As a bundle over E , the family of the fibres of charR E coincides with the equation E itself. So, the bundle charR E , in spite of its importance for the study of a given equation E , cannot be used to define E as an object pertaining to M (1). Still this can be arranged: it suffices “to project everything one step down”, so to speak. The result is precisely the characteristic cone VE : Projective sub-bundle of PL→M (2) Characteristic variety charR E (π2,1)∗−→ Projective sub-bundle of PC1 →M (1) Characteristic cone VE (5.9) It is easy to see that some characteristic lines, which are distinct entities in charR E , may collapse into VE (e.g., when the corresponding fibres of the tautological bundle have non-zero intersec- tion): hence, VE is not the most appropriate environment for the study of characteristics. On the other hand, VE is a bundle over M (1), where there is no trace of the original equation E : as such, it may effectively replace the equation itself and serves as a source of its invariants. Now we can give a rigorous definition of our main tool, the characteristic cone of E . Indeed (see Remark 4.7), the characteristic variety charR E can be regarded as a sub-bundle of PL −→ E . In turn, this makes it possible to use the commutative diagram (5.8) to map charR E to PC1 and define the sub-bundle VE := { H ∈ PC1 \| ∃m2 ∈ E : H ⊂ Lm2 and H is a characteristic hyperplane of E at m2 } ⊆ PC1 as the image of such a mapping. Definition 5.7. Above defined bundle VE −→M (1) is the characteristic cone of E . By its definition, VE fits into the commutative diagram charR E // // �� VE �� E //M (1) (5.10) revealing that, in a sense, VE is covered by charR E . It is worth observing that, from a local perspective, (1.9), (5.9), and Definition 5.7 all define the same object. For instance, by Definition 5.7, a line H = 〈ν〉 ∈ PC1 m1 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 5.3 shows that this is the case if and only if the generator ν = (ν1, ν2) of H satisfies equation (1.7). 3rd Order Monge–Ampère Equations 17 5.3 The irreducible component VE I of the characteristic cone of a 3rd order PDE E Let E = {F = 0} and identify Ann(V E) with SmblF as in Remark 5.4. As a homogeneous cubic tensor on L (see (5.5)), SmblF possesses a linear factor, i.e., a section PL // E SmblI Ftt such that SmblF = SmblI F � SmblII F, (5.11) where SmblII F is a section of PS2L→ E . Plugging (5.11) into (5.7) shows that, in order to have〈 SmblI,m2 F � SmblII,m2 F, α� α2 〉 = 〈SmblI,m2 F, α〉 〈 SmblII,m2 F, α2 〉 = 0 it suffices that 〈SmblI,m2 F, α〉 = 0. (5.12) In other words, if (5.12) is satisfied, then 〈α〉 ∈ PL∗m2 is a characteristic direction (incidentally revealing that 3rd order PDEs always possess a lot of them). Moreover, in view of the identification of characteristic directions with characteristic lines (Remark 4.7), (5.12) shows also that the line SmblI,m2 F ∈ PLm2 is always a characteristic line for E , since there always is an α such that (5.12) is satisfied. It is clear now how diagram (5.8) can be made use of and how to define, much as we did in Section 4.2, a subset VEI ⊆ VE fitting into the commutative diagram SmblI F // �� VEI �� E //M (1) parallel to (5.10). This gives a solid background to the statement of Theorem 3.4, where VEI was mentioned without exhaustive explanations. 6 Proof of the main results Now we are in position to prove the main results, Theorem 3.3, concerned with the structure of 3rd order MAEs, Theorem 3.4 and Theorem 3.5, concerned with the structure of Goursat-type 3rd order MAEs. Since the former are special 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 (1.9), the main object of our interest has been defined again in a less direct way, which passes through the characteristic variety (see above Definition 5.7). The reason behind this choice is revealed by diagram (5.10): the characteristic variety charR E plays the role of a “minimal covering object” for both the equation E itself and its characteristic cone VE . 18 G. Manno and G. Moreno It is easy to guess10 that such a minimal covering must be defined in terms of Ω-isotropic flags on M (1), in the sense of Section 2.2, i.e., by means of the commutative diagram Fliso2,1(C1) %1 $$ %2 zz M (2) // //M (1) PC(1)oooo (6.1) where Fliso2,1(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 (6.1) allows to pass from a sub- bundle of M (2) (e.g., a 3rd order PDE E) to a sub-bundle of PC1 (e.g., its characteristic cone VE), and vice-versa. Lemma 6.1 indicates how to proceed in one direction. Lemma 6.1. Let E be a PDE. 1. Take the pre-image %−1 2 (E) of E and select the points of tangency with the %1-fibres: the result is charR E. 2. Project charR E onto PC1 via %1: the result is VE . Proof. Item 1 follows from the fact that the %1-fibres are the prolongations H(1) of hyperplanes H ∈ PC(1), and the tangency condition means that they are determined by characteristics (see (4.11)). Item 2 is just a paraphrase of Definition 5.7. � Coming back is easier. Indeed, given a sub-bundle V ⊆ PC1, the same definition (3.2) of EV given earlier can be recast as EV = %2 ( %−1 1 (V) ) . (6.2) In spite of the name “double fibration transform”, performing (6.2) first, and then applying Lemma 6.1 to the resulting equation EV , does not return, as a rule, the original sub-bundle V. Corollary 6.2. Let V ⊆ PC1 be a sub-bundle. Then V ⊆ VEV . (6.3) Proof. Directly from Lemma 6.1 and formula (6.2). � We stress that, for any odd-order PDE E , it holds the inclusion E ⊆ EVE . (6.4) This paper begins to tackle the problem of determining those PDEs which are recoverable from their characteristics in the sense that (3.3) is valid. Lemma 6.3 below provides a simple evidence that, as a matter of fact, not all PDEs are recoverable (in the sense of Definition 3.2). Lemma 6.3. V is made of strongly characteristic lines for EV . Proof. Let H ∈ V be a line belonging to V. Then %2(%−1 1 (H)) is nothing but H(1) (see Definition 5.2) and (6.2) tells precisely that H(1) is entirely contained in EV . Hence, H is a strongly characteristic line for EV in any point of %2(%−1 1 (H)). � 10The structural relationship between the rank-one cone and the characteristic variety, which is the backbone of the present paper, made its appearance in many discussions concerning the characteristic variety, including [11, Chapter V], the late-1960’s work by Guillemin on the characteristics of Pfaffian systems, and the more more recent work by Malgrange on the homological algebra of characteristic varieties. 3rd Order Monge–Ampère Equations 19 The meaning of Theorem 3.4 is that 3rd order MAEs of Goursat-type are, in a sense, those PDEs whose characteristic cone takes the simplest form, namely that of a linear projective sub- bundle, and, moreover, they are also recoverable from it. Observe that (3.4) is a particular case of (6.2), so that Goursat-type 3rd order MAEs constitute a remarkable example of equations determined by a projective sub-bundle of PC1. Then Lemma 6.3 reveals that all the lines lying in D are strongly characteristic lines for ED, in strict analogy with the classical case [2]. 6.2 Proof of Theorem 3.3 As we already underlined, we called “3rd order MAEs” the equations (3.1) defined by means of a 2-form, disregarding the fact that, actually, Boillat introduced them by imposing the condition of complete exceptionality (which also allowed him to write down the local form (3.15)). Fix now a point m1 ∈ M (1) and recall (see Lemma 4.1) that the fibre M (2) m1 is an affine space modeled over S3L∗m1 . Let ω ∈ Λ2C1 ∗ and regard the corresponding 2-form ωm1 on C1 m1 as a linear map Λ2C1 m1 φ−→ R. The linear projective subspace H := P kerφ ⊂ PΛ2C1 m1 (6.5) is the so-called hyperplane section determined by ωm1 . Now we show that H corresponds pre- cisely to the fibre Eω,m1 of the 3rd order MAE determined by ω, via the Plücker 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 polarisation/Spencer operator. In turn, L∗m1 ⊗ S2L∗m1 is an affine neighborhood of Lm1 in the Grassmannian Gr(2, C1 m1), which is sent to PΛ2C1 m1 by the Plücker embedding. Hence, the subspace Eω,m1 of M (2) m1 can be regarded as a subspace of PΛ2C1 m1 , and (3.1) tells precisely that such a subspace coincides with H defined by (6.5). We are now in position to generalize a result about classical MAEs [2, Theorem 3.7], to the context of 3rd order MAEs. Proposition 6.4. A characteristic direction for Eω is also strongly characteristic. Proof. Let H ∈ PC1 m1 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 in Eω,m1 . To this end, recall that H(1) ⊂M (2) m1 is a 1D affine subspace modeled over S3 AnnH, passing through Lm2 (see, e.g., the proof of Theorem 1 in [7]). By the above arguments, H(1) can be embedded into PΛ2C1 m1 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 6.4 allows us to prove Theorem 3.3. Proof of Theorem 3.3. Inclusion (6.3) is valid for any 3rd order PDE. Conversely, ifm2 ∈ EVE , then there is a line H ∈ VE , such that Lm2 ⊃ H. But H is a strong characteristic line for E thanks to Proposition 6.4, so that all Lagrangian planes passing through H and, in particular, Lm2 ≡ m2 itself, belong to E . � 6.3 Proof of Theorem 3.4 As it was outlined in Section 3, Goursat-type MAEs are MAEs which correspond to decom- posable forms, modulo a certain ideal. Before proving Theorem 3.4, we make rigorous this statement. To this end, we shall need the submodule of contact 2-forms Θ = { ω ∈ Λ2M (1) \|ω\|Lm2 = 0 ∀m2 ∈M (2) } ⊂ Λ2M (1), (6.6) 20 G. Manno and G. Moreno and the corresponding projection Λ2M (1) −→ Λ2M (1) Θ . (6.7) Observe that the quotient bundle Λ2M(1) Θ is canonically isomorphic to the rank-one bundle Λ2L∗ over M (1). Hence, (6.7) can be thought of as (Λ2L∗)-valued. Remark 6.5. Two 2-forms have the same projection (6.7) if and only if they differ by an element of Θ. Proposition 6.6. For a 3D distribution D ⊂ C1, it holds ED = Eω ⇔ ω = ρ1 ∧ ρ2 mod Θ. Moreover, the 3D sub-distribution D ⊆ C1 is given by D = ker ρ1\|C1 ∩ ker ρ2\|C1 . (6.8) Proof. If AnnD = 〈ρ1, ρ2〉 ⊂ C1 ∗ is the annihilator of D in C1, then ED can be written as Eρ̃1∧ρ̃2 , where ρ̃1, ρ̃2 ∈ Λ1M (1) are extensions of ρ1, ρ2, respectively. The result follows from Remark 6.5. Conversely, in light of (3.1) and (6.6), ω and ρ1 ∧ ρ2 + Θ give rise to the same equation, i.e., Eω = Eρ1∧ρ2 . (6.9) It remains to be proved that the right-hand side of (6.9) is of the form ED. To this end, it suffices to define D as in (6.8), ED = E ρ̃1\|C1∧ρ̃2\|C1 , (6.10) and observe that the right-hand sides of (6.9) and (6.10) coincide since ρ̃i\|C1 −ρi ∈ Ann C1 ⊂ Θ, i = 1, 2. � Before starting the proof of Theorem 3.4, we provide a meta-symplectic analog of the well- known formula AnnD⊥ = Dyω. Proposition 6.7. 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, AnnDi = D1yΩ Hi ⊆ C1 ∗, i = 2, 3, (6.11) where H2 and H3 are the characteristic lines of Dv1. Proof. Let D1 = 〈aD1 + bD2, X1, X2〉, with X1 and X2 as in (4.13). Then, the quasi-linear 3rd order MAE determined by D1 is ED1 = ap111 + (b− a(k1 + k2))p112 + (ak1k2 − b(k1 + k2))p122 + bk1k2p222. In order to obtain the right-hand side of (6.11), compute first aD1 + bD2yΩ = (adp11 + bdp12)⊗ ∂p1 + (adp12 + bdp22)⊗ ∂p2 , and factor it by Hi, aD1 + bD2yΩ = (adp11 + (b− kia)dp12 − kibdp22)⊗ ∂p1 mod Hi. (6.12) 3rd Order Monge–Ampère Equations 21 Combining (6.12) above with (4.15), yields D1yΩ Hi ' 〈 kjdx 1 + dx2, adp11 + (b− kia)dp12 − kibdp22 〉 . (6.13) Finally, the wedge product of the two 1-forms spanning the module (6.13) 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 6.6. � Now we turn back to Theorem 3.4, and deal separately with its two implications. 6.3.1 Proof of the suff icient part of Theorem 3.4 The sufficient part of Theorem 3.4 will be proved through Lemma 6.8 below. Lemma 6.8. Let E = {F = 0}, where F is given either by (3.16), or F = ap111 + bp112 + cp122 + dp222 + e, (6.14) 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 VEI = PD. Moreover, E = ED, i.e., according to (3.4), it is a 3rd order MAE of Goursat-type. We recall that Lemma 6.8 is to be interpreted in fibre-wise perspective (see Section 4.5), i.e., as a collection of statements, each of which corresponds to the fixation of a point m1 ∈ M (1). But, since the formula (6.14) must correspond to an affine hyperplane in the space of the pijk’s over m1, then the vector (a(m1), b(m1), c(m1), d(m1)) ∈ R4 must be non-zero. This means that for any m1 ∈M (1), there is always one of the functions a, b, c, d, which is non-zero at m1. Proof of Lemma 6.8. Let us begin with the non-linear case. In order to verify that E = ED, it suffices to put D := 〈 D1 + f111∂p11 + f112∂p12 + f122∂p22 , D2 + f211∂p11 + f212∂p12 + f222∂p22 , R∂p11 + S∂p12 + T∂p22 〉 , (6.15) where (R,S, T ) = A is the same appearing in (3.16), and observe that (see also (3.11)) Lm2 = 〈D1 + p111∂p11 + p112∂p12 + p122∂p22 , D2 + p112∂p11 + p122∂p12 + p222∂p22〉 belongs to ED (see (3.4)) if and only if the 5× 5 determinant det ∣∣∣∣∣∣∣∣∣∣ 1 0 p111 p112 p122 0 1 p112 p122 p222 1 0 f111 f112 f122 0 1 f211 f212 f222 0 0 R S T ∣∣∣∣∣∣∣∣∣∣ = det ∣∣∣∣∣∣∣∣∣∣ 0 0 p111 − f111 p112 − f112 p122 − f122 0 0 p112 − f211 p122 − f212 p222 − f222 1 0 f111 f112 f122 0 1 f211 f212 f222 0 0 R S T ∣∣∣∣∣∣∣∣∣∣ = −F is zero. We now prove that VEI = PD in the case the coefficient of p111 of equation (3.16) is not zero. The other cases are formally analog. So, let us solve equation (3.16) with respect to p111 and 22 G. Manno and G. Moreno take the remaining coordinates p112, p122, p222 as local coordinates on E . Thus SmblF (see (5.6)) is, up to a factor, equal to(( S(p222 − f222)− T (p122 − f212) ) ξ1 + ( T (p112 − f112)− S(p122 − f122) ) ξ2 ) × (( T (p122 − f212)− S(p222 − f222) ) ξ2 1 + ( R(p222 − f222)− T (p112 − f211) ) ξ1ξ2 + ( S(p112 − f211)−R(p122 − f212) ) ξ2 2 ) , (6.16) so that (see also Section 5.3) VEI = {(( S(p222 − f222)− T (p122 − f212) ) D1 + ( T (p112 − f112)− S(p122 − f122) ) D2 ) m2 \|m2 ∈ E } . (6.17) A direct computation shows that VEI = {( T (p122 − f212)− S(p222 − f222) ) (D1 + f111∂p11 + f112∂p12 + f122∂p22) + ( S(p122 − f122)− T (p112 − f112) ) (D2 + f211∂p11 + f212∂p12 + f222∂p22) + (−p112p222 + p112f222 + f112p222 − f112f222 + p2 122 − p122f212 − f122p122 + f122f212) × (R∂p11 + S∂p12 + T∂p22) \| p112, p122, p222 ∈ R } so that VEI turns out to be the 3D linear space (6.15). Let us now pass to the quasi-linear case. One of the coefficients of the third derivatives of (6.14) must be non-zero. Assume that a 6= 0; the remaining cases can be treated similarly. Again, we solve equation (6.14) with respect to p111, so that p112, p122, p222 become local coordinates on it, and SmblF = aξ3 1 + bξ2 1ξ2 + cξ1ξ 2 2 + dξ3 2 = (kξ1 + hξ2)q(ξ1, ξ2), (6.18) where q(ξ1, ξ2) is a homogeneous quadratic function in ξ1 and ξ2. Thus, we have that (see again Section 5.3). VEI = 〈 (kD1 + hD2)m2 \|m2 ∈ E 〉 . (6.19) Let us assume d = 0. In order to verify that E = ED, it is enough to put D := 〈 D1 − e a ∂p11 ,− b a ∂p11 + ∂p12 ,− c a ∂p11 + ∂p22 〉 . (6.20) To prove that VEI = PD, observe that the symbol (6.18) contains the linear factor ξ1, and VEI = 〈D1\|m2〉m2∈E = { D1 − e a ∂p11 + p112 ( − b a ∂p11 + ∂p12 ) + p122 ( − c a ∂p11 + ∂p22 ) \| p112, p122 ∈ R } is the projectivization of (6.20). Finally assume d 6= 0. Observe that the lines in VEI , which, in view of (6.19), are generated by kD1 + hD2 − ke a ∂p11 + p112 (( −kb a + h ) ∂p11 + k∂p12 ) + p122 ( −kc a ∂p11 + h∂p12 + k∂p22 ) + p222 ( −kd a ∂p11 + h∂p22 ) , 3rd Order Monge–Ampère Equations 23 p112, p122, p222 ∈ R, fill a 3D linear space since ξ1 ξ2 = −h k is a solution to (6.18), i.e., −ah 3 k3 + b h2 k2 − ch k + d = 0. (6.21) So, if we put D := 〈 D1 + h k D2 − e a ∂p11 , ( − b a + h k ) ∂p11 + ∂p12 , k h d a ∂p11 − ∂p22 〉 with h, k 6= 0 such that (6.21) is satisfied, we obtain E = ED and VEI = PD. � 6.3.2 Proof of the necessary part of Theorem 3.4 Being Em1 a closed submanifold of codimension 1, in the neighborhood of any point m2 ∈ Em1 , we can always present E in the form E = {F = 0}, with F := pijk − G, with G not depending on pijk, for some (i, j, k) which we shall assume equal to (1, 1, 1), since the other cases are formally analog. Then the symbol of F at m1 is Smblm1 F = ξ3 1 −Gp112ξ2 1ξ2 −Gp122ξ1ξ 2 2 −Gp222ξ3 2 . (6.22) The right-hand side of (6.22) is a 3rd order homogeneous polynomial with unit leading coefficient: hence, there exist unique β, A, B such that Smblm1 F = (ξ1 +βξ2)(ξ2 1 +Aξ1ξ2 +Bξ2 2). Following the general procedure (see also formula (5.9) and Definition 5.7), to construct the characteristic cone VE , one easily sees that VE contains the following 3-parametric family of lines: VEI := { D1 + βD2 + (G+ βp112)∂p11 + (p112 + βp122)∂p12 + (p122 + βp222)∂p22 \| p112, p122, p222 ∈ R } . Observe that, with the same notation as (1.8), 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 4.7). Suppose now that there exists a 3D sub-distribution D ⊂ C1 such that VEI = PD. (6.23) Since D has codimension 2 in C1, condition (6.23) can be dualized as follows: there are two independent forms ρ1, ρ2 ∈ C1 ∗, ρ1 = k1dx 1 + k2dx 2 + k11dp11 + k12dp12 + k22dp22, ρ2 = h1dx 1 + h2dx 2 + h11dp11 + h12dp12 + h22dp22, such that AnnD = 〈ρ1, ρ2〉, i.e., D = ker ρ1 ∩ ker ρ2. In other words, (6.23) holds true if and only if ρ1(v) = ρ2(v) = 0 for all v ∈ VEI , i.e., k1 + k12p112 + k22p122 + ( k2 + k11p112 + k12p122 + k22p222 ) β + k11G = 0, (6.24) h1 + h12p112 + h22p122 + ( h2 + h11p112 + h12p122 + h22p222 ) β + h11G = 0, (6.25) identically in p112, p122, p222, where the 2× 5 matrix∣∣∣∣k1 k2 k11 k12 k22 h1 h2 h11 h12 h22 ∣∣∣∣ (6.26) of functions on M (1) has rank two everywhere. 24 G. Manno and G. Moreno Now (6.24), (6.25) must be regarded as linear system of two equations in the unknowns β and G. Its discriminant is easily computed, ∆ := (h11k22−k11h22)p222 +(h11k12−k11h12)p122 + h11k2 − k11h2, and ∆ is a polynomial in p122, p222. Suppose ∆(m2) = 0, and interpret (6.24), (6.25) 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 (6.26) is of rank 1 in m1. This contradicts the hypothesis that dimDm1 = 3, so that ∆ must be everywhere non-zero. Then, in the points with ∆(p122, p222) 6= 0 one can find ∆G as a polynomial expression of p112, p122, p222, whose coefficients turn out to be minors of the matrix (6.26). In particular, p111 −G can be singled out from the so-obtained expression, namely −∆(p111 −G) = −h1k2 + k1h2 − ( h11k2 − k11h2 ) p111 − ( k11h1 − h11k1 + k2h 12 − k12h2 ) p112 − ( k2h 22 + k12h1 − k22h2 − k1h 12 ) p122 − ( k22h1 − k1h 22 ) p222 + ( k12h22 − h12k22 )( p112p222 − p2 122 ) + ( k22h11 − k11h22 ) (−p111p222 + p112p122) + ( k11h12 − h11k12 )( p111p122 − p2 112 ) . (6.27) We need to show that (6.27) is satisfied if and only if there is a nowhere zero factor λ such that F = λ(p111 −G), (6.28) where F is given by (3.16), i.e., equation (6.28) needs to be solved with respect to fijk, R, S, T . By equating the coefficients of p111, one obtains λ = T (p122 − f212)− S(p222 − f222), (6.29) and replacing (6.29) into (6.28) yields F − (T (p122 − f212)− S(p222 − f222))(p111 −G) = 0. (6.30) Direct computations show that the left-hand side of (6.30) is a rational function whose numerator is a 2nd order polynomial in p112, p122, p222. From the vanishing of this polynomial, one can express fijk, R, S, T as functions of the entries of the matrix (6.26). 6.4 Proof of Theorem 3.5 It will be accomplished in steps. As a preparatory result, we show that a 3D sub-distribution of C1 with 2D vertical part determines a quasi-linear 3rd order MAE (Lemma 6.9 below). The converse statement, i.e., that the characteristic cone of a generic quasi-linear 3rd order PDE possesses a linear sheet determined by a 3D sub-distribution D ⊆ C1 with 2D vertical part, has been proved by Lemma 6.8 above. Then we pass to generic 3D sub-distributions of C1, and derive the expression of the corresponding equation ED (Lemma 6.11). As a consequence of these results (Corollary 6.14), we prove the initial statement of Theorem 3.5. The next Sections 6.4.1, 6.4.2, 6.4.3 are devoted to the specific proofs of the items 1, 2, 3 of Theorem 3.5, respectively. Besides the proof of Theorem 3.5 itself, a few interesting byproducts will be pointed out. For instance, Corollary 6.14 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 6.17 is a non-trivial and unexpected generalization of a phenomenon firstly observed in the classical case (see [2, Theorem 1.1]). 3rd Order Monge–Ampère Equations 25 Lemma 6.9. Let D1 = 〈h1,Dv1〉, with h1 = aD1 + bD2 + f11∂p11 + f12∂p12 + f22∂p22 , Dv1 = 〈Ri∂p11 + Si∂p12 + Ti∂p22 \| i = 1, 2〉 . Then ED1 is of the form (3.15) with A = 0, (6.31) C = det ∣∣∣∣∣∣ f11 f12 f22 R1 S1 T1 R2 S2 T2 ∣∣∣∣∣∣ , (6.32) 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 ∣∣∣∣) . (6.33) 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∣∣∣∣∣∣∣∣∣∣ 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 ∣∣∣∣∣∣∣∣∣∣ (6.34) is zero (see also (3.11) and (3.4)). Standard matrix manipulations reveal that the determinant of (6.34) coincides in turn with (3.15) under conditions (6.31), (6.32) and (6.33). � Remark 6.10. Since dim Gr(3, 5) = 6, it takes 6 parameters from C∞(M (1)) to identify a 3D sub-distribution D ⊂ C1 and, hence, an equation of the form ED. Above Remark 6.10 shows that the description (3.16) of the ED’s has some redundancies, and Lemma 6.13) below is a way to refine it. Lemma 6.11. The local form of a generic ED with D ∈ Gr(3, C1), in the case dimDv = 1, is given by (3.16). Proof. A distribution D ∈ Gr(3, C1) with dimDv = 1 is locally given by D = 〈 D1 + f111∂p11 + f112∂p12 + f122∂p22 , D2 + f211∂p11 + f212∂p12 + f222∂p22 , R∂p11 + S∂p12 + T∂p22 〉 , where fijk, R, S, T are C∞ functions (defined in some neighborhood of M (1)). Now it is clear that a Lagrangian plane 〈D1 + p111∂p11 + p112∂p12 + p122∂p22 , D2 + p112∂p11 + p122∂p12 + p222∂p22〉 non-trivially intersects D iff det  1 0 p111 p112 p122 0 1 p112 p122 p222 1 0 f111 f112 f122 0 1 f211 f212 f222 0 0 R S T  = p111 − f111 p112 − f112 p122 − f122 p112 − f211 p122 − f212 p222 − f222 R S T  = 0 that is equal to (3.16) with A = (R,S, T ). � 26 G. Manno and G. Moreno Now we discuss the possibility of reducing the number of redundant parameters in (3.16) (see Remark 6.10). In turn, this is linked to the natural question whether there exists a Lagrangian horizontal part H of D ∈ Gr(3, C1) in the case that dimDv = 1, i.e., the existence of a splitting D = H⊕Dv, with Lagrangian H. Lemma 6.12. Let D ⊂ C1 be a 3D distribution such that dimDv = 1. If there exists a Lag- rangian horizontal part H of D, then ED can be put in the form E = {F = 0}, where F is given by (3.16) with f112 = f211 and f122 = f212. Proof. It follows easily taking into account the computations of Lemma 6.11. � Lemma 6.13. Let D ⊂ C1 be a 3D distribution such that dimDv = 1. Let m1 ∈ M (1). If rank(Dvm1) 6= 1 (see Definition 4.6 concerning rank-one lines), then ED can be put, in a neigh- borhood of m1, in the form E = {F = 0}, where F is given by (3.16) with f112 = f211 and f122 = f212. Proof. Let the vertical part Dv of D be spanned by V = R∂p11 + S∂p12 + T∂p22 where R,S, T ∈ C∞(M (1)). Let D = 〈D1 + b111∂p11 + b112∂p12 + b122∂p22 , D2 + b211∂p11 + b212∂p12 + b222∂p22 ,V〉, with bijk ∈ C∞(M (1)). For any α, β ∈ C∞(M (1)) we have that D = 〈 D1 + b111∂p11 + b112∂p12 + b122∂p22 + αV, D2 + b211∂p11 + b212∂p12 + b222∂p22 + βV,V〉. (6.35) If rank(Vm1) 6= 1, then α and β can be chosen in such a way that, in a neighborhood of m1, b112 + αS = b211 + βR, b122 + αT = b212 + βS. (6.36) In fact system (6.36) is always compatible for any R, S, T , b112, b211, b122, b212 since rank(Vm1) 6=1 iff RT − S2 6= 0 at m1. To conclude this part of the lemma it is enough to set f111 = b111, f112 = b112 + αS, f122 = b122 + αT, f222 = b222 and the corresponding ED, with D given by (6.35), is precisely described by (3.16) with f112 = f211 and f122 = f212. � Corollary 6.14. 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 correspon- ding to a fixed linear factor of the symbol, i.e., an element H belonging to VEI (see Section 5.3). In other words, inclusion (6.4) can be made more precise: EVEI ⊇ E . (6.37) It remains to prove that the inverse of inclusion (6.37) is valid when VEI = PD1. Indeed, Lemma 6.9 and Lemma 6.11 together guarantee that the left-hand side of (6.37) is a closed submanifold (possibly with singularities, see Section 4.5) of codimension one. But the right-hand side of (6.37) is a closed submanifold (again with possible singularities) of the same dimension of the left-hand side. Hence, the two of them must coincide as well. � 3rd Order Monge–Ampère Equations 27 6.4.1 Proof of the statement 1 of Theorem 3.5 The proof of its first claim is contained in the next corollary. Corollary 6.15. A 3rd order MAE ED is quasi-linear if and only dimDv = 2. Proof. A direct consequence of Lemmas 6.8, 6.9 and 6.11. � The remainder of statement 1 is concerned with the “other component” VEII of VE , i.e., the one associated with the quadratic factor of AnnV E (see Section 5.3). Recall that VEII might be empty, in which case there is nothing to prove. At the far end, there is the case when VEII is, in its turn, decomposable, i.e., it consists of two linear sheets, which is dealt with by Lemma 6.16 below. Lemma 6.16. Let E = {F = 0} be a 3rd order quasi-linear PDE such that SmblF is completely decomposable, i.e., SmblF = (ξ + k1η)� (ξ + k2η)� (ξ + k3η). (6.38) If Di := 〈 D1\|m2 + ki D2\|m2 \|m2 ∈ Em1 〉 (6.39) is the 3D sub-distribution of C1 corresponding11 to the ith linear factor of (6.38) and D = 〈 h,Dv 〉 ⊆ C1 (6.40) is a generic 3D sub-distribution of C1 with dimDv = 2, then D satisfies condition (3.6) if and only if D = Dj with j 6= i. Proof. To begin with, (6.38) dictates some restrictions on F , which must be of the form F = p111 + (k1 + k2 + k3)p112 + (k1k2 + k1k3 + k2k3)p122 + k1k2k3p222 + c. (6.41) We begin with the “homogeneous” case, i.e., we assume c = 0 since, as we shall see at the end of the proof, the general case can be easily brought back to this one. Equating (6.41) to zero allows to express p111 as a linear combination of p112, p122, and p222, i.e., to identify Em1 with R3 ≡ {(p112, p122, p222)}. In turn, this makes it possible to parametrize the space of vertical elements of (6.39) by three real parameters, viz., Dvi = 〈 −((k1 + k2 + k3 − ki)p112 + (k1k2 + k1k3 + k2k3)p122 + k1k2k3p222)∂p11 + (p112 + kip122)∂p12 + (p122 + kip222)∂p22 \| p112, p122, p222 ∈ R 〉 . (6.42) It is worth observing that, in compliance with Lemma 6.8, the dimension of Dvi , which equals the rank of the 3× 3 matrix Mi = ∣∣∣∣∣∣ −k1 − k2 − k3 + ki −k1k2 − k1k3 − k2k3 −k1k2k3 1 ki 0 0 1 ki ∣∣∣∣∣∣ (6.43) is 2: indeed, detMi = −(k1 − ki)(k2 − ki)(k3 − ki) vanishes for i = 1, 2, 3. In order to find a basis for (6.42), regard the matrix (6.43) as a (rank-two) homomorphism Mi : R3 −→ V = 〈∂p11 , ∂p12 , ∂p22〉 and compute its kernel: kerMi = 〈 k2 i ,−ki, 1 〉 . 11See also, on this concern, the construction of VEI in Section 5.3. 28 G. Manno and G. Moreno Then, independently on i (and on the value of ki as well), R3 = 〈(1, 0, 0), (0, 1, 0)〉 ⊕ kerMi, and Dvi = 〈Mi · (1, 0, 0),Mi · (0, 1, 0)〉 is the sought-for basis. In other words, instead of (6.39), we shall work with the handier description Di = 〈hi,Dvi 〉, where hi = D1 + kiD2, Dvi = 〈−(k1 + k2 + k3 − ki)∂p11 + ∂p12 ,−(k1k2 + k1k3 + k2k3)∂p11 + ki∂p12 + ∂p22〉 . (6.44) Concerning (6.40), introduce similar descriptions of its horizontal and vertical part: h = aD1 + bD2, Dv = 〈r1∂p11 + s1∂p12 + t1∂p22 , r2∂p11 + s2∂p12 + t2∂p22〉 . (6.45) This concludes the preliminary part of the proof. Now impose condition (3.6): Ω ( hi,D v) = Ω ( h,Dvi ) = 1D subspace. (6.46) Observe that (6.46) 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,D v) = 〈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〉 , (6.47) i.e., to the equation det ∣∣∣∣r1 + kis1 s1 + kit1 r2 + kis2 s2 + kit2 ∣∣∣∣ = k2 iA− kiB + C = 0, (6.48) with A = det ∣∣∣∣s1 t1 s2 t2 ∣∣∣∣ , B = −det ∣∣∣∣r1 t1 r2 t2 ∣∣∣∣ , C = det ∣∣∣∣r1 s1 r2 s2 ∣∣∣∣ . (6.49) Interestingly enough, above coefficients (6.49) characterize the dual direction of Dv, i.e., a non- zero co-vector ω′ = Adp11 +Bdp12 + Cdp22 ∈ V ∗, defined up to a non-zero constant, such that Dv = kerω′. (6.50) This dual perspective on Dv allows to rewrite (6.48) as ω′ ( k2 i ∂p11 − ki∂p12 + ∂p22 ) = 0. (6.51) In turn, (6.51) dictates the form of ω′: ω′ = Adp11 +Bdp12 + ( Bki −Ak2 i ) dp22, (A,B) 6= (0, 0). (6.52) Together, (6.52) and (6.50) produce a simplified expression Dv = 〈( Ak2 i −Bki ) ∂p11 +A∂p22 ,−B∂p11 +A∂p12 〉 , (6.53) which, in comparison with (6.45), depending on 6 parameters, needs only 2 of them, or even 1, if a non-zero constant is neglected. Now, thanks to (6.53), it is easier to see that the space (6.47) is 1D. Indeed, (6.47) reads Ω ( hi,D v) = 〈( Ak2 i −Bki ) ∂p1 +Aki∂p2 ,−B∂p1 +A∂p2 +Aki∂p1 〉 , 3rd Order Monge–Ampère Equations 29 where the first vector equals the second multiplied by ki, so that it can be further simplified: Ω(hi,Dv) = 〈(Aki −B)∂p1 +A∂p2〉 . (6.54) We can pass to the other condition dictated by (6.46). In particular, the subspace Ω ( h,Dv ) = 〈 a(−(k1 + k2 + k3 − ki)∂p1 + ∂p2) + b∂p1 , (−(k1k2 + k1k3 + k2k3)∂p1 + ki∂p2) + b(ki∂p1 + ∂p2) 〉 = 〈 (b− a(k1 + k2 + k3 − ki))∂p1 + a∂p2 , (bki − a(k1k2 + k1k3 + k2k3))∂p1 + (aki + b)∂p2 〉 (6.55) must be 1D, i.e., det ∣∣∣∣ b− a(k1 + k2 + k3 − ki) a bki − a(k1k2 + k1k3 + k2k3) aki + b ∣∣∣∣ = ∏ j 6=i (b− akj) = 0. (6.56) Above equation (6.56), makes it evident that, for any i = 1, 2, 3, the space (6.54) is 1D if and only if b = akj , j ∈ {1, 2, 3}\{i}, a 6= 0, (6.57) which corresponds to h = D1 + kjD2, j ∈ {1, 2, 3}\{i}. (6.58) Plugging (6.57) into (6.55), we find a unique vector (ki + kj − k1 − k2 − k3)∂p1 + ∂p2 (6.59) generating Ω(h,Dv). Introducing the complement c(i, j) of {i, j} in {1, 2, 3}, (6.59) reads Ω ( h,Dv ) = 〈 −kc(i,j)∂p1 + ∂p2 〉 , j ∈ {1, 2, 3}\{i}. To conclude the proof, recall that, besides their one-dimensionality, (6.46) also requires the equa- lity of the subspaces (6.53) and (6.54), i.e., Ω(hi,D v ) = 〈 (Aki −B)∂p1 +A∂p2 ,−kc(i,j)∂p1 + ∂p2 〉 = Ω(h,Dv) or, alternatively, det ∣∣∣∣Aki −B A −kc(i,j) 1 ∣∣∣∣ = 0. (6.60) Thanks to (6.60), B = A(ki + kc(i,j)), j ∈ {1, 2, 3}\{i}, (6.61) and (6.61) allows to eliminate B from (6.53): Dv = 〈( Ak2 i −A(ki + kc(i,j))ki ) ∂p11 +A∂p22 ,−A(ki + kc(i,j))∂p11 +A∂p12 〉 . (6.62) Being a non-zero constant12, A can also be removed from (6.62), which becomes: Dv = 〈 −kc(i,j)ki∂p11 + ∂p22 , (kc(i,j) + ki)∂p11 + ∂p12 〉 . (6.63) 12If A = 0, then from (6.53) follows dimD v = 1. 30 G. Manno and G. Moreno To enlighten the conclusions, it is useful to rewrite together (6.58) and (6.63) above: there are exactly 2 distributions D which are “compatible” (in the sense of (6.46)) with the distribution Di given by (6.39). More precisely, their horizontal and vertical parts are h = D1 + kjD2, Dv = 〈 −kc(i,j)ki∂p11 + ∂p22 , (kc(i,j) + ki)∂p11 − ∂p12 〉 , (6.64) respectively, for the only two possible values of j ∈ {1, 2, 3}\{i}. One only needs to reali- ze that (6.64) is one of the Dvi ’s from (6.44). To this end, rewrite (6.44) 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 : Dvj = 〈 −(k1 + k2 + k3 − kj)∂p11 + ∂p12 , (kj(k1 + k2 + k3 − kj)− (k1k2 + k1k3 + k2k3))∂p11 + ∂p22 〉 . (6.65) Formulas kj(k1 + k2 + k3 − kj) = kj ∑ l 6=j kl, k1k2 + k1k3 + k2k3 = kj ∑ l 6=j kl + ∏ l 6=j kl, show that the coefficient of ∂p11 in the second vector of (6.65) reduces to − ∏ l 6=j kl. Hence, (6.65) reads Dvj = 〈 − (∏ l 6=j kl ) ∂p11 + ∂p22 , (∑ l 6=j kl ) ∂p11 − ∂p12 〉 . (6.66) Comparing now (6.64) with (6.66) 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 (6.41), with c = 0, and by Ec a “inhomogeneous” 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 that ϕc(Di), i = 1, 2, 3, are precisely the three distributions associated with the factors of the symbol of Ec, and plainly Ω(h,Dv) = Ω(ϕc(h), ϕc(Dv)). So, the “inhomogeneous” case reduces to the “homogeneous” one, which has been established above. � Corollary 6.17 (proof of statement 1 of Theorem 3.5). 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 =⋃3 i=1 PDi on U , where Di ⊆ C\|U are orthogonal (see Definition 4.4) with respect to the meta- symplectic structure on M (1) and E = EDi on 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 ) m1 6= ∅, then it means that the function ∆ (see Remark 5.6) is positive on the whole fibre M (1) m1 and, hence, on the neighbouring fibres. So, U is indeed open. Proof of Corollary 6.17. Let m1 ∈ U . Then VEII 6= ∅, i.e., there is a point m2 ∈ Em1 such that Smblm2 F is fully decomposable. But, in view of the quasi-linearity of F , this means that SmblF is fully decomposable on the whole fibre Em1 , i.e., E fullfills the hypotheses of Lemma 6.16 on U . � 3rd Order Monge–Ampère Equations 31 6.4.2 Proof of the statement 2 of Theorem 3.5 The first part of this statement merely rephrases the fist part of statement 1. Proving its second part consists in completing the proof of Lemma 6.8 above with the compu- tation of VEII, which corresponds to the quadratic factor (see Section 5.3) of (6.16). Formally, the procedure to get a description of VEII out of the quadratic factor of (6.16) is the same as that for obtaining (6.17) from its linear factor, but it is more involved when it 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 quadratic factor of (6.16) admits real roots. If ∆ is the discriminant of such quadratic factor, then one its factor is given by ξ1 − T (p112 − f211)−R(p222 − f222) + √ ∆ 2(T (p122 − f212)− S(p222 − f222)) ξ2, (6.67) and the corresponding line H1 is obtained by replacing13 ξi with Di\|E in (6.67). 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 that the corresponding lines H1 do not lie all in the same 3D subspace of C1. This shows that VEII cannot contain any linear irreducible component. In order to prove that ED1 = EVEII , observe that the “⊇” inclusion follows from the fact that, by definition, VEII is made of characte- ristic lines of ED1 which are also strongly characteristic by Proposition 6.4, so that all Lagrangian planes containing a line in VEII belong also to ED1 (see also Definition 5.2). 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 plane belonging to ED1 , is of rank 2. The same result is attained by considering the line H2 obtained by changing the sign of √ ∆ in (6.67). 6.4.3 Proof of the statement 3 of Theorem 3.5 Given any 3D sub-distribution D ⊆ C1, and the corresponding Goursat-type equation ED (see (3.4)), we are now in position to clarify how many and which sort of elements can be found in the set{ D̃ \| D̃ is a 3D sub-distribution of C1 such that ED̃ = ED } . (6.68) 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 priori14; 2) in the points of M (1) where dimDv = 1, the equation ED is non-linear and the set (6.68) contains only D; 3) in the points of M (1) where dimDv = 2, the equation ED is quasi-linear and the set (6.68) contains three elements Di, i = 1, 2, 3, possibly repeated and comprising D itself, which are orthogonal each other (see Definition 4.4). 13This 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 3.6). 14Still, the description of the set (6.68) is very easy: it contains only D, which in turn coincides with VM (1). 32 G. Manno and G. Moreno Proof of 2 and 3. Let D̃ be an element of the set (6.68). Then, in particular, VED = VED̃ . (6.69) In view of Corollary 6.2, equality (6.69) implies PD ⊆ VED̃ , PD̃ ⊆ VED . (6.70) In other words, condition (3.5) is fulfilled by both equations ED̃ and ED, so that the statements 1 and 2 of Theorem 3.5 (see above Sections 6.4.1 and 6.4.2) can be made use of. To this end, rewrite (6.70) as VED̃ = PD ∪ VED̃II , (6.71) VED = PD̃ ∪ VEDII , (6.72) respectively. Because of statement 1 (reps., 2) of Theorem 3.5, from (6.71) it follows that ED̃ is quasi-linear (resp., non-linear) if and only if dimDv = 2 (resp., 1) and from (6.72) it follows that ED is quasi-linear (resp., non-linear) if and only if dim D̃v = 2 (resp., 1). Summing up, dimDv = dim D̃v = 1, 2, and the two cases can be treated separately. Directly from (6.71) and (6.72) we get PD̃ ∪ VED̃II = PD ∪ VED̃II , PD ∪ VEDII = PD̃ ∪ VEDII . (6.73) Prove now 2. If dimDv = 1, then also dim D̃v = 1 and the statement 2 of Theorem 3.5 (see Section 6.4.2) guarantees that VED̃II (resp., VEDII ) cannot contain PD (resp., PD̃). It follows from (6.73) that PD̃ = PD. Finally prove 3. If dimDv = 2, then also dim D̃v = 2 and, in view of the statement 1 of Theorem 3.5 (see Section 6.4.1), VED̃II (resp., VEDII ) is either empty, in which case PD̃ = PD, or consists of two, possibly repeated, distributions “orthogonal” to D̃ (resp., D). � The result presented in this section mirrors the analogous result for classical multi-dimensional MAEs (see [2, Theorem 1]), but displays some new and unexpected features: the threefold multiplicity of the notion of orthogonality and the distinction of the cases according to the dimension of the vertical part. 7 Intermediate integrals of Goursat-type 3rd order MAEs Theorem 3.5 established a link between 3rd order MAEs of Goursat-type, i.e., non-linear PDEs of order three, and 3D sub-distributions D of C1, i.e., linear objects involving (at most) second- order partial derivatives. Besides its aesthetic value, such a perspective also allows to formulate concrete results concerning the existence of solutions, as the first integrals of 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] and [3], where a more general concept of intermediate integral is exploited to explicitly construct solutions to 2nd order parabolic MAEs). Proposition 7.4 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 the same. To facilitate its proof, we deemed it convenient to introduce equations of the form E = EV where V = PĎ, with, for the first time in this paper, dim Ď = 2. By analogy with (3.4), we still write EĎ instead of EPĎ, but we warn the reader that, unlike all the cases considered so far, EĎ is actually a system of two independent equations (see Lemma 7.2 below). 3rd Order Monge–Ampère Equations 33 Recall that a function f ∈ C∞(M (1)) determines, in the neighborhood of an its non-singular point, the hyperplane distribution 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 7.1. A function f ∈ C∞(M (1)) is an intermediate integral of a PDE E ⊆M (2) if( M (1) )(1) f−c ⊆ E (7.1) for all c ∈ R. Inclusion (7.1) can be interpreted as follows: if f = c defines a 2nd order PDE (in the sense given in Section 2.1), then any solution of such an equation is also a solution of the 3rd order equation E . Lemma 7.2. Let Ď ⊂ C1 be a 2D non-vertical sub-distribution. Then EĎ −→ M (1) is a non- linear bundle of rank 2. Proof. To simplify the notations, let E := EĎ. Then, fix m1 ∈ M (1) and observe that Em1 = {m2 ∈M (2) \|Lm2 ∩ Ďm1 6= 0} is a 2D manifold. Indeed, Em1 is15 a rank-one bundle over the 1D manifold PĎ, its fibres being the prolongations of the lines lying in Ď (see [8, Proposition 2.5]). To prove non-linearity of E , write down Ď in local coordinates as Ď = 〈aiD1 + biD2 +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 Ď fails to be two-dimensional or when Ď is vertical, which are forbidden by the hypo- theses. � Corollary 7.3. Let Ď ⊂ C1 be a 2D sub-distribution contained in ker df . Then for any m1 ∈M (1) there is a m2 ∈M (2) m1 such that Lm2 ⊂ ker dm1f and Lm2 ∩ Ďm1 = 0. Proof. Assume for contradiction that{ m2 ∈M (2) \|Lm2 ⊂ ker dm1f } ⊆ { m2 ∈M (2) \|Lm2 ∩ Ďm1 6= 0 } (7.2) and set c := f(m1). Then (7.2) means that, over the point m1,( M (1) )(1) f−c ⊆ EĎ, (7.3) which is impossible, since the left-hand side of (7.3) is linear while its right-hand side is either empty (in the case when Ď ⊆ VM (1)) or non-linear, thanks to above Lemma 7.2. � Proposition 7.4. Let E be a Goursat-type 3rd order MAE and f ∈ C∞(M (1)). Then the following statements are equivalent: 15This step may require the restriction to an open and dense subset. 34 G. Manno and G. Moreno • f is an intermediate integral of E; • there exists a 3D distribution D ⊂ C1 such that f is a first integral of D and ED = E. Proof. If f is a first integral of D, then D ⊆ C1 ∩ ker df ⊆ C1 (7.4) is a (3, 4, 5)-type flag of distributions (C1 does not posses first integrals). By definition,( M (1) )(1) f−c = { m2 ∈M (2) \| f(π2,1(m2)) = c, df \|Lm2 = 0 } (7.5) and condition df \|Lm2 = 0 means precisely that the 2D subspace Lm2 of C1 is also contained in ker df , i.e., Lm2 lies in the 4D subspace C1 ∩ ker df of C1. Because of (7.4), the 3D subspace D is also contained in C1 ∩ ker df and, as such, it cannot fail to non-trivially intersect Lm2 . This means that m2 ∈ ED, and (7.1) follows from the arbitrariness of m2. Suppose, conversely, that f is an intermediate integral of E . In view of (7.5) and inclu- sion (7.1), one has that Lm2 ⊆ ker df ⇒ Lm2 ∩ Dm1 6= 0 (7.6) for any distribution D such that ED = E . We wish to show that, there is at least one among these distributions such that f is an its first integral, i.e., that the first inclusion of (7.4) is satisfied. Towards a contradiction, suppose that no distributionD is contained in ker df , i.e., that Ďf := 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 7.3, it is possible to find a (2D subspace) Lm2 which is contained in C1 ∩ ker df and also trivially intersects one Ďf , thus violating (7.6). � Corollary 7.5. If the derived flag of D never reaches TM (1), then ED admits an intermediate integral. Acknowledgments The authors wish to express their gratitude towards the anonymous referees whose comments contributed to shape the paper into its final form. The authors thank C. Ciliberto, E. Ferapontov and F. Russo for stimulating discussions. 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Phys. 11 (2014), 1460039, 35 pages, arXiv:1311.3477. http://dx.doi.org/10.1007/s10440-008-9204-8 http://dx.doi.org/10.1016/j.difgeo.2008.06.019 http://arxiv.org/abs/0707.0683 http://dx.doi.org/10.5167/uzh-42719 http://dx.doi.org/10.1016/j.geomphys.2014.05.006 http://arxiv.org/abs/1208.5880 http://dx.doi.org/10.1007/978-1-4613-9714-4 http://dx.doi.org/10.2478/s11533-011-0046-7 http://dx.doi.org/10.2478/s11533-011-0046-7 http://arxiv.org/abs/1102.0426 http://dx.doi.org/10.1007/s10711-007-9228-7 http://arxiv.org/abs/0710.5110 http://dx.doi.org/10.1007/BFb0094793 http://arxiv.org/abs/hep-th/9407018 http://dx.doi.org/10.1023/A:1022236709346 http://arxiv.org/abs/nlin.SI/0205056 http://dx.doi.org/10.1007/978-3-662-02950-3 http://dx.doi.org/10.1016/j.geomphys.2010.10.012 http://arxiv.org/abs/1002.0077 http://dx.doi.org/10.1016/B978-044452833-9.50015-2 http://dx.doi.org/10.1007/BF00580702 http://dx.doi.org/10.1007/BF01095431 http://dx.doi.org/10.1007/BF01095431 http://mathoverflow.net/q/138544 http://mathoverflow.net/q/138544 http://dx.doi.org/10.2478/s11533-013-0292-y http://arxiv.org/abs/1207.6290 http://arxiv.org/abs/math.DG/0510555 http://dx.doi.org/10.1016/0375-9601(95)00944-2 http://dx.doi.org/10.1142/S0219887814600391 http://dx.doi.org/10.1142/S0219887814600391 http://arxiv.org/abs/1311.3477 1 Introduction 2 Preliminaries on (prolongations of) contact manifolds, and (meta)symplectic structures 2.1 Contact manifolds, their prolongations and PDEs 2.2 The meta-symplectic structure on C1 3 Description of the main results 3.1 Local coordinate description of the main objects 3.1.1 Boillat and Goursat 3rd order MAEs 4 Vertical geometry of (prolongations of) contact manifolds and their characteristics 4.1 Vertical geometry of M(k) and three-fold orthogonality in M(1) 4.2 Rank-one lines, characteristic directions and characteristic hyperplanes 4.3 Canonical directions associated with orthogonal distributions 4.4 Some examples 4.5 Smoothness and singularities issues 5 Characteristics of 3rd order PDEs 5.1 Characteristics of a 3rd order PDE and relationship with its symbol 5.2 The characteristic variety as a covering of the characteristic cone 5.3 The irreducible component VIE of the characteristic cone of a 3rd order PDE E 6 Proof of the main results 6.1 Reconstruction of PDEs by means of their characteristics 6.2 Proof of Theorem 3.3 6.3 Proof of Theorem 3.4 6.3.1 Proof of the sufficient part of Theorem 3.4 6.3.2 Proof of the necessary part of Theorem 3.4 6.4 Proof of Theorem 3.5 6.4.1 Proof of the statement 1 of Theorem 3.5 6.4.2 Proof of the statement 2 of Theorem 3.5 6.4.3 Proof of the statement 3 of Theorem 3.5 7 Intermediate integrals of Goursat-type 3rd order MAEs References

Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics

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