L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Dmitri V. ALEKSEEVSKY, Ricardo ALONSO-BLANCO, Gianni MANNO & Fabrizio PUGLIESE

Contact geometry of multidimensional Monge-Ampère equations:

characteristics, intermediate integrals and solutions Tome 62, n^o2 (2012), p. 497-524.

<http://aif.cedram.org/item?id=AIF_2012__62_2_497_0>

© Association des Annales de l’institut Fourier, 2012, tous droits réservés.

L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

cedram

Article mis en ligne dans le cadre du

CONTACT GEOMETRY OF MULTIDIMENSIONAL MONGE-AMPÈRE EQUATIONS: CHARACTERISTICS,

INTERMEDIATE INTEGRALS AND SOLUTIONS

by Dmitri V. ALEKSEEVSKY, Ricardo ALONSO-BLANCO, Gianni MANNO & Fabrizio PUGLIESE

Abstract. — We study the geometry of multidimensional scalar 2^nd order PDEs (i.e.PDEs withnindependent variables), viewed as hypersurfacesEin the Lagrangian Grassmann bundleM⁽¹⁾over a (2n+ 1)-dimensional contact manifold (M,C). We develop the theory of characteristics ofEin terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals ofE. After specializing such results to general Monge- Ampère equations (MAEs), we focus our attention to MAEs of type introduced by Goursat in 1899:

det

∂²f

∂xⁱ∂x^j −bij(x, f,∇f)

^{= 0.}

We show that any MAE of this class is associated with an n-dimensional sub- distributionDof the contact distributionC, and viceversa. We characterize these Goursat-type equations together with their intermediate integrals in terms of their characteristics and give a criterion of local contact equivalence. Finally, we develop a method to solve Cauchy problems for this kind of equations.

Keywords:Hypersurfaces of Lagrangian Grassmannians, contact geometry, subdistribu- tions of a contact distribution, Monge-Ampère equations, characteristics, intermediate integrals.

Math. classification:53D10, 35A30, 58A30, 58A17.

Résumé. — Nous étudions la géométrie des équations aux dérivées partielles scalaires du deuxième ordre multidimensionnelles (c’est-à-dire, EDP avec nva- riables indépendantes), considérées comme hypersurfacesEdans le fibré Grassman- nien LagrangienM⁽¹⁾ sur une variété de contact (2n+ 1)-dimensionnelle (M,C).

Nous développons la théorie des caractéristiques deEen termes de la géométrie de contact et de la géométrie du fibré Grassmannien Lagrangien et étudions leur rela- tion avec les intégrales intermédiaires deE. Après avoir appliqué tels résultats aux équations de Monge-Ampère générales (EMA), nous concentrons notre attention sur les EMA du type introduit par Goursat en 1899 :

det

∂²f

∂xⁱ∂x^j −bij(x, f,∇f)

^{= 0.}

Nous montrons que toutes les EMA de cette classe sont associées à une sous- distributionn-dimensionnelleDde la distribution de contactCet vice-versa. Nous caractérisons les équations du type de Goursat avec leurs intégrales intermédiaires en fonction de leurs caractéristiques et donnons un critère d’équivalence locale de contact. Enfin, nous développons une méthode pour résoudre les problèmes de Cauchy pour ce genre d’équations.

Introduction

Characteristics of PDEs are a classic subject ([9, 10, 19, 21]), as they are related to the local existence and uniqueness of solutions of Cauchy problems. Consider the scalar second order partial differential equation with one unknown function (2^ndorder PDE)

(0.1) F(x¹, . . . , xⁿ, z, p1, . . . , pn, p11, p12, . . . , pnn) = 0

where z = z(x¹, . . . , xⁿ), pi = ∂z/∂xⁱ, pij = ∂²z/∂xⁱ∂x^j; the Cauchy problem consists in finding a solutionz=f(x¹, . . . , xⁿ) of (0.1) such that (0.2) f|_(X1(t),...,Xⁿ(t)) =Z(t), ∂f

∂xⁱ

_{(X1(t),...,Xn(t))}

=Pi(t), where

(0.3) Φ(t) = (X¹(t), . . . , Xⁿ(t), Z(t), P₁(t), . . . , P_n(t)),

t= (t1, . . . , t_n−1) is a given (n−1)-dimensional manifold,i.e.aCauchy datum(obviously, the particular choice of the parametrization is irrelevant). If submanifold (0.3) is non-characteristic, then, in the C^∞ case, Cauchy problem (0.1)–(0.2) admits a unique formal solution; if, moreover,F is real analytic, then such solution is, in fact, an ordinary one, analytical and locally unique.

As a well known example, take n= 2. In this case, Φ(t) is a curve in the space (x¹, x², z, p1, p2); given a pointm= Φ(0) = (x¹, x², z, p₁, p₂) on this

curve and a point m¹ = (x¹, x², z, p₁, p₂, p₁₁, p₁₂, p₂₂) satisfying (0.1), the tangent vectorv= ˙Φ(0) is non-characteristic for (0.1) atm¹ if

(0.4) ∂F

∂p11

_m1

(v²)²− ∂F

∂p12

_m1

v¹v²+ ∂F

∂p22

_m1

(v¹)²6= 0

wherev=v¹(∂_x1+p₁∂_z+p₁₁∂_p1+p₁₂∂_p2)+v²(∂_x2+p₂∂_z+p₁₂∂_p1+p₂₂∂_p2).

Vectorvcan be considered as an “infinitesimal Cauchy datum”. From equa- tion (0.4) it is clear that one can associate with any pointm¹satisfying (0.1) two (possibly imaginary) directions in the space (x¹, x², z, p1, p2), namely, those annihilating (0.4) (“characteristic lines”); if we letm¹ vary on (0.1) keepingm fixed, these two directions form, in general, two distinct cones atm. It is proved that the only PDEs for which these cones degenerate into two 2-dimensional planes are classical Monge-Ampère equations (MAEs) (see for instance [3, 2]).

One of the aims of this paper is to see whether a similar phenomenon occurs also in the case of MAEs with an arbitrary number of independent variables, which, of course, is considerably more complicated.

In fact, MAEs forn= 2 have been intensely studied since the second half of XIX century by many géomètres, among them Darboux, Lie, Goursat (a systematic account of such investigations can be found in [8] and [9]);

later, this classical approach was put aside in favor of more “hard analysis”

techniques. The last 40 years have witnessed a renewed interest in the differential-geometric approach to MAEs, mainly due to Lychagin and his school (see [12] and [13] for an exhaustive bibliography). However, such results are focused on the classical case (n= 2).

Up to now, no serious effort has been made to extend the classical theory to the general multidimensional case (only very special cases have been studied). In fact, the main achievements so far obtained in this direction are due to Boillat and Lychagin.

Boillat [4] noticed that MAEs with two independent variables are the only 2^nd order PDEs exceptional in the sense of Lax [14]. This property was used in [20] to find the general form of a MAE in three independent variables, and in [5] for the case of arbitrary independent variables. Such general form is

(0.5) Mn+M_n−1+· · ·+M0= 0

whereM_kis a linear combination (with functions ofxⁱ, z, p_i as coefficients) of allk×kminors of the Hessian matrixkz_xix^jk.

Lychagin [15], by introducing a new approach based on contact geome- try, defined multidimensional MAEs as the kernel of a differential operator

associated with a class ofn-differential forms on a contact manifold. Lo- cally, such PDEs are described by (0.5). In the rest of the paper, when we write “general MAEs” we mean “multidimensional MAEs in the sense of Lychagin”. In [6, 7] an interpretation of MAEs with constant coefficients is given in terms of Lagrangian Grassmannians.

As far as we know, the oldest paper about the multidimensional general- ization of classical MAEs dates back to 1899. In [10] it was noticed that clas- sical MAEs (n= 2) can be obtained by substitutingdp₁=p₁₁dx¹+p₁₂dx² anddp2=p12dx¹+p22dx² into the following pfaffian system

(dp1−b11dx¹−b12dx²= 0

dp₂−b₂₁dx¹−b₂₂dx²= 0, b_ij =b_ij(x¹, x², z, p₁, p₂)

and by requiring the linear dependence of the obtained 1-forms. Obviously, such a procedure can be extended to any number n of independent vari- ables; namely, one can consider the system

dpi−

n

X

j=1

bijdx^j = 0, i= 1, . . . , n, bij=bij(x¹, . . . , xⁿ, z, p1, . . . , pn) thus getting MAE

(0.6) det||pij−bij||= 0.

It turns out that the class of PDEs considered by Goursat is a subclass of those considered by Lychagin.

The above analytical procedure has a natural geometrical meaning, tightly connected with the fundamental notion of characteristics of a PDE.

Such a connection, which was already studied in [3, 2] forn = 2, will be extended below to the case of any number of independent variables. As we shall see, forn >2 the complexity of the problem dramatically increases.

To this purpose, as a first step we develop a coordinate free setting of the theory of characteristics of 2^ndorder PDEs (withnindependent variables) in terms of contact manifolds and Lagrangian Grassmannians. Then, we fo- cus our attention to MAEs of type (0.5) and (0.6), describe them in terms of their characteristics, study their intermediate integrals and the problem of solutions for a given Cauchy datum.

Notations and conventions

In the rest of the paper we work in the C^∞ case: the term “smooth”

meansC^∞. Latin indices will run from 1 to n, unless otherwise specified.

We will use Einstein convention. We denote byX·%the Lie derivative of the differential form % along the vector field X and by ∨ the symmetric tensor product,i.e. A∨B = ¹₂(A⊗B+B⊗A); S²(V) is the symmetric square ofV. The annihilator of a vector subspaceU will be denoted byU⁰. We denote byhviithe linear span of vectorsv₁, . . . , vn.

1. Preliminaries and description of the main results Let (M,C) be a (2n+ 1)-dimensional contact manifold, i.e. C is a com- pletely non-integrable distribution on M of codimension 1. Locally, C is the kernel of a 1-form θ, determined up to a non vanishing factor, with θ∧dθ∧· · · ∧ⁿ dθ6= 0 . The restrictionω:=dθ|_C defines on each hyperplane C_m,m∈M, a conformal symplectic structure. Lagrangian planes ofC_mare tangent to maximal integral submanifolds ofC; for this reason, such sub- manifolds are calledLagrangian(or alsoLegendrian). We denote byL(Cm) theGrassmannian of Lagrangian planes ofCmand by

π:M⁽¹⁾= [

m∈M

L(Cm)→M

the bundle of Lagrangian planes. Since points of M⁽¹⁾ are Lagrangian planes, throughout the paper we will consider the identification m¹ ≡ Lm¹ ∈M⁽¹⁾, so that thetautological bundleT(M⁽¹⁾) :=S

m¹∈M⁽¹⁾Lm¹ → M⁽¹⁾ is well defined.

A scalar 1^st order PDE with one unknown function and nindependent variables (1^st order PDE) is a hypersurface F ofM and itssolutions are integral manifolds ofCcontained inF. Ascalar2^ndorder PDE with one un- known function andnindependent variables(2^ndorder PDE) is a hypersur- faceE ofM⁽¹⁾ and itssolutions are Lagrangian submanifolds Σ⊂M such that TΣ ⊂ E. ACauchy datum forE is defined as an (n−1)-dimensional integral submanifold ofC. The restriction toEof fibre bundleπis a bundle overM whose fibre atmis the hypersurface ofL(C_m)

Em:=E ∩ L(Cm).

Acharacteristic subspace forE at m¹ is a hyperplaneU ⊂L_m1 such that the curveU⁽¹⁾⊂ L(Cm) of Lagrangian planes containingUis tangent toEm

atm¹. The tangent space T_m1U⁽¹⁾ is called acharacteristic direction forE atm¹. WhenU⁽¹⁾⊂ Em, hyperplaneU is said to bestrongly characteristic.

By means of previous geometric concepts, we are able to give an intrinsic definition of MAEs of form (0.5) and (0.6). Of these, the former describes,

locally, hypersurfacesEΩ of M⁽¹⁾ formed by Lagrangian planes which an- nihilate an n-form Ω on M (to avoid trivial equations, this form can be chosen in Λⁿ(M)rI(θ), whereI(θ)⊂Λ^∗(M) denotes the differential ideal generated by a contact formθ):

(1.1) EΩ=n

m¹∈M⁽¹⁾

Ω|L_m1 = 0o .

As to (0.6), it locally describes hypersurfaces E_D of M⁽¹⁾ whose points are Lagrangian planes which non trivially intersect ann-dimensional sub- distributionDofC:

(1.2) ED =n

m¹∈M⁽¹⁾

Lm¹∩ Dπ(m¹)6= 0o .

One of the main geometric objects associated with a 2^nd order PDE E is its conformal metric gE, which is defined by means of the canonical isomorphism g_m1: T_m^∗1L(Cm) →^∼ L_m1 ∨L_m1, ρ 7→ gρ, where m¹ ∈ M⁽¹⁾ (see Section 2 for details), and defining g_E as (g_E)_m1 = [g_(dF)

m1], where E={F = 0}.

Now, we are in the position to formulate the main result of the paper.

Theorem 1.1. — LetE ⊂M⁽¹⁾ be a2^nd order PDE. ThenE is locally of the formE_D for somen-dimensional distributionD ⊂ Ciff the following properties are satisfied:

(1) Its conformal metric is decomposable: (gE)m¹ =`m<sup>1</sup>∨`⁰_m1, where

`<sub>m</sub>1, `⁰_m1 ⊂L_m1 are lines;

(2) if we let vary the pointm¹along the fibreEm, the lines`m<sup>1</sup>, `⁰_m1 fill twon-dimensional spacesD1m, D2mofCm.

Furthermore,D₁ and D₂ are mutually orthogonal w.r.t. ω =dθ andE = E_D1=E_D2.

Essentially, we find necessary and sufficient conditions for a scalar 2^nd order PDE to be ofE_D type. In [10] the author found sufficient conditions in terms of the existence of a suitable number of intermediate integrals: we give a geometrical interpretation of this result in Corollary 6.6. Also, we would like to underline that the above theorem is the natural generaliza- tion of a well known result for n = 2: a 2^nd order PDE E ⊂ M⁽¹⁾ with two independent variables is a non-elliptic MAE iff the characteristic lines fill two 2-dimensional subdistributions D1, D2 ⊂ C which turn out to be mutually orthogonal w.r.t.ω =dθ. The equation is parabolic if D₁ =D₂ and hyperbolic otherwise.

Then, we describe some procedures for integrating equations E_D, based on the existence of classical or nonholonomic intermediate integrals (this

notion is a generalization of the ordinary one, see Section 6.4). For this kind of equations, we get an easy generalization of the Monge method stated in Theorem 6.12. As an application, in Section 7.1 we prove that MAEs of type ED (possibly, with no ordinary intermediate integral) admitting a special nonholonomic intermediate integral have (smooth) solutions. In Section 7.2 we prove that when a MAE of typeEDadmits a suitable number of independent intermediate integrals the Cauchy problem can be solved.

In Section 7.2.1 we work out all details and computations for an explicit equation by using our results, including main Theorem 1.1.

2. Geometry of the tangent and cotangent bundle of the Lagrangian Grassmannian L(V)

Lagrangian Grassmannian L(V) and its tautological bundle T(L(V)). Let (V, ω) be a symplectic 2n-dimensional vector space. A La- grangian planeis an isotropic subspaceL⊂V of maximal dimension (i.e.

ω|L= 0 and dimL=n). We shall denote byL(V) theGrassmannian of La- grangian planesinV and byT(L(V)) itstautological bundle,i.e.the fiber at pointL∈ L(V) isL. Fixed a symplectic basis{ei, eⁱ}(i.e.ω(ei, e^j) =δ^j_i), eachn-planeL ∈ L(V) transversal to he¹, . . . , eⁿi is uniquely determined by a symmetric real matrix P = kpijk: L =L_P =hei+p_ije^ji. If U is a subspace of V, we shall denote by U^⊥ the orthogonal complement of U w.r.t.ω.

The Plücker embedding ι: L = hv1, v2, . . . , vni ∈ L(V) 7→ [volL] ∈ PΛⁿ(V), where volL := v1 ∧v2∧ · · · ∧vn ∈ Vn

(V), allows to identify L(V) with its image into the projective spacePΛⁿ(V). A straight line of PΛⁿ(V) which is included inι(L(V)) is called alineofL(V). We will denote by`(L,L) the line of˙ PΛⁿ(V) passing atLwith direction ˙L∈T_LL(V).

Metrics associated with tangent and cotangent vectors ofL(V).

It is well known that there is a canonical isomorphism (2.1) g:TLL(V)−→^∼ S²(L^∗), L˙ 7−→g^L˙.

In this way, a vector field X on L(V) defines a section g^X of S² T^∗(L(V))

which we will call a metric onT(L(V)) (note that it can be degenerate). By duality, we also get a canonical isomorphismg:T_L^∗L(V)→^∼ S²(L), ρ7→ g_ρ (the use of super and subscripts eliminates the ambiguity on maps “g”).

In terms of coordinates, the metricg^L˙ onL=hwi:=e_i+p_ije^jiassociated with ˙L∼ ||p˙ij||is given by

g^L˙ = ˙p_ijeⁱ⊗e^j.

In the same way, the metric gρonL^∗associated with 1-formρ=ρ^ijdpij

is g_ρ = ρ^ijw_i ⊗w_j, In particular, a function F ∈ C^∞(L(V)) defines a metric onL^∗:

(2.2) g_(dF)L =X

i6j

∂F

∂pij

wi∨wj.

Rank of tangent vectors ofL(V).Isomorphism (2.1) allows to define the rank of a tangent vector ˙L ∈ TLL(V) as that of the corresponding symmetric bilinear formg^L˙ ∈ S²(L^∗). We call the set T¹L(V) of vectors of rank 1 thecharacteristic coneor Segre variety (see [1]). If ˙L∈T_L¹L(V), then, up to a sign,

(2.3) L˙ 'g^L˙ =η⊗η, for some η∈L^∗. From now on, we identify ˙Lwithg^L˙.

Proposition 2.1. — The straight line `(L,L)˙ of PΛⁿ(V) is a line of L(V)iffrank( ˙L) = 1.

Proof. — Assume that ˙L ∈ T_L¹L(V). Take coordinates P =||pij|| with P(L) = 0 andP( ˙L) = diag(1,0, . . . ,0). Then,

`(L,L) = [(e˙ ₁+te¹)∧e₂· · · ∧en] = [e₁∧ · · · ∧e_n+te¹∧e2∧ · · · ∧en]⊂ L(V).

The converse is derived from the following property: if a, a⁰ ∈ Λ^k(W) are twok-vectors such thatta+sa⁰ is decomposable for anyt, s∈R, then there exists a decomposable (k−1)-vectorb∈Λ^k−1(W) and vectorsv, v⁰ such thata=v∧banda⁰=v⁰∧b. Indeed, ak-vectorcis decomposable iff it satisfies the Plücker relation (γyc)∧c= 0 for anyγ∈Λ^k−1(W^∗) (see, for example [11]). By hypothesis, these relations hold forc=a,c=a⁰ and c=a+a⁰. Then we get

0 = (γya)∧a⁰+ (γya⁰)∧a, ∀γ∈Λ^k−1(W^∗).

We chooseγsuch thatv⁰:=γya6= 0 andv:=−γya⁰6= 0. Thenv⁰∧ a= v∧a⁰, so thata=v∧b,a⁰=v⁰∧b for someb∈Λ^k−1(W).

3. Hypersurfaces of the Lagrangian Grassmannian

3.1. Characteristic cone and characteristic subspaces of a hypersurfaceE ofL(V)and its conformal metric gE

Let E = {F = 0} with F ∈ C^∞(L(V)) such that dF 6= 0 be a hyper- surface of L(V). We denote by gE := [gdF|E] the conformal class of the restriction ofgdF to E; we call it theconformal metricon E. It is indepen- dent of the choice ofF and its local expression is given by (2.2).

Definition 3.1. — The setCh_L(E) =T_LE∩T_L¹L(V)of rank1tangent vectors toEat pointLis called thecharacteristic coneofEatLand its ele- ments are calledcharacteristic vectorsforEatL. The1-dimensional vector space generated by a characteristic vector is called a characteristic direc- tion. A characteristic vector L˙ for E at Lis called strongly characteristic if the line`(L,L)˙ is contained inE.

Lemma 3.2. — Characteristic vectorsL˙ ∈ChL(E) are, up to sign, the tensor squareL˙ =η⊗η ofg_E-isotropic covectorsη∈L^∗.

Proof. — By definition, ˙L is characteristic for E ={F = 0} if, besides being of the form±η⊗η (rank 1), it is tangent to E; in other words, if ˙L kills (dF)L. So,

0 =hL,˙ (dF)Li=hg^L˙, g_(dF)Li=h±η⊗η, g_(dF)Li=±g_(dF)L(η, η) and the result follows because (gE)L=g(dF)_L.

We define theprolongation U⁽¹⁾ ⊂ L(V) of a subspaceU ⊂V by:

(3.1) U⁽¹⁾:=

(L∈ L(V)|L⊇U, if dim(U)6n L∈ L(V)|L⊆U, if dim(U)>n.

SinceL=L^⊥, thenU ⊂W =⇒U⁽¹⁾ ⊃W⁽¹⁾ and also U⁽¹⁾ = U^⊥(1) . If U is isotropic, then

(3.2) U⁽¹⁾ 'U ⊕ L(W) :=

U ⊕L⁰|L⁰ ∈ L(W) ,

whereW := (U⊕U⁰)^⊥withU⁰satisfying dimU⁰ = dimU andω|_U⊕U⁰ non- degenerate. In coordinates, let{ei}be a basis ofLsuch thatU =heai16a6k; let also {e_i, eⁱ} be its extension to a symplectic basis of V and consider U⁰ :={e^a}16a6k. So,

U⁽¹⁾=n

L=hea, ei+pije^ji

16a6k , ||pij|| ∈S²(R^n−k)o ,

and its tangent space atLis given by (3.3) T_LU⁽¹⁾ =

eⁱ∨e^j, i, j=k+ 1, . . . , n

'S²(U⁰)⊂S²(L^∗), whereU⁰⊂L^∗ denotes the annihilator ofU.

Definition 3.3. — An isotropic subspaceU is calledcharacteristicfor a covectorρ∈T_L^∗L(V)ifU ⊂Landρ|_TLU(1) = 0. It is called characteristic for a hypersurfaceE ={F = 0}ofL(V)at a pointL∈Eif it is character- istic for the covector(dF)L. It is calledstrongly characteristicifU⁽¹⁾ ⊂E.

A covectorη ∈L^∗ is called characteristic for ρif the hyperplane Ker(η)is characteristic forρ.

This definition extends Definition 3.1 in the following sense. Let η∈L^∗ and U := Ker(η) ⊂ L be its associated hyperplane; equation (3.3) gives TLU⁽¹⁾=hη⊗ηi, so thatU is characteristic for (dF)L in the sense of Def- inition 3.3 iff the (one-dimensional) tangent direction toU⁽¹⁾ is generated by one characteristic vector for (dF)L in the sense of Definition 3.1.

By using again identification (3.3), and by arguing as in the proof of Lemma 3.2, we can determine the “characteristicness” of a subspace in terms of the conformal metric: this is the content of the following

Lemma 3.4. — LetU ⊂L∈ L(V)andρ∈T_L^∗L(V). ThenU is charac- teristic forρiff its annihilatorU⁰⊂L^∗ isg_ρ-isotropic.

The following proposition relates the decomposability ofgρ with the be- havior of the set of characteristic hyperplanes forρ.

Proposition 3.5. — Let ρ ∈ T_L^∗L(V). Then gρ is decomposable iff characteristic hyperplanes for ρ form two (n−2)-parametric families H andH⁰ such that

dim \

U∈H

U = dim \

U∈H⁰

U = 1.

Proof. — Letgρ=v∨wfor somev, w∈L. By Lemma 3.4, a hyperplane U = Ker(η) of Lis characteristic iff g_ρ(η, η) = η(v)η(w) = 0. This means thatv∈U orw∈U. So we get two families of characteristic hyperplanes H:={U ⊂L| v∈U}, H⁰ :={U ⊂L|w∈U} such thatT

U∈HU =hvi andT

U∈H⁰U =hwi.

Viceversa, let Hbe a (n−2)-parametric family of characteristic hyper- planes forρwhich contain a common linehvi. By dimensional reasons, the setS

U∈HU⁰ ={η∈L^∗ |η|U = 0 for someU ∈ H} contains a conic con- vex open subsetO of the annihilator v⁰ ⊂L^∗. So η, η⁰ ∈ O implies that η+η⁰ ∈ O. Lemma 3.4 shows thatgρ(η, η) =gρ(η⁰, η⁰) =gρ(η+η⁰, η+η⁰) = 0

which impliesg_ρ(η, η⁰) = 0,∀η, η⁰ ∈ O. Sincev⁰ is spanned by O, it isg_ρ-

isotropic. Thus,gρ=v∨wfor somew∈L.

3.2. Hypersurfaces of L(V)associated with n-forms and their characteristics

Anyn-form Ω∈Λⁿ(V^∗) defines the hypersurface

(3.4) E_Ω=

L∈ L(V) | Ω|_L = 0 .

For each σ ∈ Λⁿ⁻²(V^∗), the n-form Ω^σ := Ω +σ∧ω defines the same hypersurface.

Definition 3.6. — Let Ω ∈ Λⁿ(V^∗). A k-dimensional subspace U = he₁,· · ·, e_ki ⊂V is called Ω-isotropicif(e₁∧ · · · ∧e_k)yΩ = 0.

Theorem 3.7. — Let L∈E_Ω and H be a hyperplane ofL. Then the following equivalences hold:

(1) H is characteristic forE_Ωat L;

(2) H is strongly characteristic;

(3) H isΩ^σ-isotropic for someσ∈Λⁿ⁻²(V^∗).

Proof. — Implications 2⇒1 and 3⇒1 are trivial.

1 ⇒ 2.Below we will adopt the following notation: if W =hvii, then volW := v1∧v2∧ · · · ∧vn. Let {ei, eⁱ} be a symplectic basis of V such that H =he₁, . . . , e_n−1i ⊂ he₁, . . . , e_n−1, e_ni = L, so that H⁽¹⁾ ={L_t = he1, . . . , en−1, en+teⁿi}. Any Lagrangian plane in a neighborhood ofL=L0

is of the formLe=hei+p_ije^ji. Let volt:= volLt, so that volt= volL+t volL⁰

whereL⁰ =he1, . . . , e_n−1, eⁿi. In this way the tangent vector toH⁽¹⁾ at L is defined by the derivative along volL⁰. Also, letF(eL) = vol

LeyΩ, so that E_Ωis locally described by{F= 0}. The derivative ofF atLalong volL⁰ is

t→0lim

F(Lt)−F(L)

t = lim

t→0

voltyΩ−volLyΩ t

= lim

t→0

(volL+tvolL⁰)yΩ−volLyΩ t

= volL⁰yΩ =F(L⁰)

which vanishes iffL⁰ belongs to EΩ. Hence,H⁽¹⁾ is included in EΩ.

1⇒3.By using the above results, we have that H is characteristic ⇐⇒ H⁽¹⁾ ⊂EΩ

⇐⇒ voltyΩ = 0 ⇐⇒Ωa(en) = Ωa(eⁿ) = 0 where Ωa :=ayΩ, a =e₁∧ · · · ∧e_n−1. For anyσ ∈Λⁿ⁻²(V^∗), we have that

ayΩ^σ= Ωa+

n−1

X

j=1

(−1)^jσ(e1, . . . , e_j−1, ej+1, . . . , e_n−1)(ejyω).

Thus, (ayΩ^σ)|L⁰ = 0 and (ayΩ^σ)(eⁱ) vanishes ifσ(e1, . . . , e_i−1, ei+1, . . . , e_n−1) = (−1)ⁱ⁺¹Ω_a(eⁱ). For such aσ,ayΩ^σ = 0,i.e.H is isotropic for Ω^σ.

3.3. Hypersurfaces ED associated with ann-plane D and their characteristics

We associate with ann-dimensional subspaceD⊂V the following subset ofL(V):

(3.5) ED={L∈ L(V)|L∩D6= 0}. IfD={%1=%₂=· · ·=%_n = 0}, then

(3.6) ED= EΩ_D where ΩD:=%1∧ · · · ∧%n.

IfD=hei+bije^ji(for some symplectic basis{ei, eⁱ}), then ED={L= LP | det(P −B) = 0}, where P = ||pij|| and B = ||bij||. In particular, EDis an algebraic hypersurface of L(V). Below we describe the conformal metricg_ED in coordinates.

Proposition 3.8. — Let E_D be the hypersurface of L(V) associated withn-planeD=hei+bije^jiandL=LP =hwi=ei+pije^ji ∈ED. Then the conformal metricgE_D onL^∗ is given by

(3.7) g_ED =A^ijw_i∨w_j

whereA=||A^ij||andA^ij is the algebraic complement of the(i, j)-entry in matrix(P−B). Moreover

(1) A= 0ifrank(P−B)< n−1;

(2) A = ||aⁱb^j|| if rank(P −B) = n−1 where (P −B)·a = 0 and (P−B^t)·b= 0. In particular

(a) gE_D =a∨b,a=aⁱwi,b=bⁱwi;

(b) matrix ¹₂(A+A^t)has rank equal to1ifB=B^tand rank equal to2 ifB 6=B^t.

Proof. — Since _∂p^∂

ij(det(P −B)) equals Aⁱⁱ if i = j, and A^ij+A^ji if i6=j, then, for eachη ∈L^∗,

g_ED(η, η) =X

i6j

∂

∂p_ij det(P−B) ηiηj

=X

i,j

A^ijη_iη_j= 1 2

X

i,j

(A^ij+A^ji)η_iη_j,

whereηi=η(wi). This proves (3.7). The second part of the lemma follows from elementary properties of adjoint matrices.

Definition 3.9. — A pointL∈EDis calledsingularifdim(L∩D)>2 andregularotherwise. The set of regular points ofED will be denoted by E^reg_D .

Now we give a criterion to distinguish singular points.

Proposition 3.10. — A pointL_P ∈E_D is singular iff the differential ofdet(P−B)atLvanishes, that is iff the metric gE_D vanishes atL.

Proof. — SinceL∩D= Ker(P−B), we derive the equivalence dim(L∩ D) =k ⇐⇒ rank(P−B) =n−k.If k>2, then rank(P −B)6n−2, which implies that its adjoint matrix vanishes. Then _∂p^∂

ij det(P−B)

= 0

at the pointLand (gE_D)L= 0.

The following key proposition states that, given an n-dimensional sub- spaceD ⊂V, the only other subspace defining the same ED is the skew- orthogonal complementD^⊥.

Proposition 3.11. — Let(V, ω)be a2n-dimensional symplectic vector space. LetD andDe ben-dimensional planes ofV. Then

E

De= E_D ⇐⇒ De =D or De=D^⊥.

Proof. — One implication will be proved if dim(L∩D) = dim(L∩D^⊥) for any Lagrangian planeL. But this easily follows from identities

L∩D^⊥ =L^⊥∩D^⊥= (L∪D)^⊥ = (L+D)^⊥.

As to the inverse implication, we must prove that for anye∈V\(D∪D^⊥) there exists a Lagrangian subspace L 3 e such that L∩D = 0, so that L /∈ED; in this way, ife∈D, then Ee

De 6= ED. In order to get such anL, let us consider the (n−1)-dimensional subspaceD∩e^⊥ and the quotient map Π :e^⊥→e^⊥/hei(symplectic reduction). The projection Π(D∩e^⊥) is a half

dimensional space in the symplectic spacee^⊥/heiand it is elementary the existence of a Lagrangian subspaceLe⊂e^⊥/heisuch that Π(D∩e^⊥)∩Le= 0.

Then,L:= Π⁻¹(eL) is a Lagrangian subspace ofV withe∈LandL∩D= 0,

as required.

If we translate previous proposition in terms of n-forms, we get the fol- lowing

Corollary 3.12. — Up to a factor, at most two different decomposable n-forms give equationEΩ.

The proposition below describes characteristic hyperplanes for hypersur- faces ED.

Proposition 3.13. — LetD and ΩD be as in (3.6). Let alsoH ⊂V be an (n−1)-dimensional isotropic subspace and H⁽¹⁾ = {Lt}. Then the following conditions are equivalent:

(1) H⊂L0 is characteristic forED atL0∈ED; (2) H⁽¹⁾⊂ED;

(3) hΩD,volti= 0, wherevolt denotes an arbitrary element inVn

(Lt) different from zero;

(4) L_t∩D6= 0for allt;

(5) H has non trivial intersection with Dor D^⊥.

Proof. — Equivalence 1⇔2 is Theorem 3.7, taking into account (3.6).

Properties 3 and 4 are, by definition, alternative ways to write property 2.

Let us passe to equivalence 2 ⇔ 5. Let H be characteristic for ED at L, (so, it is also strongly characteristic and, hence, any Lagrangian plane containingH intersects D non trivially). We want to prove that H has a non trivial intersection with eitherDorD^⊥. LetH∩D= 0. Let{ei, eⁱ}be a symplectic basis such thatH =he₁, . . . , e_n−1iandL=he₁, . . . , e_n−1, e_ni.

By assumption,L∩D6= 0, so that the unique possibility is thatL∩Dis gen- erated by a vectore_n+Pn−1

i=1 α_ie_i. Up to a change of basis, we can assume such generator to been(in particular,en∈D). Now, the Lagrangian planes Lt:=he1, . . . , e_n−1, en+teⁿihave non trivial intersections withD. In fact, by the same reasoning as above,L_t∩D,t6= 0, is generated by a vector of the formen+teⁿ+Pn−1

i=1 αi(t)ei=en+t

eⁿ+Pn−1

i=1 t⁻¹αi(t)ei

.Taking into account thaten ∈D, we get vn :=eⁿ+Pn−1

i=1 t⁻¹αi(t)ei∈D. If we take two different values t, t we have that Pn−1

i=1

t⁻¹αi(t)−(t)⁻¹αi(t) ei ∈ D∩H = 0 which implies that v_n is independent of t. A new change of basis allows to takeeⁿ =vn, so thatLt∩D =hen+teⁿi; in particular,

D⊃ hen, eⁿiandD^⊥ ⊂ hen, eⁿi^⊥. Also,H ⊂ hen, eⁿi^⊥ and a computation gives us

dimD^⊥∩H = dimD^⊥+ dimH−dim(D^⊥+H)>n+ (n−1)−(2n−2) = 1, because D^⊥ +H ⊂ he_n, eⁿi^⊥. As a consequence, H ∩D^⊥ 6= 0, as we

wanted.

Remark 3.14. — Claims 1, 2, 3 of the above theorem remain equivalent also for hypersurfaces EΩ.

Bringing together Propositions 3.5, 3.8, 3.11, 3.13, in the theorem below we summarize the main results regarding the hypersurfaces of type E_D by pointing out how to describe them in terms of their characteristics.

Theorem 3.15. — LetE^reg_D be the set of regular points of ED. Then

• A hyperplane H of L ∈ E^reg_D is characteristic for E_D at L iff it contains one of the following straight lines:

`L:=L∩D or `⁰_L:=L∩D^⊥.

Then, if`<sub>L</sub>6=`⁰_L, there are two(n−2)-parametric familiesH(t₁, . . . , tn−2) and H⁰(t1, . . . , tn−2)of characteristic hyperplanes in L: one contains`L=T

t1,...,tn−2H(t1, . . . , t_n−2)and another contains`⁰_L= T

t1,...,tn−2H⁰(t1, . . . , t_n−2). If`L=`⁰_L, these two families coincide.

• The conformal metric of E^reg_D is decomposable and is given by (g_E^reg

D )L=`L∨`⁰_L.

• For any line`⊂D there existsL∈E<sup>reg</sup><sub>D</sub> such that`=`L=L∩D.

HenceD=S

L∈E_D`_L andD^⊥=S

L∈E_D`⁰_L.

4. Local description of PDEs and MAEs

In this section we refer to definitions given in Section 1. From now on, for simplicity, we will assume that the contact form θ is globally defined.

A diffeomorphism of M which preserves C is called a contact transfor- mation. There exist coordinates (xⁱ, z, p_i) on M, i = 1, . . . , n, such that θ=dz−pidxⁱ. Such coordinates are called contact (or Darboux) coordi- nates. Locally defined vector fields

∂bxⁱ :=∂xⁱ+pi∂z, ∂pi, i= 1, . . . , n,

span distributionC. A system of contact coordinates (xⁱ, z, pi) on M in- duces coordinates (xⁱ, z, p_i, p_ij = p_ji, 1 6 i, j 6 n) on M⁽¹⁾ as follows:

a point m¹ ≡ Lm¹ ∈ M⁽¹⁾ has these coordinates iff π(m¹) = (xⁱ, z, pi)

and the corresponding Lagrangian plane is given byL_m1 =L_P :=h∂b_xi+ pij∂p_ji ⊂ C_π(m1).

A 1^storder PDE is locally described as zero level setM_f :={f(xⁱ, z, p_i) = 0} of a function f ∈ C^∞(M), whereas a 2^nd order PDE E is locally de- scribed byE ={F(xⁱ, z, pi, pij) = 0}, withF ∈C^∞(M⁽¹⁾).

MAEs of type EΩare, taking into account the beginning of Section 3.2, the zero locus of the following n-form on the tautological bundle T(M⁽¹⁾) : m¹ 7→ Ω|L_m1. It is straightforward to check that, locally, such MAEs are described by (0.5). For a givenn-dimensional subdistributionD ofC, we have

ED=EΩD, with ΩD:=Y1·θ∧ · · · ∧Yn·θ,

(see also [16]) whereYiare vector fields generating the orthogonal distribu- tionD^⊥ (w.r.t.ω=dθ). Indeed the distributionDis defined by the system of equations{θ = 0,Yi·θ= 0}, so that the result follows from (3.6). On the other hand, it is always possible to choose a contact chart (xⁱ, z, pi) such that

D=hX1, X2, . . . , Xni, Xi=∂b_xi+bij∂p_j,

for some functionsbij (it is sufficient thatD ∩ h∂p₁, . . . , ∂p_ni= 0). In this case,

E_D=n

LP =h∂b_xi+pij∂p_ji

detkpij−bijk= 0o .

Remark 4.1. — Even ifD^⊥ and Ddefine the same equation, they are not necessarily contactomorphic.

5. Characteristics of PDEs, of MAEs and proof of Theorem 1.1

We define theprolongationN⁽¹⁾⊂M⁽¹⁾of a submanifoldN of a contact manifoldM as follows (see (3.1)):

N⁽¹⁾:=

(m¹∈M⁽¹⁾|Lm¹⊇TmN∩ Cm, if dim(N)6n m¹∈M⁽¹⁾|L_m1⊆TmN∩ Cm, if dim(N)>n.

If N is an integral submanifold of the contact manifold (M,C), then the natural projectionπN:N⁽¹⁾ → N is a fibre bundle whose typical fibre is U⊕ L(W)' L(R^2n−2k) whereU andW are as in the identification (3.2), withU =TmN andV =Cm.

Definitions given in Section 3.1 can be immediately reformulated in the language of PDEs by replacingV withCmand E withEm. In particular, a

direction inT_m1Emis called characteristic forE if it is generated by a rank 1 tangent vector (inT_m1L(Cm)). In the same way, a subspace U ⊂TmM is said to becharacteristic for the equation E at m¹ ifU⁽¹⁾ is tangent to E at m¹. If in addition U⁽¹⁾ ⊂ E, U is said to be strongly characteristic.

Also, we can introduce a conformal metric (g_E)_m1 =g_{Eπ(m1 )}∈S²(L^∗_m1) at each pointm¹ ≡L_m1 ∈ E and Lemma 3.4 is still valid mutatis mutandis.

In coordinates, a tangent vector toEmat m¹ having ˙P =||p˙ij|| as matrix of coordinates is of rank 1 iff ˙pij = ηiηj up to a sign (see also (2.3)).

Furthermore, it is characteristics forE={F= 0}if it satisfies

(5.1) X

i6j

∂F

∂pij

˙

pij=X

i6j

∂F

∂pij

ηiηj= 0

i.e. covector η is isotropic for g_E. In view of Proposition 3.5, (g_E)m¹ is decomposable iff characteristic hyperplanes ofL_m1 are divided in two (n− 2)-parametric familiesH_m1 andH_m⁰ 1 such that

dim \

U∈H_m1

U = dim \

U∈H⁰

m1

U = 1.

All results of Section 3.2 can be applied to fibersEΩmjust by replacing Ω with ΩmandEΩm with EΩ,m∈M. In fact, in view of (1.1) and (3.4), we have thatEΩ=S

m∈MEΩm.For the sake of completeness, we reformulate the results of Theorem 3.7 in the language of MAEs:

Theorem 5.1. — Let m¹ ∈ EΩ. Then a hyperplane of Lm¹ is charac- teristic forEΩ iff it is strongly characteristic. Moreover, characteristic hy- perplanes are those hyperplanes which are isotropic with respect to some n-formΩ^σ:= Ω +σ∧dθ, whereσ∈Λⁿ⁻²(M).

Furthermore, all results of Section 3.3 can be applied to fibersED_m just by replacing Dm with D and E_Dm with ED, m ∈ M. In fact, in view of (1.2) and (3.5), we have thatED =S

m∈MEDm. The following statement is a reformulation of Theorem 3.15.

Theorem 5.2. — Let m¹ ∈ EDm be a regular point. Then (gED)m¹ =

`<sub>m</sub>1 ∨`⁰_m1, where `<sub>m</sub>1 =L<sub>m</sub>1 ∩ Dm and `⁰_m1 =L_m1 ∩ D_m^⊥ are lines. Thus there exist only two (n−2)-parametric families of characteristic hyper- planes ofL_m1: one rotates around `<sub>m</sub>1, the other around `⁰_m1. Moreover, Ch_m1(E_D) ={±η⊗η, η∈`<sup>0</sup><sub>m</sub>1∪`⁰_m⁰1}where `<sup>0</sup><sub>m</sub>1, `⁰_m⁰1 ⊂L^∗_m1 are, respec- tively, the annihilators of`m<sup>1</sup> and`⁰_m1. Covectors η ∈L^∗_m1 corresponding to characteristic directions and belonging to`<sup>0</sup><sub>m</sub>1 (resp.,`⁰_m⁰1) define hyper- planes{η= 0}which contain`m<sup>1</sup> (resp.,`⁰_m1). If one let the pointm¹vary