Monge-Ampère Equations on (Para-)Kähler Manifolds: from Characteristic Subspaces to Special...

Acta Appl Math (2012) 120:3–27DOI 10.1007/s10440-012-9707-1

Monge-Ampère Equations on (Para-)Kähler Manifolds:from Characteristic Subspaces to Special LagrangianSubmanifolds

Dmitri Alekseevsky · Ricardo Alonso-Blanco ·Gianni Manno · Fabrizio Pugliese

Received: 30 September 2011 / Accepted: 2 March 2012 / Published online: 24 March 2012© Springer Science+Business Media B.V. 2012

Abstract We present the basic notions and results of the geometric theory of second orderPDEs in the framework of contact and symplectic manifolds including characteristics, for-mal integrability, existence and uniqueness of formal solutions of non-characteristic Cauchyproblems. Then, we focus our attention to Monge-Ampère equations (MAEs) and discussa natural class of MAEs arising in Kähler and para-Kähler geometry whose solutions arespecial Lagrangian submanifolds.

Keywords Contact and symplectic manifolds · Characteristics of PDEs · Monge-Ampèreequations · (Para-)Kähler manifolds · Special Lagrangian submanifolds

D. AlekseevskyA.A.Kharkevich Institute for Information Transition Problems, Bol’shoi Karetnyi 19, 127994, Moscow,Russiae-mail: [email protected]

Present address:D. AlekseevskyDepartment of Mathematics, Masarik University, Kotlarska 2, 61137 Brno, Czech Republic

R. Alonso-BlancoDepartamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca,Spaine-mail: [email protected]

G. Manno (�)Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, via Cozzi 53,20125 Milano, Italye-mail: [email protected]

F. PuglieseDipartimento di Matematica, Università di Salerno, via Ponte don Melillo, 84084 Fisciano, Italye-mail: [email protected]

mailto:[email protected]




4 D. Alekseevsky et al.

1 Introduction

Characteristics of PDEs are a classical subject [10, 11, 25, 26], as they are related to thelocal existence and uniqueness of solutions of Cauchy problems. Solving such problems fora 1st order scalar PDE in the C∞ case is relatively simple, as the so called method of char-acteristics provides a constructive way for their solutions (see also Sect. 2). For scalar PDEsof order greater than one the solution of a Cauchy problem is assured, under some analyt-icity assumption, by Cauchy-Kowalewski theorem. Broadly speaking, if a Cauchy datum isnon-characteristic, then, in the C∞ case, Cauchy problem for a given PDE admits, locally, aunique formal solution: in fact, in this case, we can put the PDE in the Cauchy-Kowalewskiform. Under the same assumptions, in the analytic case, Cauchy problem admits a locallyunique solution. This suggests to define the characteristics of a PDE as the tangent spacesto a Cauchy datum for which there is not a unique formal solution. This is an intrinsicproperty, whose geometrical meaning is explained in Sects. 2 and 3 in terms of contact ge-ometry. Thus, a natural question arises: to study PDEs whose characteristics behave in aprescribed way. It is well known that in the classical case of scalar PDEs with 2 independentvariables and one unknown function such characteristics form a distribution of cones onan appropriate contact manifold: such distribution degenerates into a distribution of planesonly for Monge-Ampère equations (MAEs) and, conversely, any 2-dimensional MAE canbe uniquely recovered from its characteristic distribution. It is natural to ask if similar resultsalso hold in the multidimensional case, i.e. for MAEs with an arbitrary number of indepen-dent variables. However [2], only a special class of multidimensional MAE, namely, thoseof the form

det

∥∥∥∥

∂2f

∂xi∂xj− bij (x, f,∇f )

∥∥∥∥= 0, i, j = 1, . . . , n, (1)

where bij are C∞ functions, behave as in the classical case: their characteristics forman n-dimensional distribution on the contact manifold locally described by coordinates(xi, u, fxi ). Such distribution turns out to be Legendrian when bij = bji . Equations (1),originally introduced by Goursat [11], can also be described by decomposable n-forms onsuch contact manifold, modulo the differential ideal generated by the contact form. It turnsout that this class of MAEs is strictly smaller, for n≥ 3, than that considered by Lychagin in[20], who defined (general) MAEs as PDEs associated to effective n-form (see also [18]) onsome contact manifold. A similar reasoning can be applied to define MAEs on symplecticmanifolds: in this case the obtained MAEs are independent of the unknown function andthey are called symplectic MAEs [18].

A key tool for the analysis in the present paper is the conformal metric associated to asecond order scalar PDE; such metric (which can be degenerate) is closely related to thenotion of symbol of PDEs. We show how the conformal metric controls many geometricaspects of the given PDE: its rank is an invariant which is responsible for its formal integra-bility (see Sect. 4) and for the given PDE to be a MAE of Goursat type (see Sect. 5). In thisspirit, in Sect. 4.3 we prove, from a geometric viewpoint, the Cauchy-Kowalewski theoremconcerning the existence of a formal solution satisfying a non-characteristic Cauchy datum.In the last two sections we shortly discuss an important class of MAEs which are associatedwith the imaginary (or real) part of a (para-)holomorphic n-form Ω on a 2n-dimensional(para-)Kähler manifold: their solutions are special Lagrangian (SLAG) submanifolds. Fol-lowing Harvey and Lawson [13], we indicate an application of such MAEs to the Monge-Kantorovich mass transport problem.

Monge-Ampère Equations on (Para-)Kähler Manifolds 5

Notations and conventions In the rest of the paper Latin indices will run from 1 to n, unlessotherwise specified. We will use Einstein convention. We denote by X · ρ the Lie derivativeof the differential form ρ along the vector field X. The symmetric tensor product will bedenoted by ∨, i.e. A∨B = 1

2 (A⊗B +B⊗A). The annihilator of a vector subspace U willbe denoted by U 0. We denote by 〈vi〉 the linear span of vectors v1, . . . , vn.

2 Contact Manifolds, Scalar 1st Order PDEs and Their Characteristics

Basic notions and results on contact manifolds can be found in [19].

Definition 1 A (2n+ 1)-dimensional smooth manifold M endowed with a completely non-integrable codimension one distribution C is called a contact manifold. A diffeomorphismΨ of M which preserves C is called a contact transformation.

Locally, C= Ker θ , where the contact form θ is defined up to a conformal factor. Thereexist coordinates (xi, u,pi), i = 1, . . . , n such that θ = du − pidxi . Such coordinates arecalled contact (or Darboux) coordinates. Locally defined vector fields

∂xidef= ∂xi + pi∂u, ∂pi

, i = 1, . . . , n (2)

span the contact distribution C. The 2-form dθ is non degenerate on Cm, ∀m ∈M , so thatthe 2-form

ω= dθ |Cis a symplectic structure on the distribution C. A contact transformation induces a conformaltransformation both of θ and of ω, so the (nonholonomic) symplectic form ω on C is definedup to a conformal factor. We will denote by U⊥ the orthogonal complement of U w.r.t. ω.

Example 1 An important example of a contact manifold is the manifold J 1(E,n) of then-dimensional contact elements of an (n+ 1)-dimensional manifold E (also known as firstorder jet of hypersurfaces [29–31] or extended jet bundle [24]). Essentially, J 1(E,n) isa bundle on E whose fiber at p ∈ E is the set formed by the equivalence classes [S]1pof hypersurfaces S of E which have a contact of order 1 at p ∈ E. Thus, such fiber co-incides with the Grassmannian of n-dimensional subspaces of TpE, so that J 1(E,n) isthe Grassmann bundle Grn(E) on E of n-dimensional subspaces of T E (see also [21]).The contact distribution on J 1(E,n) is defined as follows. Consider the vector subspaceC[S]1p ⊂ T[S]1pJ 1(E,n) formed by vectors of T[S]1pJ 1(E,n) whose projection belongs to TpS:

the distribution [S]1p → C[S]1p is a contact distribution on J 1(E,n).Another example of a contact manifold, which follows from the above construction, is

the first order jet J 1π of a line bundle π . More precisely, if π : E → Q is a line bundle,J 1π is a bundle on E whose fiber at p ∈ E is the set of the equivalence classes [S]1p ofhypersurfaces S of E which are images of local sections s of π , s(x)= p, x ∈Q [8, 24]. Inthe particular case of a trivial bundle π :E =R×Q→Q, J 1π =R× T ∗Q coincides withthe first order jet of functions J 1(Q) of Q.


2.1 Cartan and Hamiltonian Vector Fields

Definition 2 Sections of the contact distribution C are called Cartan vector fields. The typeof a Cartan field Y is defined as the rank of the sequence θ , Y · θ , Y · (Y · θ), . . . .

Any 1-form α ∈Λ1(M) determines a Cartan vector field Yα ∈ C by the relation

Yα · θ = Yα�dθ = α− α(Z)θ, θ(Yα)= 0,

where Z is the Reeb vector field (associated with θ ) defined by conditions θ(Z) = 1,Z �dθ = 0. In particular Y = Y(Y ·θ) for any Cartan field Y . Although Yα depends on thechoice of θ , its direction does not change.

Definition 3 A vector field Yf := Ydf is called a Hamiltonian vector field.

It is easy to check that Yf is of type 2 (the minimum possible). In addition, Yf is acharacteristic symmetry of the distribution {θ = 0, df = 0}. In other words, Yf coincideswith the classical characteristic vector field of the 1st order PDE f (xi, u,pi)= 0.

Two functions f and g on M are said to be in involution if ω(Yf ,Yg)= 0. This conditionis equivalent to the integrability of the distribution 〈Yf ,Yg〉. By using this property, it can beproved that any set (f1, . . . , fk) of k ≤ n independent functions on the contact manifold M

which are in involution can be extended to a contact chart.

2.2 Integral Submanifolds of the Contact Distribution

Recall that an integrable subdistribution of C is ω-isotropic, hence it has dimension lessor equal to n. An n-dimensional integral submanifold is called a Legendrian submanifold.Any Legendrian submanifold, in appropriate contact coordinates (xi, u,pi), is defined by a(generating) function g(x1, . . . , xn) and it is given by

u= g(

x1, . . . , xn)

, pi = ∂g

∂xi

(

x1, . . . , xn)

.

In order to characterize (by Proposition 1) integral subdistributions of C of dimension(n− 1), we need the following lemma.

Lemma 1 Let N be an integral submanifold of C, f ∈ C∞(M) such that f |N = 0 and ϕt

be the local flow of Yf . Then⋃

t ϕt (N) is a solution of θ = 0 and also of f = 0.

Proof We have already seen that the local flow ϕt of Yf preserves solutions of the Pfaffsystem {θ, df }. So

⋃

t ϕt (N) is a solution of both θ = 0 and f = 0. �

Proposition 1 An (n− 1)-dimensional submanifold N is an integral submanifold of C iff itis a hypersurface of an n-dimensional integral submanifold of C.

Proof The condition is obviously sufficient. As to its necessity, let us consider a functionf on M such that f |N = 0 and (Yf )m ∩ TmN = 0 for any m ∈ N . Such a function alwaysexists. In fact, if N = {f1 = 0, . . . , fn+2 = 0} then the (n+ 2) Hamiltonian vector fields Yfi

cannot be simultaneously tangent to N for dimensional reasons. The proposition follows inview of Lemma 1. �


Corollary 1 Let N be an integral (n − 1)-dimensional submanifold of C. Then, for anypoint of N there exists a neighborhood in N which is described by

{

x1, . . . , xn−1, xn = 0, u= φ(

x1, . . . , xn−1)

,

p1 = ∂φ

∂x1, . . . , pn−1 = ∂φ

∂xn−1,pn = φn

(

x1, . . . , xn−1)}

w.r.t. some local contact coordinates (xi, u,pi) of M for certain functions φ and φn. Fur-thermore, one can select a new contact chart (xi, u,pi) by taking u= u− φ so that in thisnew chart N is described by

{

x1, x2, . . . , xn−1, xn = 0, u= 0, p1 = 0, . . . , pn−1 = 0, pn = φn

(

x1, . . . , xn−1)}

.

2.3 Scalar PDEs of 1st Order and Methods of Characteristics

Definition 4 A scalar first order partial differential equation (1st order PDE) with one un-known function and n independent variables is a hypersurface F of a (2n+ 1) -dimensionalcontact manifold (M,C). A solution of F is a Legendrian submanifold contained in F.

In terms of contact coordinates, a first order PDE F can be described as a zero level set{f (xi, u,pi) = 0} of a function f ∈ C∞(M). A solution parameterized by x1, . . . , xn canbe written as u = φ(x1, . . . , xn), pi = ∂φ/∂xi(x1, . . . , xn), where the function φ satisfiesf (xi, φ, ∂φ/∂xi)= 0, which coincides with the classical notion of solution.

Remark 1 The role of coordinates “xi” as independent variables is purely extrinsic, becausex’s and p’s can be mixed by an appropriate contact transformation. For instance, a total orpartial Legendre transformation can be used in order to consider “pi” coordinates (all orsome of them) as new independent variables.

Definition 5 A Cauchy datum for a first order PDE {f = 0}, f ∈ C∞(M), is an (n− 1)-dimensional integral submanifold of C satisfying {f = 0}. It is called non-characteristic ifit is transversal to the Hamiltonian vector field Yf .

Remark 2 The name “non-characteristic” is justified since Yf coincides with the classicalcharacteristic vector field of first order PDE {f = 0} (we recall that Yf preserves the Pfaffsystem {θ, df }). The name “Cauchy datum” is justified in view of the following fact: inthe case that M is the space J 1(Rn) of 1-jets of functions on R

n, an (n− 1)-dimensionalsubmanifold N ′ of R

n+1 can be prolonged in a unique way to a Cauchy datum N for equationf = 0 without solving any differential equation. In coordinates, if (xi, u,pi) is a contactchart on M = J 1(Rn) and N ′ is locally described by

N ′ : xi = φi(t1, . . . , tn−1), u= φ(t1, . . . , tn−1),

then,

N : xi = φi(t1, . . . , tn−1), u= φ(t1, . . . , tn−1), pi =ψi(t1, . . . , tn−1),


where functions ψi are uniquely determined by the system of n algebraic equations⎧

⎨

⎩

0= (du− pidxi)|N =(

∂φ

∂th−ψi(t)

∂φi

∂th

)

dth,

0= f |N = f (φi(t), φ(t),ψi(t)).

Now, let us consider a given non-characteristic Cauchy datum N for the equation{f = 0}. Then, by Lemma 1, manifold Σ =⋃

t ϕt (N), where ϕt is the local flow of theHamiltonian vector field Yf , is a solution of f = 0. This solution is, locally, the only onecontaining N , because by Lemma 1 Yf is tangent to any maximal solution of {f = 0}. Inmore concrete terms, construction of solutions of first order PDE f = 0 goes along thefollowing steps:

1. take a non-characteristic Cauchy datum N ;2. integrate vector field Yf ;3. take the set Σ of integral curves of Yf crossing N .

The above method is called the method of characteristics (see also [8]).

3 Prolongation of Contact Manifolds, 2nd Order PDEs and Their Characteristics

Before introducing characteristics in a rigorous mathematical way, below we give the ideawhich motivates the construction of Sects. 3.1–3.4. Let

F(

x1, . . . , xn, u,p1, . . . , pn,p11,p12, . . . pnn

)= 0 (3)

where u = u(x1, . . . , xn), pi = ∂u/∂xi , pij = ∂2u/∂xi∂xj be a scalar 2nd order PDE. TheCauchy problem consists of finding a solution u= f (x1, . . . , xn) of (3) which satisfies thefollowing conditions

f |(X1(t),...,Xn(t)) =U(t),∂f

∂xi

∣∣∣∣(X1(t),...,Xn(t))

= Pi(t), (4)

where

Φ(t)= (

X1(t), . . . ,Xn(t),U(t),P1(t), . . . ,Pn(t))

, t= (t1, . . . , tn−1) (5)

is a given (n− 1)-dimensional manifold, i.e. a Cauchy datum; obviously, in (5) the choiceof the parametrization is irrelevant. It is well know that if Cauchy datum (5) is non-characteristic, then, in the C∞ case, Cauchy problem (4) for (3) admits, locally, a uniqueformal solution. Under the same hypothesis, in the analytic case it admits a unique localsolution.

In the case n= 2, non-characteristicity means that tangent direction v = Φ(0) at a pointm= Φ(0)= (x1, x2, u,p1,p2) of the (1-dimensional, in this case) Cauchy datum satisfiesthe classical condition

∂F

∂p11

∣∣∣∣m1

(

v2)2 − ∂F

∂p12

∣∣∣∣m1

v1v2 + ∂F

∂p22

∣∣∣∣m1

(

v1)2 �= 0 (6)

for each m1 = (x1, x2, u,p1,p2,p11,p12,p22) satisfying (3), where

v = v1(∂x1 + p1∂u + p1i∂pi)+ v2(∂x2 + p2∂u + p2i∂pi

).


The vector v can be considered as an “infinitesimal Cauchy datum”.From (6) it is clear that one can associate with any point m1 satisfying (3) two (possibly

imaginary) directions in the space (xi, u,pi), namely, those annihilating (6) (“characteristiclines”); if we let this point vary keeping the point m fixed, these two directions form, ingeneral, two cones at m.

It is clear that coordinates (xi, u,pi) locally describe a (2n + 1)-dimensional contactmanifold, whereas (xi, u,pi,pij ) describe the set of its Legendrian n-planes. The aim ofthe forthcoming Sects. 3.1–3.4 is to introduce characteristics of scalar 2nd order PDEs in thegeometrical context of contact manifolds and to study their elementary properties.

3.1 Prolongation of a Contact Manifold

A contact manifold (M,C) defines a conformal symplectic structure ω= dθ |C on C, whereθ is any 1-form such that Ker(θ)= C. We denote by L(Cm) the Lagrangian Grassmannianof (Cm,ωm), m ∈M :

L(Cm)def= {set of Lagrangian planes of Cm}.

Definition 6 The prolongation of a contact manifold (M,C) is the fiber bundle π :M(1) →M where

M(1) =⋃

m∈M

L(Cm).

Points of M(1) are Lagrangian planes of (Cm,ωm), m ∈M : a generic point of M(1) willbe denoted either by m1 or by Lm1 so that the tautological bundle

T(

M(1))= {(

m1, v) | v ∈ Lm1

}→M(1),(

m1, v) →m1

over M(1) is well defined. Thus, M(1) is a subbundle of the Grassmann bundle Grn(M).We have the Plücker embedding ι of M(1) into the projective bundle PΛn(C) given by

ι : L= 〈e1, e2, . . . , en〉 ∈L(Cm) → [volL] := [e1 ∧ · · · ∧ en] ∈ PΛn(Cm)

where volL = e1 ∧ e2 ∧ · · · ∧ en is the volume element associated with the basis {ei} of L.A straight line of projective space PΛn(Cm) included in ι(L(Cm)) is called a line of

L(Cm). We will denote by �(m1, v) the line of PΛn(Cm) starting from m1 in direction v ∈TLL(Cm).

From now on, where needed, we shall identify L(Cm) with ι(L(Cm)).A system of contact coordinates (xi, u,pi) on M induces coordinates

(

xi, u,pi,pij = pji, 1≤ i, j ≤ n)

(7)

on M(1) as follows: a point m1 ≡ Lm1 ∈M(1) has coordinates (7) iff m= π(m1)= (xi, u,pi)

and the corresponding Lagrangian plane Lm1 is given by

Lm1 = ⟨

∂xi + pij ∂pj

⟩⊂ Cm,

where ∂xi are defined in (2) and all vectors are taken in the point m. Note that the isotropycondition entails that pij = pji , so that the number of “second order” coordinates pij , i.e.coordinates in the fibre π−1(m)=L(Cm), is n(n+1)

2 and dimM(1) = 12 (n2 + 5n+ 2).


Remark 3 Note that, in the case M = J 1(E,n), where dimE = n+ 1 (see Example 1), thefirst prolongation M(1) is strictly bigger than the manifold J 2(E,n) of second order jetsof hypersurfaces of E (recall [29–31] that a k-th order jet at a point p ∈ E is an equiv-alence class of hypersurfaces which have a k-th order contact with each other at p). Infact, J 1(E,n)(1) is, by definition, the set of Lagrangian n-planes of distribution C whereasJ 2(E,n) is the set of Lagrangian n-planes which project on J 1(E,n) without lowering theirdimension (horizontal planes); see, for instance, [4]. Thus, J 2(E,n) turns out to be only anopen dense subset of J 1(E,n)(1).

3.2 Metrics Associated with Tangent and Cotangent Vectors of L(Cm)

Let us fix m1 ∈M(1). Denote m= π(m1) the projection of m1 on M .Let υ ∈ Tm1L(Cm) and φt a 1-parameter subgroup of the symplectic group Sp(Cm), such

that υ = dφt (m1)

dt|t=0. Let us define a symmetric bilinear form gυ on Lm1 by

gυ(v,w)def= ω

(dφt (v)

dt

∣∣∣∣t=0

,w

)

, v,w ∈ Lm1 . (8)

gυ does not depend on the arbitrary choice of φt ; in fact, any other 1-parameter subgroup ofSp(Cm) satisfying the above conditions is of the form φ′t = φt ◦ ht + o(t) where ht belongsto the stabilizer H = Spm1(Cm) of the point m1. Then,

dφ′t (v)

dt

∣∣∣∣t=0

= dφt (v)

dt

∣∣∣∣t=0

+ dht (v)

dt

∣∣∣∣t=0

and

ω

(dφ′t (v)

dt

∣∣∣∣t=0

,w

)

= ω

(dφt (v)

dt

∣∣∣∣t=0

,w

)

since ω|Lm1 = 0. Also, gυ is symmetric. Indeed, for arbitrary v,w ∈ Lm1 , taking into account

that φt preserves ω, we have that ω(φt (v),φt (w)) = ω(v,w)= 0. Thus, by differentiatingby t , we obtain

0= ω

(d(φt (v))

dt

∣∣∣∣t=0

,w

)

+ω

(

v,d(φt (w))

dt

∣∣∣∣t=0

)

= gυ(v,w)− gυ(w,v).

Then, we get the following theorem.

Theorem 1 The map

g : Tm1L(Cm)−→ S2(

L∗m1

)

, υ −→ gυ, (9)

where gυ is defined by (8), is a canonical vector space isomorphism.

In terms of coordinates, metric gυ on Lm1 = 〈∂xi + pij ∂pj〉 associated with a vector

υ = αij ∂pijis given by gυ = αij dxi ⊗ dxj . If X is a vertical vector field w.r.t. projection π :

M(1) →M , then the correspondence m1 ∈M(1) → gXm1 ∈ S2(L∗

m1) defines a metric (whichcan be degenerate) on the tautological bundle T(M(1)). By duality, we get the following


Corollary 2 There is a canonical isomorphism

g : T ∗m1L(Cm)−→ S2(Lm1), ρ −→ gρ. (10)

In fact, keeping in mind the canonical isomorphism (S2(L∗m1))

∗ � S2(Lm1), map (10) isjust the inverse of the pullback of (9). There is no ambiguity in denoting by g both the maps(9) and (10): in fact, vectors appear as superscripts whereas covectors as subscripts. If Θ is a1-form on M(1), the correspondence m1 ∈M(1) → gΘ

m1 ∈ S2(Lm1) defines a metric (which

can be degenerate) on T∗(M(1)).In coordinates, the metric gρ on L∗

m1 associated with covector ρ = ρijdpij , with ‖ρij‖ be-ing the symmetric matrix of coordinates of ρ with respect to basis {(dpij )m1} of T ∗

m1L(Cm),is

gρ = ρijwi ⊗wj ,

where Lm1 = 〈wi = ∂xi + pij ∂pj〉. In particular, a function F ∈ C∞(M(1)) defines a metric

on T∗(M(1)):

g(dF)m1 =

∑

i≤j

∂F

∂pij

wi ∨wj , (11)

where we recall that wi ∨wj = 12 (wi ⊗wj +wj ⊗wi).

Remark 4 Under conformal change ω → λω of the symplectic form, the above metricschange as gυ → λgυ , gρ → λ−1gρ .

3.3 Rank of Tangent Vectors of L(Cm) and Its Geometrical Meaning

By using Theorem 1, we define the rank of a tangent vector υ ∈ Tm1L(Cm) as the rank of thecorresponding bilinear symmetric form gυ . In view of Remark 4, this definition is invariantunder a conformal change of the symplectic form. Obviously, parallel vectors have the samerank. We denote by

T k

m1L(Cm)= {

υ ∈ Tm1L(Cm) | rank(υ)= k}

the set of vectors of rank k and define the canonical map Ker : Tm1L(Cm) → Grn−k(Cm)

which associates to any tangent vector υ ∈ Tm1L(Cm) the radical of gυ :

Ker(υ) :=Ker(

gυ)

.

The space Ker(υ) is the intersection of the Lagrangian plane Lm1 and the infinitesimallyclose Lagrangian plane Lm1+υ dt :

Ker(υ)= limt→0

Lm1 ∩L(t), L(0)= Lm1 , L(0)= υ.

In fact, if Lm1 = {x = xi ∂xi } and L(t)= {xi (∂xi + pij (t)∂pj)} with pij (0)= 0, then

Lm1 ∩L(t)= {

x = xi ∂xi | pij (t)xi = 0

}=Ker(

P (t))

and Ker(υ)= limt→0 Lm1 ∩L(t)=Ker(P (0)).


We call the set T 1m1L(Cm) of vectors of rank 1 the characteristic cone or Segre variety

(see [1]). If υ ∈ T 1m1L(Cm), then

υ � gυ =∓η⊗ η, for some η ∈ L∗m1 (12)

and the canonical map Ker takes values in Grn−1(Lm1)� PL∗m1 . From now on, unless oth-

erwise specified, we identify υ with gυ .In terms of coordinates, if Lm1 = 〈wi = ∂xi +pij ∂pj

〉 and υ ∈ T 1m1L(Cm) has coordinates

αij , then by (12) αij = ηiηj and Ker(υ)= [ηidxi] ∈ PL∗m1 .

Recall that �(m1, υ) denotes the straight line in PΛn(Cm) starting from m1 with directionυ ∈ Tm1L(Cm). A straightforward computation leads to the following proposition.

Proposition 2 ([2]) Let υ ∈ Tm1L(Cm). The straight line �(m1, υ) of PΛn(Cm) is a line ofL(Cm) (i.e. it is included in L(Cm)) if and only if rank(υ)= 1, i.e. υ ∈ T 1

m1L(Cm).

3.4 Characteristic Cone and Characteristic Subspaces of a 2nd Order PDE E and ItsConformal Metric gE

We define the prolongation U(1) ⊂L(Cm) of a vector subspace U ⊂ Cm as follows:

U(1) :={

m1 ∈M(1) | Lm1 ⊇U, if dim(U)≤ n,

m1 ∈M(1) | Lm1 ⊆U, if dim(U)≥ n.

Since Lm1 = L⊥m1 , where we recall that ⊥ stands for the orthogonal complement w.r.t. the

symplectic form, we have that U(1) = (U⊥)(1). Also, a straightforward computation showsthat if U is a hyperplane of Lm1 , then U(1) is 1-dimensional. The prolongation N(1) ⊂M(1)

of a submanifold N ⊂M is defined as follows:

N(1) =⋃

m∈N

(TmN ∩ Cm)(1). (13)

Any υ =±η⊗ η ∈ T 1m1L(Cm) defines hyperplane H =Ker(gυ)=Ker(η) of Lm1 which has

the property that Tm1H(1) = 〈υ〉, and viceversa. Thus we have the following correspondence:

hyperplanes of Lm1 (i.e. elements of PL∗m1 ) ⇐⇒ directions of Tm1L(Cm) of rank 1.

It follows that if Ker(η)=H ⊂ Lm1 is a hyperplane of a Lagrangian plane Lm1 then H(1) =�(m1, υ = η⊗ η)= {Lt } is a straight line of L(Cm) in view of Proposition 2.

Definition 7 Let (M,C) be a (2n+ 1)-dimensional contact manifold and M(1) its prolon-gation. A hypersurface E of M(1) is called a scalar second order partial differential equation(2nd order PDE) with one unknown function and n independent variables. A solution of E isa Legendrian submanifold Σ ⊂M whose prolongation Σ(1) is contained in E.

In terms of coordinates (7), a second order PDE E is locally described by E ={F(xi, u,pi,pij )= 0}, with F ∈ C∞(M(1)). A solution parameterized by x1, . . . , xn, can bewritten as u = ϕ(x1, . . . , xn), pi = (∂ϕ/∂xi)(x1, . . . , xn), pij = (∂2ϕ/∂xi∂xj )(x1, . . . , xn)

where the function ϕ satisfies the equation F(xi, ϕ, ∂ϕ/∂xi, ∂2ϕ/∂xi∂xj )= 0, which coin-cides with the classical notion of solution.


Let us assume that the restriction πE of π :M(1) →M to the equation E⊂M(1) is a fibrebundle. Locally, this is always the case if we suppose that dF is not vanishing on the verticaltangent space of M(1) at each point of E. The fibre at m of πE is denoted by Em:

Em := E∩ π−1(m).

Let E = {F(xi, u,pi,pij ) = 0} ⊂ M(1) be a 2nd order PDE. The conformal class of themetric gdF |E in the tautological vector bundle T(M(1))|E depends only on E (i.e., it does notdepend on the choice of the particular function F vanishing on E). It is called the conformalmetric of E and it is denoted by gE. For a coordinate description of the conformal metricsee (11).

Definition 8 The set

Chm1(E)= Tm1Em ∩ T 1m1L(Cm)

of rank 1 (vertical) tangent vectors to the hypersurface E at m1 is called the characteristiccone of the equation E at m1. Elements of Chm1(E) are called characteristic vectors forE at m1. The 1-dimensional vector space generated by a characteristic vector is called acharacteristic direction. A characteristic vector υ for E at m1 is called strongly characteristicif the line �(m1, υ) is contained in Em.

Definition 9 A subspace U ⊂ TmM is said to be characteristic 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.A submanifold S ⊂M is said to be characteristic for E (resp. strongly characteristic) if, forany m ∈ S, TmS is characteristic at least for a point m1 ∈ E (resp. strongly characteristic).

Definition 10 A Cauchy datum N for a second order PDE E is an (n − 1)-dimensionalintegral submanifold of the contact distribution C. It is said to be characteristic in m1 ∈ Em

if TmN is characteristic for E in m1.

Let m1 ∈ E. In view of the above definitions and taking into account the considerationswritten after formula (13), we obtain the following correspondence:

(n− 1)-dim. characteristic subspaces for E at m1

⇐⇒ characteristic directions for E at m1

Thus, a hyperplane H = Ker(η) of Lm1 , where η ∈ L∗m1 , is characteristic for E in m1 iff

covector η is isotropic for the conformal metric gE; such covector is called characteristiccovector.

We will see in Sect. 5 that strong characteristicity is a necessary condition for a 2nd orderPDEs to be of Monge-Ampère type.

We underline that all results of Sect. 3 are of pointwise nature, so that the theory ofcharacteristics of PDEs could be developed starting from a general symplectic vector spaceand considering its Lagrangian Grassmannian. This has been done in [2]. Hypersurfaces ofa Lagrangian Grassmannian associated with a symplectic vector space can be interpreted as2nd order PDEs depending only on second derivatives (see also [9]).

Example 2 Here we consider the classical case n = 2. Let E = {F = 0} be a second orderscalar PDE and m1 ∈ E. Then, η = (η1, η2) is a characteristic covector for E at m1 if it


satisfies

∂F

∂p11η2

1 +∂F

∂p12η1η2 + ∂F

∂p11η2

2 = 0,

where ∂F∂pij

are computed in m1. We note that (η11, η1η2, η

22) is a vector of the characteristic

cone of E at m1. Dually, v = (v1, v2) spans a 1-dim. characteristic subspace for E at m1 iff

∂F

∂p11

(

v2)2 − ∂F

∂p12v1v2 + ∂F

∂p22

(

v1)2 = 0

(see also (6)). Previous equations have 2, 1 or no real solutions, according to the sign ofΔ= ( ∂F

∂p12)2 − 4 ∂F

∂p22

∂F∂p11

.

4 Full Prolongation of Contact Manifolds and 2nd Order PDEs: Formal Integrabilityand Cauchy-Kowalevski Theorem

In this section we study formal integrability and existence of formal solutions of non-characteristic Cauchy problems of a 2nd order PDE E. We will treat this subject in theframework of contact manifolds by using, in addition, the conformal metric gE.

4.1 The Full Prolongation of a Contact Manifold

One can define the k-prolongation M(k) of a contact manifold (M,C) iteratively as follows.To start with, we put M(0) =M , C(0) = C and π1,0 = π . Then we define

M(k+1) = {

Lagrangian planes of M(k)}

,

where Lagrangian planes of M(k) are defined iteratively in the following way. The manifoldM(k) is endowed with the distribution

C(k) = {

v ∈ TmkM(k) | πk,k−1∗(v) ∈ Lmk

}

, (14)

where Lmk ≡mk is a point of M(k) considered as a Lagrangian plane in C(k−1)

mk−1 and

πk,k−1 :M(k) →M(k−1), mk →mk−1 (15)

is the natural projection. It is known [18] that (15) are affine bundles for any k > 1 (for k = 1the bundle π1,0 = π is a subbundle of the Grassmann bundle Grn(M), see Definition 6).Denote by θ(k) the system of 1-forms on M(1) (which we can consider as one vector-valued1-form) which defines the distribution (14): C(k) =Ker θ(k).

Definition 11 An n-dimensional subspace L ⊂ TmkM(k) is called a Lagrangian plane if itis horizontal w.r.t. πk,k−1 and forms θ(k) and dθ(k) vanish on it.

As in Sect. 3, a contact chart (xi, u,pi) of M defines a chart (xi, u,pi,pi1i2 , . . . , pi1···ik+1)

of M(k) in a way that a point mk ≡ Lmk ∈M(k) is given by

Lmk =⟨

∂xi +∑

|I |≤k

pI,i∂pI

⟩

,


where I = (i1 · · · i�), 1≤ i1 ≤ i2 ≤ · · · ≤ i� ≤ n is a multi-index of length |I | = � and I, idef=

(i1, . . . , i�, i) (which will be reordered if necessary). The set of 1-forms θ(k) is spanned by

θI = dpI − pI,idxi, |I | ≤ k.

Integral manifolds of C(k) project onto integral manifolds of C(k−1) through πk,k−1 (under acondition of constant rank in order to assure the smoothness of the image set). In particu-lar, Legendrian submanifolds S ⊂M(k) (i.e. submanifolds such that TsS ∈M(k+1),∀s ∈ S)project onto Legendrian submanifolds of M(k−1).

We define the full prolongation M(∞) as the inverse limit of the tower of projec-

tions . . . −→ M(k)πk,k−1−→ M(k−1) −→ . . . so that a point m∞ ∈ M(∞) is a sequence (m =

m0,m1, . . . ,mk, . . . ) where mk ∈M(k) and πk,k−1(mk)=mk−1.

We underline that, if we consider M = J 1(E,n) (see also Example 1), J 1(E,n)(∞) isstrictly bigger than J∞(E,n) in view of Remark 3.

4.2 The Full Prolongation of a Second Order PDE E⊂M(1) and Its Formal Integrability

The 1st-prolongation of a submanifold S ⊂M(k) is the submanifold S(1) ⊂M(k+1) definedas follows:

S(1) = The set of points mk+1 ∈M(k+1) such that Lmk+1

{⊆ TmkS ∩ C(k)

mk if dimS ≥ n,

⊇ TmkS ∩ C(k)

mk if dimS ≤ n,

where mk = πk+1,k(mk+1). Iteratively, we define the h-prolongation S(h) ⊂M(k+h) of S.

Similarly to the construction of M(∞), we can define the full prolongation S(∞) of anysubmanifold S ⊂M(k).

A system of (resp. scalar) PDEs of order k, with one unknown function, is a submanifold(resp. hypersurface) E of M(k−1).

Definition 12 A formal solution of a k-th order PDE E is a point of E(∞).

Now we describe the k-th prolongation E(k) ⊂M(k+1) of a second order PDE

E= {

F(

xi, u,pi,pij

)= 0}⊂M(1).

We denote by

Di = ∂xi + pi∂u + pij ∂pj+ · · ·

the total derivative w.r.t. xi and for I = (i1, . . . , i�) we put DI =Di1 ◦ · · · ◦Di� . It is straight-forward to check that the k-th prolongation E(k) of E is locally described by the system

E(k) = {

F = 0, DIF = 0, 1≤ |I | ≤ k}

.

As a corollary, we can describe the fibre E(k)

mk = π−1k+1,k(m

k)∩ E(k) of the projection

πk,k−1|E(k) : E(k) → E(k−1)


in terms of the coordinates pI , |I | = k + 2, of the fibre M(k+1)

mk = π−1k+1,k(m

k) and of themetric

gij

E =1

2− δij

∂F

∂pij

.

We will consider coordinates pI = pi1···i� as symmetric tensor of S�(Rn).

Corollary 3 Let m1 = (xi, u,pi,pij ) ∈ E. Then E(1)

m1 is defined by the following system oflinear equations

E(1)

m1 ={(

2− δj�)

gj�pij� = ci

}

where ci = ci(m1)=−( ∂F

∂xi + pi∂F∂u+ pij

∂F∂pj

)(m1).

More generally, if mk = (xi, u,pi, . . . , pi1···ikj ) ∈ E(k−1), then E(k)

mk = {(2 − δj�)gj� ×pi1···ikj� = ci1···ik } where

ci1···ik = ci1···ik(

mk)= [

Dik ci1···ik−1 −(

Dikgj�)

pi1···ik−1j�

](

mk)

.

Recall [18] the following

Definition 13 An equation E⊂M(1) is called formally integrable if the prolongations E(k)

are smooth submanifolds of M(k+1) and πk+1,k|E(k) : E(k) → E(k−1) are smooth fibre bundles.

Usually, the above definitions and constructions are given in terms of jet spaces [8, 18,24, 29–31]. The construction of the infinite prolongation of a system of PDEs allows tohave a nice geometric description of many objects which are important for its study suchas higher (or generalized [24]) symmetries, conservation laws, non-local symmetries, Bäk-lund transformations. Among the PDEs that can be efficiently studied in the context of jetbundles, there are well known examples coming from physics (KdV, Burger, ecc. [8]). Anexample of PDE that can be naturally formulated in the framework of jets of submanifoldsis the equation of minimal submanifolds [22]. If one wants to study, for instance, the prob-lem of finding normal forms of Legendrian distributions up to contact-morphisms (see [6]),then the PDEs to be studied (in this case they are MAEs, see Sect. 5) can be formulated in anatural way on a “pure” contact manifold.

Below we prove the following theorem.

Theorem 2 Let E = {F = 0} ⊂ M(1) be a smooth hypersurface of M(1). The equation E

is formally integrable if the associated conformal metric gE does not vanish (i.e. for anym1 ∈ E, (gdF )m1 �= 0).

To prove the theorem we need the following lemma.

Lemma 2 Let b = bj�(y) ∈ S2V ∗ (resp., c = ci1···ik−1(y) ∈ Sk−1V ) be a symmetric bilinearform (resp., symmetric contravariant (k − 1)-tensor) in the vector space V = R

n = {v =(v1, . . . , vn)} which smoothly depends on coordinates y = (y1, . . . , yq) ∈R

q . If b �= 0 for ally ∈R

q , then the equation

bj�(y)pi1···ik−1j� = ci1···ik−1(y) (16)

defines a smooth submanifold H ⊂Rq×Sk+1V such that the natural projection π :H →R

q

is an affine fibration with a fibre of dimension d(k,n) := dimSk+1R

n − dimSk−1R

n.


Proof First of all, one can easily check that the contraction

ιb : Sk+1V → Sk−1V, pi1···ik−1j� → bj�pi1···ik−1j�

is surjective if b �= 0. This shows that π−1(y) is an affine space of dimension d(k,n). To con-struct local coordinates in H , we consider a linear change of coordinates vi → v′i =A

j

i (y)vj ,with the matrix A(y) depending on y, which transforms the bilinear form b into the standardform b= εiδ

ij , εi ∈ {±1,0}. We can assume that ε1 = 1. The components pi1···ik−1j�, ci1···ik−1

transform like tensors. In terms of the new components p′i1···ik−1j�, c′i1···ik−1equation (16)

takes the form p11I = cI −∑

j>1 εjpjjI . This is a system of linear equations with free vari-ables pJ ,p1J where the multi-index J does not contain 1. These free variables togetherwith y form a coordinate system of H such that the projection π : H → R

q is given byπ(y,pJ ,p1J )= y. �

Proof of Theorem 2 Now we can prove the theorem by induction. We will assume thatE(k−1) ⊂M(k) is a smooth submanifold. Then the restriction of the affine bundle M(k+1) →M(k) to E(k−1) is a locally trivial bundle which locally can be identified with the trivial bundle

E(k−1) × Sk+1R

n → E(k−1), (y,pi1···ik+1) → y

where y are local coordinates of E(k−1). Then E(k) is defined by the system of equations

(

2− δj�)

gj�(y)pi1···ik−1j� = ci1···ik−1(y)

where gj�(y), cI (y) are smooth functions of y. Now the theorem follows from lemma. �

4.3 Formal Solution of a Non-characteristic Cauchy Problem

In this subsection an explicit formal solution of a second order PDE E⊂M(1) once a (non-characteristic) Cauchy datum N is given. The reader can guess that the proof of the followingtheorem is related to the possibility of writing the equation E in the Cauchy-Kowalewskinormal form. In fact, this is a particular instance of a classical result (see for instance [25]);a general statement, showing that the existence of non-characteristic covectors allows towrite a system of PDEs in the Cauchy-Kowalewski normal form, was proved in [23].

Theorem 3 Let N ⊂M be a Cauchy datum of E which is not characteristic in m1 ∈ Em,m=m0 ∈N . Then, there exists exactly one point m∞ = {mk}k∈N0 ∈ E(∞) such that, for anyk ∈N0, it holds

Lmk+1 ⊃ TmkN(k)

E , (17)

with manifolds N(k)

E ⊂M(k) recursively defined by formulas

N(k)

E := (

N(k−1)

E

)(1) ∩ E(k−1), N(0)

E :=N.

Proof (A sketch) Non-characteristicity condition in m1 means

Tm1(TmN)(1)� Tm1Em. (18)


Without entering into details, the proof consists in fixing in the neighborhood of m a Dar-boux chart (xi, u,pi) such that N is represented by

⎧

⎪⎨

⎪⎩

xn = u= 0

ph = 0, h < n,

pn =Φn(x)

(19)

for some suitable function Φn(x), x = (x1, . . . , xn−1) (see Corollary 1), and showing by arecursive scheme that, in such a chart, N

(k−1)

E is described by

⎧

⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

xn = u= 0

pI =

⎧

⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

0 if ia ≤ n− 1∀a∂ |J |∂xJ Φn · · ·n

︸︷︷︸

h

(x) if I = (J,n · · ·n︸︷︷︸

h

), h < �, jb ≤ n− 1∀b

Φn · · ·n︸︷︷︸

�

(x) if I = (n · · ·n︸︷︷︸

�

)

, (20)

with � running from 1 to k, where I = (i1 · · · i�), J = (j1 · · · j�−h), ∂xJ = ∂xj1 · · · ∂xj�−h

and function Φn · · ·n︸︷︷︸

�

(x) is obtained by expliciting the variable pn · · ·n︸︷︷︸

�

in the equation

(Dn · · ·n︸︷︷︸

�−2

F)|(N

(�−2)E )(1) = 0, where E= {F = 0}. This can be done at any step, since the co-

efficient of the higher order term of Dn · · ·n︸︷︷︸

�−2

F (i.e. the coefficient of pn · · ·n︸︷︷︸

�

), is ∂F∂pnn

(m1),

and ∂F∂pnn

(m1) �= 0 in view of non-characteristicity condition (18). Indeed, let U = TmN . Bycomputing the Jacobian matrix of (19) one gets U = 〈ξ1, . . . , ξn−1〉, with

ξh = ∂xh |m + ∂Φn

∂xh∂pn |m = ∂xh |m +

n∑

j=1

phj (m)∂pj|m

for h= 1, . . . , n− 1 (with functions phj given by (20)). But vectors ξh are exactly the firstn− 1 vectors of the canonical basis of Lagrangian plane Lm1 , for any m1 ∈ π−1(m)∩N(1);hence, U(1) = π−1(m) ∩N(1) and this curve is described by the free parameter pnn, so thatTm1U(1) = 〈∂pnn |m1〉; thus, non-characteristicity condition (18) is exactly ∂F

∂pnn(m1) �= 0. �

Once (20) is proved, it can be used to check (17) by simple computations.Note that Theorem 3 is, substantially, an infinitesimal formal analogue of Cauchy-

Kowalewski theorem, and that m∞ corresponds to the Taylor expansion of the unique formalsolution of Cauchy problem (E,N,m).

5 Monge-Ampère Equations and Their Characteristics

As we said in the introduction, it would be interesting to study PDEs whose characteristicsbehave in a prescribed way. For instance, it is known that, in the case of 2 independent vari-ables, the only PDEs for which characteristic cones degenerate in two 2-dimensional planesare classical Monge-Ampère equations (see [5, 6, 28]). In this section, which is essentialfor introducing MAEs in the context of (para)-Kähler manifolds, we recall some results


contained in [2] regarding the multi-dimensional generalization of the above property. Let(M,C) be a contact manifold and I(θ) ⊂ Λ∗(M) be the differential ideal generated by acontact form θ . Following V.V. Lychagin (see [18, 20]), we give the following definition

Definition 14 Let Ω ∈ Λn(M)\I(θ). We associate with Ω the hypersurface EΩ of M(1)

defined by

EΩdef= {

m1 ∈M(1) s.t. Ω|Lm1 = 0

}

,

where we recall that Lm1 ⊂ Tπ(m1)M is the Lagrangian plane associated with m1. Equationsof this form are called general Monge-Ampère equations (MAEs).

In other words, EΩ is the differential equation corresponding to the exterior differentialsystem {θ = 0, Ω = 0}. The correspondence m1 ∈ M(1) → Ω|L

m1 ∈ Λn(L∗m1) defines an

n-form on the tautological bundle T(M(1)). Two n-forms Ω,Ω ′ define the same equationEΩ = EΩ ′ iff, up to a non vanishing factor, they are related by

Ω ′ =Ω + α ∧ dθ + β ∧ θ for some α ∈Λn−2(M), β ∈Λn−1(M). (21)

Note that hypersurfaces EΩ (or, rather, their Plücker images) coincide with hyperplane sec-tions of M(1) ↪→ PΛn(C). More precisely, hyperplanes of PΛn(Cm) biunivocally correspondto hyperplanes of Λn(Cm), which in their turn can be identified with lines in Λn(Cm)∗. Onthe other hand, one can associate to any Ω ∈Λn(C∗m) the covector Ω ∈Λn(Cm)∗ given by

Ω(v1 ∧ · · · ∧ vn) :=Ω(v1, . . . , vn), v1, . . . , vn ∈ Cm,

so that Λn(C∗m) is canonically isomorphic to Λn(Cm)∗. Therefore,

(EΩ)m =L(Cm)∩ {

L ∈L(Cm)⊂ PΛn(Cm) | Ω(L)= 0}

,

i.e. EΩ it is a hyperplane section of PΛn(C) via the Plücker embedding. A remarkable prop-erty of a general MAE EΩ is that if a vector υ ∈ Tm1(EΩ)m is characteristic then it is stronglycharacteristic (see [2] for a proof). Below we will see that if we require further restrictionson the behavior of characteristics, we get a geometrical class of MAEs which can be easilyreconstructed by their characteristics.

Let D⊂ C be an n-dimensional subdistribution of the contact distribution C and ED theset of Lagrangian planes in Cm ⊂ TmM, m ∈M which have non-zero intersection with Dm:

ED = {

m1 ∈M(1) | Lm1 ∩Dπ(m1) �= 0}

. (22)

Then ED = EΩ is a MAE associated with the volume form Ω = vol(D)= η1 ∧ · · · ∧ ηn ofD= {θ = η1 = · · · = ηn = 0} (here η1, . . . , ηn are (local)1-forms which define D).

Definition 15 Let D⊂ C be an n-dimensional subdistribution of the contact distribution C.The PDE ED defined by (22) is called a MAE of Goursat type.

In view of the reasoning leading to formula (21), MAEs of type ED are associated withdecomposable n-forms on M up to I(θ). Locally, if D = 〈∂xi + bij ∂pj

〉i=1...n, where bij ∈C∞(M), it is easy to see that the Legendrian submanifold (xi, f (xi),

∂f

∂xk (xi)) is a solutionof ED iff f = f (x1, . . . , xn) satisfies (1). The following theorem describes some propertiesof MAEs of Goursat type ED.


Theorem 4 ([2]) ED = ED iff D = D or D⊥ = D. Let m1 ∈ (ED)m be such that(gED

)m1 �= 0. Then gEDis decomposable: (gED

)m1 = �m1 ∨ �′m1 , where �m1 = Lm1 ∩Dm

and �′m1 = Lm1 ∩D⊥

m are lines. Then there exist only two (n−2)-parametric families of char-acteristic hyperplanes of Lm1 : one rotates around �m1 , the other around �′

m1 . Moreover, thecharacteristic cone is given by Chm1(ED)= {±η⊗η, η ∈ �0

m1 ∪�′0m1} where �0

m1 , �′0m1 ⊂ L∗

m1

are, respectively, the annihilators of �m1 and �′m1 . Covectors η ∈ L∗

m1 which correspond tocharacteristic directions and belong to �0

m1 (resp., �′0m1 ) define hyperplanes {η = 0} which

contain �m1 (resp., �′m1 ). If one varies the point m1 on EDm, the line �m1 (resp., �′

m1 ) fills then-dimensional space Dm (resp. D⊥

m).

As an explaining example, consider the case n = 3 and the PDE E : {p12 = f }, f ∈C∞(M), being M locally described by (xi, u,pi), i = 1, . . . ,3. the conformal metric of E

at a point m1 ∈ E is equal to (gE)m1 = �m1 ∨ �′m1 where

�m1 = 〈∂x1 + p11∂p1 + f ∂p2 + p13∂p3〉, �′m1 = 〈∂x2 + f ∂p1 + p22∂p2 + p23∂p3〉.

Covector η =∑31 ηidxi is characteristic for E if it is isotropic for the above metric, i.e. if

η1η2 = 0. If we let vary the point m1 on the fibre Em, m = π(m1), lines �m1 and �′m1 fill,

respectively, the following mutually orthogonal 3-dimensional planes at m

Dm = 〈∂x1 + f ∂p2 , ∂p1 , ∂p3〉, D⊥m = 〈∂x2 + f ∂p1 , ∂p2 , ∂p3〉,

so that we obtain distributions D and D⊥ on M . Thus, E= ED.

6 Prolongation of Symplectic Manifolds, Their Contactization and Symplectic PDEs

In analogy with Sect. 2, first order scalar PDEs which do not depend on the unknown func-tion are hypersurfaces of a 2n-dimensional symplectic manifold (W,ω), where n is thenumber of the independent variables. For this reason, following [18], we call such equationssymplectic PDEs. We recall that, by Darboux theorem, there always exists a system of co-ordinates (xi,pi) on W such that ω= dxi ∧ dpi . We use the notation ‘pi ’ as we can regardsuch functions as momenta. Thus, locally, a first order scalar symplectic PDE is given by{f (xi,pi) = 0}. Following Sect. 3, we can introduce the prolongation W(1) of a symplec-tic manifold W as the set of Lagrangian planes of W . More precisely, we have the bundleW(1) → W whose fiber at w ∈ W is the Lagrangian Grassmannian L(TwW) of TwW . Aswe did for contact manifolds, we denote by T(W(1)) the tautological bundle of W(1), i.e.the vector bundle on W(1) whose fibre at a point w1 ∈W(1) is the (Lagrangian) vector spaceLw1 ≡w1. Smooth coordinates on W(1) can be constructed exactly as in Sect. 3.1.

A second order scalar PDE with n independent variables which is independent of theunknown function is a hypersurface of W(1), where W is a 2n-dimensional symplectic man-ifold. Thus, locally, it is given by EF = {F(xi,pi,pij )= 0}.

An n-form Ω on the symplectic manifold W defines the set {w1 ∈W(1) | Ω|Lw1 = 0}.

Such a set is a hypersurface of W(1) which is a (symplectic) MAE. This will be the keyobservation for introducing and studying MAEs on Kähler and para-Kähler manifolds.


7 Contactization of Symplectic Manifolds

Let Q be a smooth n-dimensional manifold. The space of 1-jets J 1(Q) of smooth functionson Q is canonically isomorphic to R× T ∗Q, the trivial projection π : J 1(Q)→ T ∗Q is atrivial principal R-bundle and the canonical contact form θ is a connection 1-form in π . Thecurvature of this connection is the standard symplectic form ω = dθ on T ∗Q. In terms ofcoordinates (xi, u,pi), θ = du− pidxi and ω= dxi ∧ dpi .

It is well known that any Lagrangian submanifold Lf transversal to fibers of T ∗Q→Q

is locally the graph of an exact form df :Q→ T ∗Q where f = f (x1, . . . , xn):

Lf = Γdf =(

x1, . . . , xn,∂f

∂x1

(

x1, . . . , xn)

, . . . ,∂f

∂xn

(

x1, . . . , xn))

. (23)

It is evident that Lf = Lf+c with c ∈R, so that once we fix an f defining Lf , it canonicallylifts to a Legendrian submanifold Lf = (xi, u= f (xi),pi = ∂xi (f )) of J 1(Q). Accordingto the Darboux theorem, any symplectic manifold is locally isomorphic to T ∗Q and anycontact manifold to J 1(Q)=R× T ∗Q. This suggests that the relation between Legendrianand Lagrangian manifolds can be established in a more wide context, which we describebelow.

Definition 16 A contactization of a symplectic manifold (W,ω) is an R-principal bundleπ :M →W with connection θ : T M →R such that dθ = π∗(ω).

Note that (M, θ) is a regular contact manifold, i.e., θ is a globally defined contactform and the corresponding Reeb vector field Z defines a free proper action of the groupR= {exp tZ}. So, the group Aut(M, θ) contains R= {exp tZ} as a central subgroup so thatAut(M, θ)/R � Aut(W,ω). Of course J 1(Q) is a contactization of T ∗Q: since any sym-plectic manifold is locally isomorphic to some T ∗Q, a contactization always exists at leastlocally. More generally, a symplectic manifold (W,ω) admits a contactization if the Cechcohomology class [ω] ∈ H 2(W,R) is trivial (see [7] for further details).

Theorem 5 Let (W,ω) be a symplectic manifold with [ω] = 0 and π : (M, θ) → (W,ω)

the associated contactization. Let L⊂W be a Lagrangian submanifold. Then

(1) the restriction of π to π−1(L)→ L is a principal R-bundle with flat connection θ |π−1(L);(2) for any point m ∈ π−1(L) there exists a unique Legendrian submanifold L through m

such that the restriction of π on L→ L is a covering whose structure group Γ is theholonomy group of the flat connection θ .

Proof (1) We have the decomposition Tmπ−1(L) = RZm + Hm of the tangent space ofπ−1(L), where H := C ∩ T π−1(L). Since dθ(Hm) = ω(π∗(Hm)) = ω(Tπ(m)L) = 0, thecurvature of the connection θ |π−1(L) is zero.

(2) The manifold L is the maximal integral submanifold passing at m of the involutivedistribution H. More precisely, it consists of all points of π−1(L) which can be joined withm by a horizontal path. A classical result of the theory of principal connections states thatπ |L : L→ L is a covering with the holonomy group Γ as the structure group. �

Since the holonomy group Γ is a subgroup of the structure group R of the principalbundle π |L and there exists a natural epimorphism from the fundamental group π1(L) of L

to Γ , we get the following


Corollary 4 Assume that the fundamental group π1(L) of a Lagrangian submanifold L⊂W is either trivial or has no normal subgroups K with commutative quotient π1(L)/K . Thenthe holonomy group Γ is equal to zero and the principal bundle π |L : L = R× L→ L istrivial.

Remark 5 There exists a more general definition of contactization of a symplectic manifoldW as a principal bundle on W with the structure group equal to R or S1 [7].

8 Monge-Ampère Equations on Kähler Manifolds and Special LagrangianSubmanifolds

An important natural class of n-forms on symplectic manifolds (hence MAEs) arises whenwe assume that a symplectic manifold (W,ω) is equipped with an ω-compatible com-plex or para-complex structure J . Compatibility means that the endomorphism J is skew-symmetric w.r.t. ω, i.e. the composition g = ω ◦ J is a (pseudo-)Riemannian metric. Hereand in what follows pseudo-Riemannian means a non-degenerate metric with arbitrary sig-nature.

In this and in the next section we shortly discuss the arising important class of MAEs,whose solutions are called special Lagrangian submanifolds (SLAG) [12, 14–17].

Consider at first the case when J is a skew-symmetric complex structure on (W,ω). Thestructure (ω,J, g = ω ◦J ) is a pseudo-Kähler structure and the complex structure J and thesymplectic form ω are parallel w.r.t. the Levi-Civita connection of the pseudo-Riemannianmetric g. In terms of holomorphic coordinates (z1, . . . , zn), where zj := xj + ipj , we havethat

ω= i

2

n∑

j=1

dzj ∧ dzj =n

∑

j=1

dxj ∧ dpj .

Denote by Ωn,0 the canonical line bundle of holomorphic n-forms on W . A local (holomor-phic) section of Ωn,0, in holomorphic coordinates (z1, . . . , zn), has the form

Ω =Ω1 + iΩ2 = f (z)dnz := f(

z1, . . . , zn)

dz1 ∧ · · · ∧ dzn

The real n-forms Ω1,Ω2 defines MAEs EΩ1 ,EΩ2 . A solution of EΩi, that is a Lagrangian

submanifold of W which annihilates Ωi , is called a special Ωi -Lagrangian manifold. In thecompact case, a holomorphic section Ω exists globally if and only if W is a Calabi-Yaumanifold and then the real forms Ωi, i = 1,2 are parallel. Then Ω2-SLAG submanifolds arecalibrated manifolds for Ω1, see [12].

Consider the simplest case when (W = Cn,ω = i/2

∑dzi ∧ dzi =∑

dxi ∧ dpi), zj =xj + ipj is the Hermitian vector space and

Ω = dnz= dz1 ∧ · · · ∧ dzn =Ω1 + iΩ2

is the standard holomorphic n-form. Then all equations EΩφwhere Ωφ = Re(eiφΩ) =

cos(φ)Ω1+ sin(φ)Ω2 are equivalent. Indeed, one can easily check that the symplectic trans-formation z→ eiφz maps Ω0 into Ω−nφ . This implies that Laplace equation p11 + p22 = 0and the standard elliptic MAE p11p22 − p2

12 = 1 are equivalent. Indeed, in the case n = 2,we have that

Ω =Ω1 + iΩ2 = dx1 ∧ dx2 − dp1 ∧ dp2 + i(

dp1 ∧ dx2 + dx1 ∧ dp2)


and by restricting Ω to the tautological bundle we get the following 2-form:((

1− p11p22 + p212

)+ i(p11 + p22))

dx1 ∧ dx2.

Thus, our assertion follows in view of the above result.The following proposition gives the coordinate form of the MAEs associated with Ω1

and Ω2.

Proposition 3 The equations EΩ1 and EΩ2 can be written respectively as

0= Re det(

iHess(f )+ Id)= Re

∑

k ikσk

(

Hess(f ))

,

0= Im det(

iHess(f )+ Id)= Im

∑

k ikσk

(

Hess(f ))

,(24)

where σk(H) is the k-elementary symmetric function of the eigenvalues of a symmetric ma-trix H .

Proof A straightforward computations shows that

Ω =Ω1 + iΩ2

=∞∑

m=0

∑

k1<···<k2m

(−1)mdx1 ∧ · · · ∧ dpk1 ∧ · · · ∧ dpk2m∧ · · · ∧ dxn

+ i

( ∞∑

m=1

∑

k1<···<k2m−1

(−1)m+1dx1 ∧ · · · ∧ dpk1 ∧ · · · ∧ dpk2m−1 ∧ · · · ∧ dxn

)

.

The proposition follows by taking into account that Lagrangian submanifolds are locallydescribed by (23). �

Note that (24), which defines SLAG submanifolds, look very similar to the Goursat typeof MAEs, but they are much more complicated. We will investigate their characteristics inthe future.

9 MAEs on Para-Kähler Manifolds and Monge-Kantorovich Problem

In this section we recall shortly the basic notions of para-Kähler geometry and, following[13], we describe the relationship between SLAG of para-Kähler manifolds and solutions ofthe Monge-Kantorovich mass transport problem.

Para-complex geometry is essentially the geometry of a 2n-dimensional manifold pro-vided with a field K ∈ Γ (End(T M)) of endomorphisms such that K2 = IdT M and the twoeigendistributions T ±M := ker(Id ∓ K) have the same rank (see [3] for a survey on thistopic). Such a K is called an almost para-complex structure. A para-complex structure is anintegrable almost complex structure, i.e. the two distributions T ±M are integrable. Equiva-lently, the Nijenhuis tensor NK associated with K is zero:

NK(X,Y )= [X,Y ] + [KX,KY ] −K[KX,Y ] −K[X,KY ] = 0.

Another possible approach is to develop para-complex by analogy with complex geometryfrom the very beginning by considering para-complex numbers as the elements of the 2-dimensional real algebra generated by 1 and τ with τ 2 = 0 [13]. We will denote such algebra


by D. Any number z ∈D can be written as z= x+τp, with x,p ∈R. Conjugation is definedas in the complex case: z = x − τp. The essential difference with complex numbers is theexistence of a distinguished coordinates (u, v), called null coordinates, defined as follows:

z= x + τp = ue+ ve, e= 1

2(1− τ), e= 1

2(1+ τ)

we underline that

ez = exeτp = ex(

cosh(p)+ τ sinh(p))= eeu + eev.

The vector space Dn, together with its null coordinates, is defined in a obvious way. The

multiplication of vectors of Dn by τ turns out to be a para-complex structure on D

n. Analo-gously to the complex case, we can define ∂ := dzk ∧ ∂zk and ∂ := dzk ∧ ∂zk and a functionF :Dn →D is called para-holomorphic if ∂F = ∂F

∂zj dzj = 0. In null coordinates,

∂ = edu + edv, ∂ = edv + edu

where du and dv indicate the usual differentiation w.r.t. u and v. If we write also F in null co-ordinates, i.e. F = ef + eg with f and g real-valued functions, then it is para-holomorphicif and only if f = f (uj ) and g = g(vj ). So, roughly speaking, para-holomorphic geome-try is less rigid than complex geometry as f and g are not necessarily analytic. Also, it ismore simple to study as para-holomorphic functions are explicitly given, whereas the com-plex case is described by Cauchy-Riemann equations. Extension of the above definitions onmanifolds is straightforward.

Analogously to the complex case, we can introduce the canonical bundle Ωn,0 of para-holomorphic n-forms. By considering the vector notation z= (z1, . . . , zn), u= (u1, . . . , un),v = (v1, . . . , vn), a para-holomorphic n-form Ω can be written as

Ω =Ω1 + τΩ2 = α(z)dnz= eξ(u)dnu+ eη(v)dnv,

Ω1 = ξ(u)dnu+ η(v)dnv, Ω2 = ξ(u)dnu− η(v)dnv.(25)

An interesting class of para-complex manifolds is that of para-Kähler manifolds, which aredefined as symplectic manifolds (W,ω) equipped with a para-complex structure K com-patible with ω. This means that K is skew-symmetric with respect to ω, or , equivalently,that g := ω ◦ K = ω(K·, ·) is a pseudo-Riemannian metric. This metric has neutral sig-nature (n,n) and the associated Levi-Civita connection preserves ω and K ; in particular,the eigendistributions T ±W of K are parallel (and isotropic). Thus, locally, a para-Kählermanifold is the direct product of two Lagrangian submanifolds. The symplectic form can belocally written as follows:

ω := τ

2

n∑

j=1

dzj ∧ dzj =n

∑

j=1

dxj ∧ dpj =n

∑

j=1

1

2duj ∧ dvj . (26)

As in the Kähler case, we can define a para-Kähler manifold also as a pseudo-Riemannianmanifold (W,g) with a parallel para-complex structure K . Also, the real n-forms Ω1,Ω2

of (25) defines MAEs EΩ1 ,EΩ2 and a solution of EΩiis called a special Ωi -Lagrangian

manifold.A remarkable difference from the Kähler case is that we can associate a distinguished

MAE with a para-holomorphic n-form Ω ∈Ωn,0. Essentially, this comes from the fact that


in the para-complex case we have two distinguished “directions”, i.e. the null direction.More precisely, we have the following

Proposition 4 Let Ω as in (25). Then EΩ1∓Ω2 do not depend on the conformal class of Ω .

Proof If we multiply Ω by a para-holomorphic function f + τg, we have that

(f + τg)Ω = f Ω1 + gΩ2 + τ(gΩ1 + f Ω2)= Ω1 + τΩ2,

so that Ω1 + Ω2 = (f + g)(Ω1 +Ω2) and Ω1 − Ω2 = (f − g)(Ω1 −Ω2). Therefore thesum and the difference of real and imaginary part is well defined up to a multiplication by apara-holomorphic function, so that the associated MAEs are the same. �

Consider the simplest example of a para-Kähler manifold: the para-Hermitian vectorspace D

n =Rn + τR

n with the standard para-complex structure (defined by the multiplica-tion by τ ) and the standard symplectic form given by (26).

As in the Hermitian case, we can use the symplectic transformation z →±eταz (hy-perbolic changing of phase) and also the symplectic transformation ui → vi, vi →−ui totransform an equation associated with n-form Ωφ := cosh(φ)Ω1 + sinh(φ)Ω2 into anotherof the same family. In coordinates (xi,pi), a Lagrangian submanifold Lf = {xi,pi = fxi }is Ωi -special if it satisfies (24) provided we substitute i with τ . Taking into account thatin null coordinates Lf = {xi,pi = fxi } = {ui, vi = fui

}, the following proposition give adescription of special Ω2-Lagrangian submanifolds in terms of such coordinates.

Proposition 5 Let Ω as in (25). Lagrangian submanifolds Lf (u) is a SLAG iff f satisfiesthe following MAE:

det Hess(f )= ξ(u)

η(fui)

(27)

Proof The n-form Ω2 = ξ(u)dnu− η(v)dnv vanishes on Lf (u) = {ui, vi = fui} if

0 = ξ(u)− η(fui)(f11du1 ∧ · · · ∧ f1ndun)∧ · · · ∧ (fn1du1 ∧ · · · ∧ fnndun)

= (

ξ(u)− η(fui)det

(

Hess(f )))

dnu

which implies (27). �

Solutions of MAE (27) are related to solutions of the Monge-Kantorovich optimal masstransport problem [27] (MK problem). Indeed, if ξ(u) and η(v) are smooth positive func-tions, respectively, on open domains U and V with compact closure, then ξ(u)dnu andη(v)dnv are positive densities and the MK problem is to find a map F : U → V whichpreserves such densities and that minimizes the functional

∫

Uc(u,F (u))ξ(u)dnu where

c : U × V → R is a “cost” function. If we choose the usual cost function c = 1/2|u− v|2,then the problem is equivalent to find a solution of (27). Above proposition suggests a geo-metrical interpretation of this problem in terms of para-Kähler manifolds (see [13] for furtherdetails). In fact, let us consider a para-complex structure on D

n, i.e. a couple of transversalLagrangian integrable distributions. Locally, D

n = U × V where U and V are Lagrangiansubmanifolds. Let c be a function on U ×V which gives a global Kähler potential, i.e. such


that

ω=−τ∂∂c= dudvc= ∂2c

∂ui∂vj

dui ∧ dvj

is a symplectic form on U ×V . Then (Dn,ω) turns out to be a para-Kähler manifold so thatwe can consider SLAG submanifolds associated to the para-holomorphic n-form (25). If wechoose the usual cost function c = 1/2|u− v|2, the symplectic form ω is the standard one,so that such SLAG submanifolds are described by equation (27). If U and V are compactand without boundary, existence and uniqueness of solutions of (27) are guaranteed.

Acknowledgements The first and third author thank the University of Cagliari and the University ofSalento for financial support and hospitality. They also thank the Erwin Schrödinger Institute of Wien forfinancial support, for hospitality and for the opportunity to participate to the conference “Cartan connections,geometry of homogeneous spaces, and dynamics” during that part of this research has been made.

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