Lobachevskii Journal of Mathematics Vol. 3, 1999, 185–196 \begin{matrix}c \\ ◯ \end{matrix}Y. Machida and T. Morimoto

Y. Machida and T. Morimoto

ON DECOMPOSABLE MONGE-AMPÈRE EQUATIONS

(submitted by B.N.Shapukov)

Introduction

Monge-Ampère equations have been studied by Monge, Ampère, Lie, Goursat and others (cf.[G]). An intrinsic geometric formulation was given in [M1] by defining a Monge-Ampère equation as a certain kind of exterior differential system on a contact manifold called a Monge-Ampère system, and geometric studies of Monge-Ampère systems have been developed in relation to contact geometry, in which an important role is played by the characteristic systems associated with a Monge-Ampère system.

However, these studies have been essentially limited to the Monge-Ampère equations in two independent variables (or the Monge-Ampère systems on 5dimensional contact manifolds), mainly because it is only in this case that the characteristic systems have been well defined: It is in general difficult to attach to a Monge-Ampère equation in more than three independent variables linear objects such as characteristic systems.

In this paper we introduce a remarkable class of Monge-Ampère systems on contact manifolds of arbitrary odd dimensions which we call decomposable Monge-Ampère systems. We show that we can associate to a decomposable Monge-Ampère system the characteristic systems enjoying nice properties, and that most of the results in the case of two independent variable as discussed in [M2] can be naturally generalized to this class of decomposable Monge-Ampère systems.

The details and further discussion ([M-M]) will appear elsewhere.

1. Definition

We first recall the definition of a Monge-Ampère system on a contact manifold (See [M2]). Let M be a differential manifold and let A denote the sheaf of germs of differential forms on M. An exterior differential system on M is a subsheaf Σ of A such that

(1) Each stalk Σ_{x}, x ∈ M, is an ideal of A_{§},

(2) Σ is closed under exterior differentiation, i.e., dΣ ⊂ Σ,

(3)Σ is locally finitely generated.

An integral manifold of the exterior differential system Σ is an immersed submanifold ι : N → M such that ι^{∗}ϕ = 0 for any section ϕ of Σ.

Consider now a contact manifold (M, D) of dimension 2n+1. By definition, a contact structure D is a subbundle of the tangent bundle T M of M of codimension 1 defined locally by a 1-form ω satisfying ω ∧ (dω)^{n} ≠ 0 everywhere. Such a 1-form ω is called a contact form of the contact structure D.

Definition 1.1. An exterior differential system Σ on a contact manifold (M, D) of dimension 2n + 1 is called a Monge-Ampère exterior differential system (or simply M-A system) if Σ is locally generated by a contact form ω of D and an n-form θ. Such an n-form θ is called a local representative of the M-A system Σ and we sometimes say that Σ is defined by θ.

We remark that if θ is a local representative of a M-A system then so is an n-form θ′ given by θ′ = ρθ + φ ∧ dω + ψ ∧ ω, where ρ is a non-zero function, and φ and ψ are any (n − 2)-form and (n − 1)-form respectively.

By a solution of a M-A system Σ we mean an integral manifold of Σ of dimension n. Hence a solution of a M-A system is a Legendre submanifold of (M, D) on which a local representative θ vanishes. A solution of a M-A system Σ turns out to be a solution of a so-called Monge-Ampère equation when expressed in terms of a suitable canonical coordinate system (cf.[M2]).

Let (M, D) and (M′, D′) be contact manifolds. Let Σ, Σ′ be M-A systems on M, M′ respectively. Then we say that Σ and Σ′ are equivalent ( or isomorphic ) if there exists a contact diffeomorphism f : M ⟶ M′ such that f^{∗}Σ′ = Σ. The notion of locally equivalence is defined in the obvious manner.

Now we define a decomposable of M-A system:

Definition 1.2. A M-A system Σ on a 2n + 1 dimensional contact manifold (M, D) is called decomposable if at any point of M Σ has, as its local representative, a decomposable n-form: θ = θ_{1} ∧ θ_{2} ∧ ⋯ ∧ θ_{n}, where θ_{i}(i = 1, 2, ⋯, n) are 1-forms locally defined in a neighborhood of the point.

It should be remarked that any M-A system on a 5-dimensional contact manifold (Monge-Ampère equation in two independent variables) is always decomposable in the complex category. However, it is not always the case for M-A systems on higher dimensional contact manifolds.

2. Characteristic systems

If a M-A system is decomposable we can canonically associate with it nice linear objects called characteristic systems. This fact is based on the following fundamental Proposition:

Proposition 2.1. Let (V, Ω) be a symplectic vector space of dimension 2n. Given two non-zero decomposable n-covectors θ = α_{1} ∧ ⋯ ∧ α_{n} and θ′ = α′_{1} ∧ ⋯ ∧ α′_{n} with α_{i}, α′_{i} ∈ V^{∗}. Then θ and θ′ are equivalent modulo Ω up to a constant factor, that is, θ′ = λθ + ϕ ∧ Ω for a nonzero scalar λ and (n − 2)- covector ϕ, if and only if Ann(α_{1}, ⋯, α_{n}) and Ann(α′_{1}, ⋯, α′_{n}) (the subspaces of V annihilated by α_{1}, ⋯, α_{n} and α′_{1}, ⋯, α′_{n}respectively) are either identical or perpendicular with respect to Ω.

We shall give a proof of this Proposition in the last section. Now admitting it, we proceed to define the characteristic systems.

Let Σ be a M-A system on a contact manifold (M, D) of dimension 2n + 1, decomposable and non-trivial everywhere. For x ∈ M let θ = θ_{1} ∧ ⋯ ∧ θ_{n} be a decomposable local representative of Σ in a neighborhood of x, and put

E_{x} = Ann((ω)_{x}, (θ_{1})_{x}, ⋯, (θ_{n})_{x}),

where ω is a contact form of D. Then E_{x} is an n-dimensional subspace of T_{x}M contained in the contact distribution D_{x}. Put F_{x} = E_{x}^{⊥}, the orthogonal complement of E_{x} with respect to (dω)_{x}, and put V(Σ)_{x} = E_{x} ∪ F_{x}.

By the Proposition above, V(Σ)_{x} does not depend on the choice of a decomposable local representative. Moreover E_{x} and F_{x} are canonically determined up to a permutation. Thus the bundle V(Σ) = ⋃_{x∈M} V(Σ)_{x} is written locally (globally under some topological condition) as

V(Σ) = E ∪ F,

where E, F are subbundles of D of rank n and perpendicular with respect to dω.

The distributions E, F are called the characteristic systems and V(Σ) the characteristic variety of Σ.

Conversely, given a subbundle E of D of rank n, it is clear that there exists a unique decomposable M-A system Σ such that V(Σ) = E ∪ E^{⊥}. Thus the correspondence between the decomposable M-A systems Σ and the characteristic varieties V(Σ) is of one-to-one.

We say Σ is of Lagrangian type at x if E_{x} = F_{x} (in this case E_{x} is a Lagrangian subspace of D_{x} ) and of Lagrangian type if so is everywhere. We say Σ is of non-Lagrangian type if E_{x} ≠ F_{x} for all x.

Let us give a remark on the singular integral elements of Σ. Clearly, for 1 ≤ k ≤ n − 1 a k-dimensional subspace U of T_{x}M is an integral element of Σ if and only if U is an isotropic subspace of D_{x}. For 1 ≤ k ≤ n − 2 all integral elements are regular. For k = n − 1 an isotropic subspace U is singular if and only if

u_{1} ∧ ⋯ ∧ u_{n−1} ⌋ θ ≡ 0 (mod ω, u_{1}⌋dω, ⋯, u_{n−1}⌋dω),

where ω is a contact form, θ a local representative n-form and u_{1}, ⋯, u_{n−1} a basis of U. The latter condition is equivalent to saying that

U ∩ V(Σ)_{x} ≠ 0.

Thus an (n − 1)-dimensional integral submanifold N is non-characteristic at x ∈ N if

T_{x}N ∩ V(Σ)_{x} = 0.

We can then study solutions of a decomposable M-A system Σ, on the basis of the following:

Proposition 2.2. An n-dimensional submanifold S of M is a solution of Σ if and only if the following two conditions are satisfied for each x ∈ S,

1. T_{x}S ⊂ D, 2. T_{x}S ∩ V(Σ)_{x} ≠ 0, or equivalently T_{x}S ∩ E_{x} ≠ 0 and T_{x}S ∩ F_{x} ≠ 0.

Note that under the condition 1, T_{x}S ∩ E_{x} ≠ 0 if and only if T_{x}S ∩ F_{x} ≠ 0. It should be also remarked that the distribution on a solution S of Σ which assigns T_{x}S ∩ E_{x} to x ∈ S is not in general completely integrable, contrary to the case of 5-dimensional contact manifolds.

We can also generalize Monge’s integration method to decomposable M-A systems of higher dimensions by slightly generalizing the proof in [M2].

Theorem 2.1. Let Σ be a decomposable M-A system on a contact manifold of dimension 2n + 1 with the characteristic variety V(Σ) = E ∪ F. If one of the characteristic systems of Σ admits independent n first integrals, then any non-characteristic integral submanifold of dimension n−1 can be extended to a solution of the M-A system. This solution is locally unique and can be obtained by integrating a characteristic vector field.

3. Two typical examples and their characterizations

The equivalence problem of decomposable M-A systems Σ can be studied as that of geometric structures (M, D, E) (cf.[M2]). Here we will only touch on two typical examples. Their proofs will be given in [M-M].

We denote by (x_{1}, x_{2}, ⋯, x_{n}, z, p_{1}, p_{2}, ⋯, p_{n}) a canonical coordinate system on the 1-jet space J^{1}(ℝ^{n}, ℝ) ≅ ℝ^{2n+1}. The p_{i} is regarded as the partial derivative z_{x_{i}} of z with respect to x_{i} and p_{ij} as the second partial derivative z_{x_{i}x_{j}} of z with respect to x_{i} and x_{j}. The canonical contact form ω is represented by ω = dz − ∑_{i=1}^{n}p_{i}dx_{i}.

3.1. As the simplest example of decomposable M-A systems of Lagrangian type, we have the following.

Theorem 3.1. If one of the characteristic systems of a decomposable M-A system Σ is completely integrable, then the two characteristic systems coincide (therefore Σ is of Lagrangian type), and Σ is locally equivalent to the M-A system defined by dp_{1} ∧ dp_{2} ∧ ⋯ ∧ dp_{n}. The corresponding Monge-Ampère equation may be written as

|\begin{matrix}p_{11} & p_{12} & … & p_{1n} \\ p_{12} & p_{22} & … & p_{2n} \\ ⋮ & ⋮ & ... & ⋮ \\ p_{1n} & p_{2n} & … & p_{nn} \end{matrix}| = 0.

3.2. As a simplest example of decomposable M-A systems of non-Lagrangian type, we have the following.

Theorem 3.2. If the characteristic systems E, F of a decomposable M-A system Σ on M of dimension 2n + 1(n = 2k) satisfies

1. D = E ⊕ F, 2. the derived systems E^{2}, F^{2} of E, F are both vector bundles of rank n + 1

and completely integrable, then Σ is locally equivalent to the M-A system defined by dp_{1} ∧ ⋯ ∧ dp_{k} ∧ dx_{1} ∧ ⋯ ∧ dx_{k} (≡ dp_{k+1} ∧ ⋯ ∧ dp_{n} ∧ dx_{k+1} ∧ ⋯ ∧ dx_{n} (mod ω, dω)). The corresponding Monge-Ampère equation may be written as

|\begin{matrix}p_{1,k+1} & p_{1,k+2} & … & p_{1,n} \\ p_{2,k+1} & p_{2,k+2} & … & p_{2,n} \\ ⋮ & ⋮ & ... & ⋮ \\ p_{k,k+1} & p_{k,k+2} & … & p_{k,n} \end{matrix}| = 0.

Here by the derived system E^{2} of a subbundle E of T M we mean the sheaf defined by E^{2} = E + [E, E], where E denotes the sheaf of the germs of sections of E.

3.3. We give a global model of the theorem in 3.1.

Let V be an (n + 2)-dimensional vector space and V^{∗} its dual space. Denote by P (V) = P = P^{n+1} the (n + 1)-dimensional projective space of V and by P^{∗}(V)(≅ P (V^{∗})) = P^{∗} = P^{∗}^{n+1} the dual projective space of P. Put

Q = Q^{2n+1} = {([x], [ξ]) ∈ P × P^{∗} |< x, ξ >= 0}.

With the canonical projections ρ : Q ⟶ P and ρ′ : Q ⟶ P^{∗}, we can define Q with the projective cotangent bundle P T^{∗}P^{n+1} of P^{n+1} and with P T^{∗}P^{∗}^{n+1} of P^{∗}^{n+1}. Then we have the double fibration:

Q

ρ ↙ ↘ ρ′
P  P^{∗}.

There is a canonical contact structure D on Q:

D = ker ρ_{∗} ⊕ ker ρ′_{∗}.

Let E be the subbundle ker ρ′_{∗} of D. Since E is a Legendre distribution, a decomposable M-A system Σ of Lagrangian type is defined on Q. Moreover E is completely integrable. Therefore Σ can be regarded as a global model of the theorem in 3.1.

3.4. We give a global model of the theorem in 3.2.

Let π : S^{2k+1} ⟶ P^{k}(ℂ) be the Hopf fibering with fiber S^{1}. There is a natural contact form ω_{0} on S^{2k+1}. Put

M^{4k+1=2n+1} = S^{2k+1} × S^{2k+1}/ ∼

the quotient space by the action of the diagonal Δ of S^{1} × S^{1}. It follows that the 1-form ω = ω_{0} ⊕ −ω_{0} induces a contact structure D on M. Let ρ_{1} : M ⟶ P^{k}(ℂ) and ρ_{2} : M ⟶ P^{k}(ℂ) be the natural projections to the left and the right respectively. Then we have the double fibration:

M^{2n+1}

ρ_{1} ↙  ↘ ρ_{2}
P^{k}(ℂ)  P^{k}(ℂ).

We can regard this double fibration as the following principal bundle with fiber S^{1} :

S^{1} ⟶  M^{2n+1}
↓π̄
P^{k}(ℂ) × P^{k}(ℂ).

Here the base space P^{k}(ℂ) × P^{k}(ℂ) has the symplectic form Ω = Ω_{0} ⊕ −Ω_{0} defined by a natural Kähler form Ω_{0} on P^{k}(ℂ). It follows that dω = π̄^{∗}Ω. The contact form ω is regarded as the connection form of the S^{1} principal bundle and the symplectic form Ω as the curvature form of the bundle.

Put

E = D ∩ ker ρ_{1}_{∗}, F = D ∩ ker ρ_{2}_{∗}.

Since E, F are n-dimensional non-Legendre distributions and perpendicular with respect to dω, a decomposable M-A system Σ of non-Lagrangian type is defined on M. Moreover it follows that D = E ⊕F and E, F have independent n first integrals given by ρ_{1}, ρ_{2} respectively. Therefore Σ can be regarded as a global model of the theorem in 3.2.

4. Proof of the fundamental Proposition

In this section we give a proof of the following Proposition stated in section 2:

Proposition 4.1. Let (V, Ω) be a symplectic vector space of dimension 2n. Given two non-zero decomposable n-covectors θ = α_{1} ∧ ⋯ ∧ α_{n} and θ′ = α′_{1} ∧ ⋯ ∧ α′_{n} with α_{i}, α′_{i} ∈ V^{∗}. Then θ and θ′ are equivalent modulo Ω up to a constant factor, that is, θ′ = λθ + ϕ ∧ Ω for a nonzero scalar λ and (n − 2)- covector ϕ, if and only if Ann(α_{1}, ⋯, α_{n}) and Ann(α′_{1}, ⋯, α′_{n}) (the subspaces of V annihilated by α_{1}, ⋯, α_{n} and α′_{1}, ⋯, α′_{n}respectively) are either identical or perpendicular with respect to Ω.

In the following, (V, Ω) denotes a symplectic vector space of dimension 2n unless otherwise stated. Thus Ω is a non-degenerate skew-symmetric bilinear form (2-covector) on V. As usual V^{∗} denotes the dual space of V. Now we begin with some elementary properties of symplectic vector spaces.

Lemma 4.1. If α_{1}, ⋯, α_{n} ∈ V^{∗} are linearly independent, then there exists a basis of V^{∗}, {ξ_{1}, ⋯, ξ_{r}, ξ′_{1}, ⋯, ξ′_{r}, η_{1}, ⋯, η_{r}, η′_{1}, ⋯, η′_{r}, ζ_{1}, ⋯, ζ_{s}, ζ′_{1}, ⋯, ζ′_{s}} (2r + s = n), such that

1. α_{1} ∧ ⋯ ∧ α_{n} = λξ_{1} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{1} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s} with non-zero constant λ,

2. Ω = ∑_{i=1}^{r}(ξ_{i} ∧ ξ′_{i} + η_{i} ∧ η′_{i}) + ∑_{j=1}^{s} ζ_{j} ∧ ζ′_{j}.

Proof. Let U = Ann(α_{1}, ⋯, α_{n}) and U^{⊥} the orthogonal complement of U with respect to Ω, and consider the chain of subspaces:

V ⊃ U + U^{⊥} ⊃ U ∩ U^{⊥} ⊃ 0.

Here note that codim(U +U^{⊥}) = dim(U ∩U^{⊥}). Since the null space of Ω|_{U+U^{⊥}}, the restriction of Ω to U + U^{⊥}, is just U ∩ U^{⊥}, (U + U^{⊥})/U ∩ U^{⊥} becomes a symplectic vector space with the induced bilinear form \overline{Ω}|_{U+U^{⊥}}, moreover we have the symplectic direct sum decomposition:

(U + U^{⊥})/U ∩ U^{⊥} = U/U ∩ U^{⊥} ⊕ U^{⊥}/U ∩ U^{⊥}.

Take a symplectic basis of V,

{e_{1}, ⋯, e_{r}, e′_{1}, ⋯, e′_{r}, f_{1}, ⋯, f_{r}, f′_{1}, ⋯, f′_{r}, g_{1}, ⋯, g_{s}, g′_{1}, ⋯, g′_{s}}

such that

\begin{matrix}\{e_{1}, ⋯, e_{r}, e′_{1}, ⋯, e′_{r}, g′_{1}, ⋯, g′_{s}\}, \{f_{1}, ⋯, f_{r}, f′_{1}, ⋯, f′_{r}, g′_{1}, ⋯, g′_{s}\}, \\ \{g′_{1}, ⋯, g′_{s}\} \end{matrix}

are bases of U^{⊥}U, U ∩ U^{⊥} respectively. Then its dual basis

{ξ_{1}, ⋯, ξ_{r}, ξ′_{1}, ⋯, ξ′_{r}, η_{1}, ⋯, η_{r}, η′_{1}, ⋯, η′_{r}, ζ_{1}, ⋯, ζ_{s}, ζ′_{1}, ⋯, ζ′_{s}}

satisfies the properties stated.

Lemma 4.2. If α_{1} ∧⋯∧α_{n} ≡ 0 (mod Ω) for α_{1}, ⋯, α_{n} ∈ V^{∗}, then α_{1} ∧⋯∧ α_{n} = 0 .

Proof. Clear from Lemma 4.1.

Lemma 4.3. If an l-covector ψ satisfies ψ ∧ Ω = 0 and if l ≤ n − 1, then ψ = 0.

Proof. Take a basis {ξ_{1}, ⋯, ξ_{n}, ξ_{1}, ⋯, ξ′_{n}} of V^{∗} such that Ω = ∑_{i=1}^{n} ξ_{i} ∧ ξ′_{i}, and put X_{i} = ξ_{i} ∧ ξ′_{i} (i = 1, ⋯, n). Then a covector is uniquely written in the following form:

ψ = ∑_{\begin{matrix}\{i_{1},··· ,i_{p}\}∩\{j_{1},··· ,j_{q}\}=∅ \\ 1≤i_{1}<···<i_{p}≤n,1≤j_{1}<···<j_{q}≤n \end{matrix}} a_{i_{1}···i_{p},j_{1}···j_{q}}(X) ∧ ξ_{i_{1}} ∧ ⋯ ∧ ξ_{i_{p}} ∧ ξ′_{j_{1}} ∧ ⋯ ∧ ξ′_{j_{q}}

where a_{i_{1}···i_{p},j_{1}···j_{q}}(X) are polynomials in X_{1}, ⋯, X_{n}. Now the assertion is a consequence of the following Lemma:

Lemma 4.4. In the polynomial ring in X_{1}, ⋯, X_{n} with additional relations X_{1} ^{2} = ⋯ = X_{n} ^{2} = 0, let Ω_{n} = X_{1} + ⋯ + X_{n} and let P_{r} be a polynomial of degree r. If P_{r}Ω_{n} ^{s} = 0 and if χ(n, r, s) = n − 2r − s ≥ 0, then P_{r} = 0.

Proof. We proceed by induction on n. Write

Ω_{n} = X_{n} + Ω_{n−1}, P_{r} = P_{r−1}X_{n} + Q_{r},

where P_{r−1}, Q_{r} are polynomials in X_{1}, ⋯, X_{n−1}. Then we have

P_{r}Ω_{n} ^{s} = (P_{r−1}Ω_{n−1} ^{s} + sQ_{r}Ω_{n−1} ^{s−1})X_{n} + Q_{r}Ω_{n−1} ^{s}.

Now suppose that P_{r}Ω_{n} ^{s} = 0 and χ(n, r, s) = n − 2r − s ≥ 0. Then

\{ \begin{matrix}P_{r−1}Ω_{n−1} ^{s} + sQ_{r}Ω_{n−1} ^{s−1} = 0, \\ Q_{r}Ω_{n−1} ^{s} = 0. \end{matrix}

Multiplying the first equation by Ω_{n−1}, we have

P_{r−1}Ω_{n−1} ^{s+1} = 0.

Note that χ(n−1, r −1, s+1) = χ(n, r, s) ≥ 0. Now the assertion of Lemma is true for Ω_{n−1} by induction assumption, so that P_{r−1} = 0, which then implies

Q_{r}Ω_{n−1} ^{s−1} = 0.

Then again by induction we have Q_{r} = 0. Hence P_{r} = 0, which completes the induction.

Lemma 4.5. If

Ω = ∑_{i=1}^{r}(ξ_{i} ∧ ξ′_{i} + η_{i} ∧ η′_{i}) + ∑_{j=1}^{s} ζ_{j} ∧ ζ′_{j},

then

ξ_{1} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{1} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s}
≡ (−1)^{r}η_{1} ∧ ⋯ ∧ η_{r} ∧ η′_{1} ∧ ⋯ ∧ η′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s} (mod Ω).

Proof. This follows from the identity:

X^{r} − (−Y)^{r} = (X + Y) ∑_{i=1}^{r−1} X^{i}(−Y)^{r−i−1}

with X_{i} = ξ_{i} ∧ ξ′_{i}, Y_{j} = η_{j} ∧ η′_{j} and X = X_{1} + ⋯ + X_{r}, Y = Y_{1} + ⋯ + Y_{r}. It should be remarked that

Ann(ξ_{1}, ⋯, ξ_{r}, ξ′_{1}, ⋯, ξ′_{r}, ζ_{1}, ⋯, ζ_{s}) ⊥ Ann(η_{1}, ⋯, η_{r}, η′_{1}, ⋯, η′_{r}, ζ_{1}, ⋯, ζ_{s})

with respect to Ω.

Now we are in a position to prove the proposition. By our preceding discussion it suffices to prove the following:

Lemma 4.6. Let the symplectic form be written in the form (cf. Lemma 4.1) :

Ω = ∑_{i=1}^{r}(ξ_{1} ∧ ξ′_{i} + η_{i} ∧ η′_{i}) + ∑_{j=1}^{s} ζ_{j} ∧ ζ′_{j}.

If

α_{1} ∧ ⋯ ∧ α_{n} ≡ ξ_{1} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{1} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s} (mod Ω),

then either

α_{1} ∧ ⋯ ∧ α_{n} = ξ_{1} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{1} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s}

or

α_{1} ∧ ⋯ ∧ α_{n} = (−1)^{r}η_{1} ∧ ⋯ ∧ η_{r} ∧ η′_{1} ∧ ⋯ ∧ η′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s}.

Proof. If n = 1 the assertion obviously holds. We proceed by induction on n. We thus assume that the assertion is true if the dimension of the symplectic vector space is less than 2n. Now suppose that the hypothesis of the lemma is satisfied.

Case 1: where one of α, say α_{1}, contains some η_{j} or η′_{j}, that is α_{1} ≢ 0(mod ξ, ξ′, ζ, ζ′). By changing the suffix if necessary, we may assume α_{1} contains η_{1}. Then we deduce that one of α_{2}, ⋯, α_{r} should contain η′_{1}, say α_{2}. Then we can write

α_{1} = λ_{1}η_{1} + f_{1}, α_{2} = λ_{2}η′_{1} + f_{2} (λ_{1}, λ_{2} ≠ 0),

where f_{1}, f_{2}, α_{3}, ⋯, α_{n} are linear combinations of

ξ_{1}, ⋯, ξ_{r}, ξ′_{1}, ⋯, ξ′_{r}, η_{2}, ⋯, η_{r}, η′_{2}, ⋯, η′_{r}, ζ_{1}, ⋯, ζ_{s}, ζ′_{1}, ⋯, ζ′_{s}

. By our assumption we have

α_{1} ∧ ⋯ ∧ α_{n} = ξ_{1} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{1} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s} + Φ ∧ Ω

for some Φ ∈ ⋀^{n−2} V^{∗}. Writing

\{ \begin{matrix}Ω & = η_{1} ∧ η′_{1} + Ω′, \\ Φ & = A ∧ η_{1} ∧ η′_{1} + B ∧ η_{1} + C ∧ η′_{1} + D, \end{matrix}

where A, B, C, D and Ω′ do not contain η_{1}, η′_{1}, we have

(λ_{1}η_{1} + f_{1}) ∧ (λ_{2}η′_{1} + f_{2}) ∧ α_{3} ∧ ⋯ ∧ α_{n}
= ξ_{1} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{1}, ⋯, ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s}
+(A ∧ η_{1} ∧ η′_{1} + B ∧ η_{1} + C ∧ η′_{1} + D)(η_{1} ∧ η′_{1} + Ω′).

Hence

\{ \begin{matrix}λ_{1}λ_{2}η_{1} ∧ η′_{1} ∧ α_{3} ∧ ⋯ ∧ α_{n} & = (D + A ∧ Ω′) ∧ η_{1} ∧ η′_{1}, \\ λ_{1}η_{1} ∧ f_{2} ∧ α_{3} ∧ ⋯ ∧ α_{n} & = B ∧ η_{1} ∧ Ω′, \\ λ_{2}η′_{1} ∧ f_{2} ∧ α_{3} ∧ ⋯ ∧ α_{n} & = −C ∧ η′_{1} ∧ Ω′, \\ f_{1} ∧ f_{2} ∧ α_{3} ∧ ⋯ ∧ α_{n} & = D ∧ Ω′+ \\ & +ξ_{1} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{1} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s} . \end{matrix}

In particular, we have

η_{1} ∧ (λ_{1}f_{2} ∧ α_{3} ∧ ⋯ ∧ α_{n} − (−1)^{n−3}B ∧ Ω′) = 0.

Since f_{2}, B, Ω′ do not contain η_{1}, we have

λ_{1}f_{2} ∧ α_{3} ∧ ⋯ ∧ α_{n} = (−1)^{n−3}B ∧ Ω′.

Then applying Lemma 4.2 to the symplectic form Ω′ in 2(n − 1)-dimensional space, we see that

ξ_{1}, ⋯, ξ_{r}, ξ′_{1}, ⋯, ξ′_{r}, η_{2}, ⋯, η_{r}, f_{2} ∧ α_{3} ∧ ⋯ ∧ α_{n} = 0.

Similarly we have

f_{1} ∧ α_{3} ∧ ⋯ ∧ α_{n} = 0.

Hence, for some λ ≠ 0

α_{1} ∧ ⋯ ∧ α_{n} = λη_{1} ∧ η′_{1} ∧ α_{3} ∧ ⋯ ∧ α_{n}.

Now in this case let us show that α_{3}, ⋯, α_{n} do not contain ξ and ξ′- components. Suppose that it were not true, for instance, that α_{3} contained ξ_{1}. Then one of α_{4}, ⋯, α_{n}, say α_{4}, must contain ξ′_{1}, from which we deduce by the same argument as above that we can write as

α_{1} ∧ ⋯ ∧ α_{n} = λη_{1} ∧ η′_{1} ∧ ξ_{1} ∧ ξ′_{1} ∧ α_{5} ∧ ⋯ ∧ α_{n}

where α_{5}, ⋯, α_{n} do not contain ξ_{1}, ξ′_{1}.

Hence, if we put

P = λη_{1} ∧ η′_{1} ∧ α_{5} ∧ ⋯ ∧ α_{n} − (−1)^{r−1}ξ_{2} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{2} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s},

we see that

ξ_{1} ∧ ξ′_{1} ∧ P = Φ ∧ Ω

for an (n − 2)-covector Φ. If we write

\{ \begin{matrix}Ω & = ξ_{1} ∧ ξ′_{1} + Ω′, \\ Φ & = A ∧ ξ_{1} ∧ ξ′_{1} + B ∧ ξ_{1} + C ∧ ξ′_{1} + D, \end{matrix}

where A, B, C, D and Ω′ do not contain ξ_{1}, ξ′_{1}, we have

B ∧ ξ_{1} ∧ Ω′ = C ∧ ξ′_{1} ∧ Ω′ = D ∧ Ω′ = 0.

Then

B ∧ Ω′ = C ∧ Ω′ = D ∧ Ω′ = 0.

But it then follows from Lemma 4.3 that B = C = D = 0. Hence Φ = A∧ξ_{1}∧ξ′_{1} and

ξ_{1} ∧ ξ′_{1} ∧ (P − A ∧ Ω) = 0.

Therefore we have

P ≡ 0 (mod Ω, ξ_{1}, ξ′_{1}).

In particular, we have

ξ_{2} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{2} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s} ≡ 0 (mod Ω, ξ_{1}, ξ′_{1}, η_{1}, η′_{1}).

By Lemma 4.2 we then have

ξ_{2} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{2} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s} = 0,

which leads to a contradiction. Thus we have shown that

α_{1} ∧ ⋯ ∧ α_{n} = λη_{1} ∧ η′_{1} ∧ α_{3} ∧ ⋯ ∧ α_{n}

with α_{3}, ⋯, α_{n} not containing ξ and ξ′-components.

Since we have by Lemma 4.5

λη_{1} ∧ η′_{1} ∧ α_{3} ⋯ ∧ α_{n}
≡ ξ_{1} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{1} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s}
≡ (−1)^{r}η_{1} ∧ ⋯ ∧ η_{r} ∧ η′_{1} ∧ ⋯ ∧ η′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s} (mod Ω),

we have

\begin{matrix}η_{1} ∧ η′_{1} ∧ (λα_{3} ∧ ⋯ ∧ α_{n} ± η_{2} ∧ ⋯ ∧ η_{r} ∧ η′_{2} ∧ ⋯ ∧ η′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s}) \\ ≡ 0 (mod Ω). \end{matrix}

By the same argument as above, we have

\begin{matrix}λα_{3} ∧ ⋯ ∧ α_{n} ± η_{2} ∧ ⋯ ∧ η_{r} ∧ η′_{2} ∧ ⋯ ∧ η′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s} \\ ≡ 0 (mod Ω, η_{1}, η′_{1}). \end{matrix}

Taking into account that α_{3}, ⋯, α_{n} do not contain ξ_{1}, ξ′_{1}, we have

λα_{3} ∧ ⋯ ∧ α_{n} ± η_{2} ∧ ⋯ ∧ η_{r} ∧ η′_{2} ∧ ⋯ ∧ η′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s} ≡ 0 (mod Ω′′),

where Ω′′ = Ω−ξ_{1} ∧ξ′_{1} −η_{1} ∧η′_{1}. Since our assertion is valid for Ω′′ by induction assumption, we conclude that

λα_{3} ∧ ⋯ ∧ α_{n} = ±η_{2} ∧ ⋯ ∧ η_{r} ∧ η′_{2} ∧ ⋯ ∧ η′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s}.

Hence

α_{1} ∧ ⋯ ∧ α_{n} = cη_{1} ∧ ⋯ ∧ η_{r} ∧ η′_{1} ∧ ⋯ ∧ η′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s}

with a constant c. But the constant is uniquely determined by Lemma 4.5 and Lemma 4.2. Thus finally we have

α_{1} ∧ ⋯ ∧ α_{n} = (−1)^{r}η_{1} ∧ ⋯ ∧ η_{r} ∧ η′_{1} ∧ ⋯ ∧ η′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s}.

Case 2: where one of α_{1}, ⋯, α_{n} contains ξ_{j} or ξ′_{j}. In view of Lemma 4.5 our discussion being symmetric in ξ and η, we have in this case

α_{1} ∧ ⋯ ∧ α_{n} = ξ_{1} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{1} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s}.

Case 3: where none of α_{1}, ⋯, α_{n} contains ξ, ξ′, η, η′, that is each α_{i} is a linear combination of ζ, ζ′. In this case we shall show that all α_{1}, ⋯, α_{n} are linear combination only of ζ. Suppose that it were not the case. We then may assume that

α_{1} = ζ′_{s} + h

and that α_{2}, ⋯, α_{n} and h are linear combinations of ζ_{1}, ⋯, ζ_{s}, ζ′_{1}, ⋯, ζ′_{s−1}. Writing down the condition

(ζ′_{s} + h) ∧ α_{2} ∧ ⋯ ∧ α_{n}
= ξ_{1} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{1} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s} + Φ ∧ Ω

with

Φ = A ∧ ζ′_{s} + B, Ω = Ω′ + ζ_{s} ∧ ζ′_{s},

we have

\{ \begin{matrix}h ∧ α_{2} ∧ ⋯ ∧ α_{n} & = ξ ∧ ξ′ ∧ ζ + B ∧ Ω′, \\ ζ′_{s} ∧ α_{2} ∧ ⋯ ∧ α_{n} & = A ∧ ζ′_{s} ∧ Ω′ + B ∧ ζ_{s} ∧ ζ′_{s}. \end{matrix}

Hence

ζ′_{s} ∧ (α_{2} ∧ ⋯ ∧ α_{n} − (−1)^{n−3}A ∧ Ω′ + B ∧ ζ_{s}) = 0.

Therefore

α_{2} ∧ ⋯ ∧ α_{n} − (−1)^{n−3}A ∧ Ω′ + B ∧ ζ_{s} = 0.

In particular, we have

ζ_{s} ∧ α_{2} ∧ ⋯ ∧ α_{n} ≡ 0 (mod Ω).

It then follows from Lemma 4.2 that

ζ_{s} ∧ α_{2} ∧ ⋯ ∧ α_{n} = 0.

If necessary by replacing α_{1}, ⋯, α_{n} by their linear combinations, we may assume α_{n} = ζ_{s}. Again from the relation

α_{1} ∧ ⋯ ∧ α_{n−1} ∧ ζ_{s}
= ξ_{1} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{1} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s} + Φ ∧ Ω,

we obtain

α_{1} ∧ ⋯ ∧ α_{n−1} ≡ ξ_{1} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{1} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s−1} (mod Ω′(= Ω − ζ_{s} ∧ ζ′_{s})).

Then by induction assumption

\begin{matrix}α_{1} ∧ ⋯ ∧ α_{n−1} = & ξ_{1} ∧ ⋯ ∧ ξ_{r} ∧ ξ′_{1} ∧ ⋯ ∧ ξ′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s−1}, \\ or & (−1)^{r}η_{1} ∧ ⋯ ∧ η_{r} ∧ η′_{1} ∧ ⋯ ∧ η′_{r} ∧ ζ_{1} ∧ ⋯ ∧ ζ_{s−1}, \end{matrix}

Y. Machida and T. Morimoto 196

which contradicts the fact that α_{1} contains ζ′_{1}. We conclude therefore that α_{1}, ⋯, α_{n} is a linear combination of ζ. Hence we have

α_{1} ∧ ⋯ ∧ α_{n} = ζ_{1} ∧ ⋯ ∧ ζ_{s}.

In this case it turns out that n = s. This completes the induction and proves the Lemma.

References

[G] E. Goursat, Leçon sur l’intégration des équations aux derivées partielles du second ordre à deux variables independentes, I, II, Hermann, 1896, 1898.

[M1] T. Morimoto, La géométrie des équations de Monge-Ampère, C. R. Acad. Sci. Paris, 289, 2 juillet 1979, A-25 –A-28.

[M2] T. Morimoto, Monge-Ampère equations viewed from contact geometry, in Symplectic singularities and geometry of gauge fields, Banach Center Publ.39, Inst. Math. Polish Academy Sci., Warszawa, (1997), pp. 105–121.

[M-M] Y. Machida and T. Morimoto, Decomposable Monge-Ampère equations and related geometric structures, in preparation.

Numazu College of Technology, 3600 Ooka Numazu-shi Shizuoka 410-8501 Japan E-mail address: machida@la.numazu-ct.ac.jp

Department of Mathematics, Faculty of Science, Nara Women’s University, Nara 630-8506 Japan E-mail address: morimoto@cc.nara-wu.ac.jp

Received July 27, 1999