Π = » ω0 § (ω1 § ω2 ’ ω3 § ω4 )

is that this Π be closed for some function » on B1 , which we can assume satis¬es

» > 0. Di¬erentiating then gives

0 = (d» ’ 2»•0 ) § ω0 § (ω1 § ω2 ’ ω3 § ω4 ).

0

Exterior algebra shows that this is equivalent to d» ’ 2»•0 being a multiple of

0

ω0 , say

d» ’ 2»•0 = σ » ω0

0

for some function σ, or in other words,

d(log ») ’ 2•0 = σ ω0 .

0

Such an equation can be satis¬ed if and only if d•0 is equivalent modulo {I} to

0

0

a multiple of dω . But we know that

dω0 ≡ ω1 § ω2 + ω3 § ω4 ,

´

52 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

and from (2.16) we see that d•0 is a multiple of this just in case S2 = 0.

0

This result may be thought of as follows. For any hyperbolic Monge-Ampere

system, d•0 ∈ „¦2 (B1 ) is both closed and semibasic for B1 ’ M . This means

0

that it is the pullback of a 2-form on M canonically associated with E.2 We

showed that this 2-form vanishes if and only if S2 = 0, which is equivalent to E

being locally Euler-Lagrange. This condition is reminiscent of the vanishing of

a curvature, when •0 is viewed as a connection in the contact line bundle I.

0

2.2 Neo-Classical Poincar´-Cartan Forms

e

We now turn to the geometry of Poincar´-Cartan forms in case n ≥ 3. In the

e

preceding section, we emphasized the corresponding Monge-Ampere system;

from now on, we will instead emphasize the more specialized Poincar´-Cartan

e

form.

Let M 2n+1 be a manifold with contact line bundle I, locally generated by a

1-form θ. Let Π ∈ „¦n+1 (M ) be a closed (n + 1)-form locally expressible as

Π = θ § Ψ,

¯

where Ψ ∈ P n(T — M/I) is primitive modulo {I}. As in the preceding section,

the pointwise linear algebra of this data involves the action of the conformal

n

R2n. When n = 2,

symplectic group CSp(n, R) on the space P n(R2n) ‚

there are four orbits (including {0}) for this action, but for n > 2, the situation

is more complicated. For example, when n = 3, the space of primitive 3-forms

on R6 has two open orbits and many degenerate orbits, while for n = 4 there

are no open orbits.

Which orbits contain the Poincar´-Cartan forms of most interest to us?

e

Consider the classical case, in which M = J 1(Rn , R), θ = dz ’ pi dxi, and

Λ = L(x, z, p)dx. We have already seen that

d(L dx + θ § Lpi dx(i))

Π= (2.19)

’θ § (d(Lpi ) § dx(i) ’ Lz dx)

= (2.20)

’θ § (Lpi pj dpj § dx(i) + (Lpi z pi + Lpi xi ’ Lz )dx).

= (2.21)

This suggests the following de¬nition, which singles out Poincar´-Cartan forms

e

of a particular algebraic type; it is these”with a slight re¬nement in the case

n = 3, to be introduced below”whose geometry we will study. Note that

non-degeneracy of the functional is built in to the de¬nition.

De¬nition 2.3 A closed (n + 1)-form Π on a contact manifold (M 2n+1, I) is

almost-classical if it can locally be expressed as

Π = ’θ § (H ij πi § ω(j) ’ Kω) (2.22)

2 This

statement also requires one to verify that d•0 is invariant under the action of some

0

element of each connected component of G1 ; this is easily done.

´

2.2. NEO-CLASSICAL POINCARE-CARTAN FORMS 53

for some coframing (θ, ω i , πi) of M with θ ∈ “(I), some invertible matrix of

functions (H ij ), and some function K.

Later, we will see the extent to which this de¬nition generalizes the classical

case. We remark that the almost-classical forms Π = θ § Ψ are those for which

¯

the primitive Ψ lies in the tangent variety of the cone of totally decomposable3

n-forms in P (T — M/I), but not in the cone itself.

n

Applying the equivalence method will yield di¬erential invariants and geo-

metric structures intrinsically associated to our Poincar´-Cartan forms. This

e

will be carried out in §2.4, but prior to this, it is best to directly look for some

naturally associated geometry. The preview that this provides will make easier

the task of interpreting the results of the equivalence method.

First note that the local coframings and functions appearing in the de¬nition

of an almost-classical form are not uniquely determined by Π. The extent of

the non-uniqueness of the coframings is described in the following lemma, which

prepares us for the equivalence method.

Lemma 2.1 If (θ, ωi , πi) is a coframing adapted to an almost-classical form

¯¯ ¯

Π as in De¬nition 2.3, then (θ, ωi, πi) is another if and only if the transition

matrix is of the form (in blocks of size 1, n, n)

« « «

¯ a 0 0

θ θ

ω i = C i Ai 0 ωj .

¯ j

j

πi

¯ πj

Di Eij Bi

Proof. That the ¬rst row of the matrix must be as shown is clear from the

¯

requirement that θ, θ ∈ “(I). The real content of the lemma is that Pfa¬an

system

def

JΠ = Span{θ, ω1 , . . . , ωn}

is uniquely determined by Π. This follows from the claim that JΠ is character-

ized as the set of 1-forms ξ such that ξ § Π is totally decomposable; this claim

we leave as an exercise for the reader.

The Pfa¬an system JΠ = {θ, ω1 , . . . , ωn } associated to Π is crucial for all

that follows. It is canonical in the sense that any local di¬eomorphism of M

preserving Π also preserves JΠ . In the classical case described previously we have

JΠ = {dz, dx1, . . . , dxn}, which is integrable and has leaf space J 0 (Rn , R).

Proposition 2.2 If n ≥ 4, then for any almost-classical form Π on a contact

manifold (M 2n+1, I), the Pfa¬an system JΠ is integrable.

Proof. We need to show that dθ, dω i ≡ 0 (mod {JΠ}), for some (equivalently,

any) coframing (θ, ω i , πi) adapted to Π as in the de¬nition. We write

Π = ’θ § (H ij πi § ω(j) ’ Kω),

3A n-form is totally decomposable if it is equal to the exterior product of n 1-forms.

´

54 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

and

dθ ≡ aij πi § πj (mod {JΠ }).

Then taking those terms of the equation dΠ ≡ 0 (mod {I}) that are cubic in

πi , we ¬nd

aij πi § πj § H kl πl = 0.

Then the 2-form aij πi §πj has at least n ≥ 3 linearly independent 1-forms as di-

visors, which is impossible unless aij πi § πj = 0. Therefore, dθ ≡ 0 (mod {JΠ })

(with only the hypothesis n ≥ 3).

For the next step, it is useful to work with the 1-forms

def

πi = H ij πj ,

and write

dωi ≡ Pjkπj § πk

i i i

(mod {JΠ }), Pjk + Pkj = 0.

From the form of Π (2.22), we have

0 = ωi § ωj § Π

for any pair of indices 1 ¤ i, j ¤ n. Di¬erentiating, we obtain

(dωi § ωj ’ ωi § dωj ) § Π

0=

j

(Pkl πk § πl § πj ’ Pklπk § πl § πi ) § θ § ω

i

=

j

(δm Pkl ’ δm Pkl )πk § πl § πm § θ § ω.

j i i

=

It is now an exercise in linear algebra to show that if n ≥ 4, then this implies

i i i

Pjk = 0. The hypotheses are that Pjk = ’Pkj and

(δm Pkl ’ δm Pkl ) + (δk Plm ’ δk Plm ) + (δlj Pmk ’ δli Pmk ) = 0.

j ji ij j

j i i i

(2.23)

By contracting ¬rst on jk and then on il, one ¬nds that for n = 2 the contraction

i i

Pik vanishes. Contracting (2.23) only on jk and using Pik = 0, one ¬nds that

i

for n = 3, all Pjk vanish.

There do exist counterexamples in case n = 3, for which (2.23) implies only

that

dωi = P ij π(j), P ij = P ji.

For example, if we ¬x constants P ij = P ji also satisfying P ii = 0, then there

is a unique simply connected, 7-dimensional Lie group G having a basis of left-

invariant 1-forms (ω i , θ, πi ) satisfying structure equations

dωi = P ij π(j), dθ = ’πi § ωi , dπi = 0.

In this case, θ generates a homogeneous contact structure on G, and the form

def

Π = ’θ § πi § ω(i)

´

2.2. NEO-CLASSICAL POINCARE-CARTAN FORMS 55

is closed, giving an almost-classical form for which JΠ is not integrable.

These counterexamples cannot arise from classical cases, however, and this

suggests that we consider the following narrower class of Poincar´-Cartan forms.

e

De¬nition 2.4 An almost-classical Poincar´-Cartan form Π is neo-classical if

e

its associated Pfa¬an system JΠ is integrable.

So the preceding Proposition states that in case n ≥ 4, every almost-classical

Poincar´-Cartan form is neo-classical, and we have narrowed the de¬nition only

e

in case n = 3.

The foliation corresponding to the integrable Pfa¬an system JΠ is the be-

ginning of the very rich geometry associated to a neo-classical Poincar´-Cartan

e

form. Before investigating it further, we justify the study of this class of objects

with the following.

Proposition 2.3 Every neo-classical Poincar´-Cartan form Π on a contact

e

manifold (M, I) is locally equivalent to that arising from some classical vari-

ational problem. More precisely, given such (M, I, Π), there are local coordi-

nates (xi , z, pi) on M with respect to which the contact system I is generated by

dz ’ pi dxi , and there is a Lagrangian of the form L(xi , z, pi)dx whose Poincar´-

e

Cartan form is Π.

Note that we have already observed the converse, that those non-degenerate

Poincar´-Cartan forms arising form classical variational problems (in case n ≥ 3)

e

are neo-classical.

Proof. We ¬x a coframing (θ, ω i , πi) as in the de¬nition of an almost-classical

form. Using the Frobenius theorem, we take independent functions (xi, z) on

M so that

JΠ = {ωi, θ} = {dxi, dz}.

By relabelling if necessary, we may assume θ ∈ {dxi}, and we ¬nd that there

/

are functions pi so that

θ ∈ R · (dz ’ pi dxi).

The fact that θ § (dθ)n = 0 implies that (xi , z, pi) are local coordinates on M .

We now introduce a technical device that is often useful in the study of

exterior di¬erential systems. Let

F p „¦q ‚ „¦p+q (M )

be the collection of (p+q)-forms with at least p factors in JΠ ; this is well-de¬ned.

With this notation, the fact that JΠ is integrable may be expressed as

d(F p „¦q ) ‚ F p „¦q+1 .

There is a version of the Poincar´ lemma that can be applied to each leaf of the

e

foliation determined by JΠ, with smooth dependence on the leaves™ parameters;