They are to be interpreted as giving

d

· gt ),

dt t=0 Ui (u

where gt is a path in G0 passing through the identity matrix at t = 0. Expo-

nentiated, we see that the vector-valued functions (U1 , U2 ) and (U3 , U4) on B0

each transform by an a¬ne-linear action of G0 along the ¬bers; that is, they

vary by a linear representation composed with a translation.1 It is the “nilpo-

i

tent” part of the group, with components g0, which gives rise to the translation.

Speci¬cally, we have for g0 as in (2.5)

U 1(u · g0) U 1 (u)

aA’1 ’ A’1C,

= (2.12)

U 2(u · g0) U 2 (u)

U 3(u · g0) U 3(u)

aB ’1 ’ B ’1 D.

= (2.13)

U 4(u · g0) U 4(u)

1 Strictlyspeaking, we have only shown that the torsion function (U i ) varies by an a¬ne-

linear action under the identity component of G0 . What will be important, however, is that

if u ∈ B0 satis¬es U i (u) = 0, then U i (u · J) = 0 as well, and likewise for some matrix in each

component where a < 0. These claims can be veri¬ed directly.

´

48 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

Now de¬ne a 1-adapted coframe to be a 0-adapted coframe u ∈ B0 satisfying

i

U (u) = 0 for 1 ¤ i ¤ 4. It then follows from the above reasoning that the subset

B1 ‚ B0 of 1-adapted coframes is a G1-subbudle of B0 , where the subgroup

G1 ‚ G0 is generated by the matrix J of (2.6), and by matrices of the form

(again, in blocks of size 1, 2, 2)

«

a00

g1 = 0 A 0 , (2.14)

00B

with a = det(A) = det(B) = 0. The structure equation (2.8) on B0 still holds

when restricted to B1 , with „ i|B1 = 0; but the pseudo-connection forms •i |B1

0

are semibasic for B1 ’ M , and their contribution should be regarded as torsion.

With everything now restricted to B1 , we write

•i = P0 ω0 + Pjiωj

i

0

and then have

dω = ’• § ω + „

with

« «

ω1 § ω2 + ω3 § ω4

•0 0 0 0 0

0

¬ · ¬ ·

’Pj1ωj § ω0

•1 •1

0 0 0

¬ · ¬ ·

1 2

•=¬ · and „ = ¬ ·.

’Pj2ωj § ω0

•2 •2

0 0 0

¬ · ¬ ·

1 2

•3 •3 ’Pj3ωj § ω0

0 0 0 3 4

•4 •4 ’Pj4ωj § ω0

0 0 0 3 4

As before, we can absorb some of this torsion into the pseudo-connection form,

respecting the constraint •0 = •1 + •2 = •3 + •4 , until the torsion is of the

0 1 2 3 4

form

«

ω1 § ω2 + ω3 § ω4

¬ ’(P ω1 + P3 ω3 + P4 ω4 ) § ω0 ·

1 1

¬ ·

dω + • § ω = ¬ ’(P ω2 + P3 ω3 + P4 ω4 ) § ω0 · .

2 2

(2.15)

¬ ·

’(Qω3 + P1 ω1 + P2 ω2 ) § ω0

3 3

’(Qω4 + P1 ω1 + P2 ω2 ) § ω0

4 4

We can go further: recall that •0 was uniquely determined up to addition of

0

a multiple of ω0 . We now exploit this, and take the unique choice of •0 =

0

•1 + •2 = •3 + •4 that yields a torsion vector of the form (2.15), with

1 2 3 4

P + Q = 0.

Now that •0 is uniquely determined, it is reasonable to try to get information

0

about its exterior derivative. To do this, we di¬erentiate the equation

dω0 = ’•0 § ω0 + ω1 § ω2 + ω3 § ω4 ,

0

2.1. THE EQUIVALENCE PROBLEM FOR n = 2 49

which simpli¬es to

(’d•0 + 2P ω1 § ω2 + 2Qω3 § ω4

0 = 0

+(P3 ’ P1 )ω1 § ω3 ’ (P3 + P2 )ω2 § ω3

2 4 1 4

+(P4 + P1 )ω1 § ω4 ’ (P4 ’ P2 )ω2 § ω4 ) § ω0 .

2 3 1 3

This tells us the derivative of •0 modulo the algebraic ideal {ω 0} = {I}, which

0

we now use in a somewhat unintuitive way.

We easily compute that

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

0

With knowledge of d•0 § ω0 from above, we can di¬erentiate this equation to

0

¬nd

0 = 2(P ’ Q)ω 0 § ω1 § ω2 § ω3 § ω4 .

This implies that P ’ Q = 0, and combined with our normalization P + Q = 0,

we have

P = Q = 0,

which somewhat simpli¬es our structure equations (2.15). In particular, we have

modulo {I}

d•0 ≡ (P3 ’ P1 )ω1 § ω3 ’ (P3 + P2 )ω2 § ω3 + (P4 + P1 )ω1 § ω4 ’ (P4 ’ P2 )ω2 § ω4 .

2 4 1 4 2 3 1 3

0

(2.16)

1 1 2 2

As before, the next step is to study the 8 torsion coe¬cients P3 , P4 , P3 , P4 ,

3 3 4 4

P1 , P2 , P1 , P2 . We can again obtain a description of how they vary along the

connected components of the ¬bers using in¬nitesimal methods, and then get a

full description of their variation along ¬bers by explicitly calculating how they

transform under one representative of each component of the structure group

G1 .

We state only the result of this calculation. The torsion in each ¬ber trans-

forms by an 8-dimensional linear representation of the group G1 , which decom-

poses as the direct sum of two 4-dimensional representations. Motivated by

(2.16), we de¬ne a pair of 2 — 2 matrix-valued functions on B1

1 4 1 3 1 4 1 3

P3 ’ P 2 P4 ’ P 2

P4 + P 2 P3 + P 2

S1 (u) = (u), S2 (u) = (u).

2 4 2 3 2 4 2 3

P4 ’ P 1 P3 ’ P 1

P3 + P 1 P4 + P 1

Now, for g1 ∈ G1 as in (2.14), one ¬nds that

S1 (u · g1 ) = aA’1 S1 (u)B, S2 (u · g1) = aA’1 S2 (u)B.

In particular, the two summand representations for our torsion are the same,

when restricted to the components of G1 of (2.14). However, one may also verify

that

0 ’1

01 t

S1 (u · J) =’ S1 (u) ,

’1 0 10

0 ’1

01 t

S2 (u · J) = S2 (u) .

’1 0 10

´

50 CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

An immediate conclusion to be drawn from this is that if S1 (u) = 0 at some

point, then S1 (u) = 0 everywhere on the same ¬ber of B1 ’ M , and likewise

for S2 .

If the torsion vector takes its values in a union of non-trivial orbits having

conjugate stabilizers, then we can try to make a further reduction to the subbun-

dle consisting of those coframes on which the torsion lies in a family of normal

forms. However, it is usually interesting in equivalence problems to consider the

case when no further reduction is possible; in the present situation, this occurs

when all of the invariants vanish identically.

We ¬rst claim that S2 = 0 identically if and only if the uniquely determined

form •0 is closed. To see this, note ¬rst that from (2.16) we have S2 = 0 if and

0

only if

d•0 = µ § ω0

0

for some 1-form µ. We di¬erentiate modulo {ω 0} to obtain

0 ≡ ’µ § dω0 (mod {ω0})

which by symplectic linear algebra implies that

µ ≡ 0 (mod {ω0 }).

But then d•0 = 0, as claimed. Conversely, if d•0 = 0, then obviously S2 = 0.

0 0

Now suppose that S1 = S2 = 0 identically. Then because d•0 = 0, we can

0

locally ¬nd a function » > 0 satisfying

•0 = »’1 d».

0

We can also compute in case S1 = S2 = 0 that

d(ω1 § ω2 ) = ’•0 § ω1 § ω2,

0

so that

d(» ω1 § ω2 ) = 0.

Now, by a variant of the Darboux theorem, this implies that there are locally

de¬ned functions p, x such that

’dp § dx = » ω1 § ω2 .

Similar reasoning gives locally de¬ned functions q, y such that

’dq § dy = » ω 3 § ω4 .

In terms of these functions, note that

d(» ω0 ) = »(ω1 § ω2 + ω3 § ω4 ) = ’dp § dx ’ dq § dy,

which by the Poincar´ lemma implies that there is another locally de¬ned func-

e

tion z such that

»ω0 = dz ’ p dx ’ q dy.

2.1. THE EQUIVALENCE PROBLEM FOR n = 2 51

The linear independence of ω 0 , . . . , ω4 implies that pulled back by any 1-adapted

coframe (that is, any section of B1 ), the functions x, y, z, p, q form local coordi-

nates on M . In terms of these local coordinates, our hyperbolic Monge-Ampere

system is

{ω0 , ω1 § ω2 + ω3 § ω4 , ω1 § ω2 ’ ω3 § ω4 }

E = (2.17)

{dz ’ p dx ’ q dy, dp § dx + dq § dy, dp § dx ’ dq § dy}. (2.18)

=

In an obvious way, transverse local integral surfaces of E are in one-to-one

correspondence with solutions to the wave equation for z(x, y)

‚2z

= 0.

‚x ‚y

This establishes the following.

Theorem 2.1 A hyperbolic Monge-Ampere system (M 5 , E) satis¬es S1 = S2 =

0 if and only if it is locally equivalent to the Monge-Ampere system (2.18) for

the linear homogeneous wave equation.

This gives us an easily computable method for determining when a given second-

order scalar Monge-Ampere equation in two variables is contact-equivalent to

this wave equation.

Looking at the equation (2.16) for d•0 (mod {I}), it is natural to ask about

0

the situation in which S2 = 0, but possibly S1 = 0. This gives an alternative

version of the solution to the inverse problem discussed in the previous chapter.

Theorem 2.2 A hyperbolic Monge-Ampere system (M 5, E) is locally equivalent

to an Euler-Lagrange system if and only if its invariant S2 vanishes identically.

Proof. The condition for our E to contain a Poincar´-Cartan form

e