d ωi + • § ωj = ’(Sj ωj + U ij πj ) § θ + T ijk πj § ωk ,

i

πi πj 0

(3.35)

where enough torsion has been absorbed so that

j

T ijk = T jik = T kji, T iik = 0; U ij = U ji; Sj = Si , Si = 0.

i i

(3.36)

We also recall the structure equation (2.48)

(n ’ 2)dρ = ’δi § ωi ’ Sj πi § ωj +

i

U ij σij § θ ’ ti πi § θ.

n’2

(3.37)

2n

The equations (3.34, 3.35, 3.36, 3.37) uniquely determine the forms ρ, ±i , and

j

we are still free to alter our pseudo-connection by

δi ; δi + bi θ + tij ωj , with tij = tji and tii = 0,

(3.38)

σij ; σij + tij θ + tijk ωk , with tijk = tjik = tkji and tiik = 0,

requiring also

2nbi + (n ’ 2)U jk tijk = 0.

We set up our example (3.33) by taking coordinates (xi , u, qi) on M =

J 1 (Rn, R), with contact form

˜ def

θ = du ’ qidxi.

Then transverse Legendre submanifolds which are also integral manifolds of

˜ def

Ψ = ’dqi § dx(i) + f(x, u)dx

correspond locally to solutions of (3.33). One can verify that the form

˜ def ˜ ˜

Π = θ§Ψ

104 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

is closed, so in particular our Poisson equation is an Euler-Lagrange equation.

We ¬nd a particular 1-adapted coframing of J 1 (Rn, R) as in Lemma 2.3 by

writing

˜ ˜

Π = ’θ § (dqi ’ f dxi ) § dx(i) ,

n

and then setting « «

˜ du ’ qidxi

θ

ωi = .

dxi

˜

f

dqi ’ n dxi

πi

˜

It turns out that this coframing is actually a section of B ’ J 1 (Rn, R), as one

discovers by setting

˜

ρ = 0, ±i = 0, δi = ’ n fu ωi ,

1

˜ ˜j ˜

and noting that the structure equations (3.34, 3.35) hold (with some compli-

cated choice of σij which we will not need). In fact, (3.35) holds with torsion

˜

coe¬cients Sj , U ij , T ijk all vanishing, and we will see the signi¬cance of this

i

presently.

In the general setting, we seek conditions under which the quadratic form

on B

def

(ωi )2

q=

can be regarded as de¬ning a conformal structure on some quotient of B. For

the appropriate quotient to exist, at least locally, the necessary and su¬cient

condition is that the Pfa¬an system I = {ω 1 , . . . , ωn } be integrable; it is easily

seen from the structure equations (and we noted in §2.4) that this is equivalent

to the condition

U ij = 0.

We assume this in what follows, and for convenience assume further that the

foliation of B by leaves of I is simple; that is, there is a smooth manifold N and

a surjective submersion B ’ N whose ¬bers are the leaves of I. Coordinates

on N may be thought of as “preferred independent variables” for the contact-

equivalence class of our Euler-Lagrange PDE, as indicated in §2.4.

We can now compute the Lie derivative of q under a vector ¬eld v which is

vertical for B ’ N , satisfying v ω i = 0; using the hypothesis U ij = 0 and the

structure equations, we ¬nd

Lv q = 2 T ijk(v πj )ωi ωk + Sj (v

i

θ)ωi ωj + 4(v ρ)q.

It follows that if T ijk = 0 and Sj = 0, then there is a quadratic form on N

i

which pulls back to a non-zero multiple of q on B. A short calculation shows

that the converse as true as well, so we have the following.

Proposition 3.3 The conditions U ij = T ijk = Sj = 0 are necessary and suf-

i

¬cient for there to exist (locally) a conformal manifold (N, [ds 2]) and a map

B ’ N such that the pullback to B of [ds2] is equal to [q] = [ (ωi )2 ].

3.3. CONFORMAL BRANCH OF THE EQUIVALENCE PROBLEM 105

From now on, we assume U ij = Sj = T ijk = 0.

i

From the discussion of the conformal equivalence problem in §3.1.2, we know

that associated to (N, [ds2]) is the second-order conformal frame bundle P ’ N

¯ ¯ ¯j ¯

with global coframing ω i , ρ, ±i , βi satisfying structure equations

± i i i

¯j

d¯ ’ 2¯ § ω + ±j § ω = 0,

ω 1 ρ ¯i ¯

d¯ + β § ω = 0,

ρ 2 ¯i ¯

(3.39)

¯¯ ¯ 1¯

d¯ i + ±i § ±k + βi § ωj ’ βj § ωi = 2 Ai ωk § ωl ,

±j ¯ k ¯ j ¯ jkl ¯ ¯

¯

ρ¯ ¯ 1¯

j

dβi + 2¯ § βi + βj § ±i = 2 Bijk ωj § ωk .

¯ ¯ ¯

Our goal is to directly relate the principal bundle B ’ M associated to the

Poincar´-Cartan form Π on M to the principal bundle P ’ N associated to the

e

induced conformal geometry on N . We shall eventually ¬nd that under some

further conditions stated below, the main one of which re¬‚ects the non-linearity

of the Euler-Lagrange system associated to Π, there is a canonical reduction

B3 ’ M of the G-structure B ’ M such that locally B3 ∼ P as parallelized

=

4

manifolds. Because the canonical coframings on B3 and P determine the bun-

dle structure of each, we will then have shown that the subbundle B3 ’ N

of B ’ N can be locally identi¬ed with the bundle P ’ N associated to the

conformal structure (N, [ds2]).

In the special case of our Poisson equation, we have ω i = dxi as part of a

˜

section of B ’ M , so we can already see that our quotient space N ∼ Rn is

=

conformally ¬‚at. This re¬‚ects the fact that the di¬erential operator ∆ in (3.33)

is the conformal Laplacian for ¬‚at conformal space.

We return to the general case, and make the simplifying observation that

under our hypotheses,

0 = d2(ω1 § · · · § ωn ) = ’ n’2 tiπi § θ § ω1 § · · · § ωn ,

2n

so that ti = 0 in the equation (3.37) for dρ. We now have on B the equations

dωi 2ρ § ωi ’ ±i § ωj ,

= (3.40)

j

’ n’2 δi § ωi .

1

dρ = (3.41)

With the goal of making our structure equations on B resemble the conformal

structure equations (3.39), we de¬ne

def 2

βi = δi .

n’2

The equations for dω i and dρ are now formally identical to those for d¯ i and d¯,

ω ρ

and computing exactly as in the conformal equivalence problem, we ¬nd that

d±i + ±i § ±k + βi § ωj ’ βj § ωi = 1 Ai ωk § ωl ,

j k j 2 jkl

4 As in the characterization in §2.5 of prescribed mean curvature systems, we will denote

by B1 , B2 , etc., reductions of the bundle B ’ M associated to Π, and these are unrelated to

the bundles of the same names used in the construction of B.

106 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

for some functions Ai on B having the symmetries of the Riemann curvature

jkl

tensor.

¯

Of course, we want Ai to correspond to the Weyl tensor Ai of (N, [ds2]),

jkl jkl

so we would like to alter our pseudo-connection forms (3.34) in a way that will

give

Al = 0.

jkl

Again, reasoning exactly as we did in the conformal equivalence problem, we

know that there are uniquely determined functions tij = tji such that replacing

βi ; βi + tij ωj

accomplishes this goal. However, these may have tii = 0, meaning that we

cannot make the compensating change in σij (see (3.38)) without introducing

torsion in the equation for dπi. We proceed anyway, and now have structure

equations

« « « «

’πi § ωi

(n ’ 2)ρ 0 0

θ θ

d ωi + § ωj =

±i ’2ρδj

i

0 0 0

j

j j i

n’2 Aω § θ

πi πj

nρδi ’±i

βi σij

2

(3.42)

where A = n’2 tii is a component of the original Ai , analogous to scalar

jkl

2n

curvature in the Riemannian setting. Also, we have

dρ = ’ 1 βi § ωi , (3.43)

2

d±i + ±i § ±k + βi § ωj ’ βj § ωi = 2 Ai ωk § ωl ,

1

(3.44)

j k j jkl

with Al = 0. These uniquely determine the pseudo-connection forms ρ, ±i , j

jkl

βi , and leave σij determined only up to addition of terms of the form tijkωk ,

with tijk totally symmetric and trace-free.

Now that βi is uniquely determined, we can once again mimic calculations

from the conformal equivalence problem, deducing from (3.40, 3.43, 3.44) that

dβi + 2ρ § βi + βj § ±j = 2 Bijk ωj § ωk ,

1

(3.45)

i

with Bijk + Bikj = 0, Bijk + Bjki + Bkij = 0.

In the case of our non-linear Poisson equation (3.33), a calculation shows that

˜ 2˜

the modi¬cation of βi = n’2 δi = 0 is not necessary, and that with everything

de¬ned as before, we have not only (3.42), but also (3.43, 3.44, 3.45) with

Ai = Bijk = 0. This gives us another way of seeing that the conformal

jkl

structure associated to (3.33) is ¬‚at. What will be important for us, however,

is the fact that along this section of B ’ J 1 (Rn, R), the torsion function A is

˜ 1

A= f (x, u).

nu