Proposition 3.2 The Euler-Lagrange equation corresponding to the Poincar´-

e

Cartan form (3.27) is locally equivalent to

n+2

∆u = Cu n’2 . (3.28)

The meaning of “locally equivalent” will come out in the proof. It includes

an explicit and computable correspondence between integral manifolds of the

Monge-Ampere system and solutions to the PDE.

We remark that the PDEs corresponding to higher Poincar´-Cartan forms

e

Πk , with k > 1, have been computed and analyzed by J. Viaclovsky in [Via00].

Proof. We begin by de¬ning a map σ : J 1 (Rn , R) ’ P , which can be projected

to M to give an open inclusion of contact manifolds with dense image. This

map will be expressed in terms of the usual contact coordinates (xi , z, pi) on

J 1 (Rn, R), except that z is replaced by u = e»z for some undetermined constant

» = 0, so that in particular,

dz ’ pi dxi = (»u)’1du ’ pi dxi.

100 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

When using coordinates (xi , u, pi), we denote our jet space by J 1 (Rn, R+ ). We

then pull back Ψ via σ, and consider its restriction to a transverse Legendre

submanifold. With a convenient choice of », we will obtain a non-zero multiple

n+2

of ∆u ’ Cu n’2 , implying the Proposition.

We de¬ne σ as a lift of the following map Rn ’ P , to be extended to

J 1 (Rn , R+) shortly:

«

« «

0

1 0

¬.·

x1

¬ · ¬ ·

¬.· 0

¬.·

¬ · ¬ ·

.

¬ · ¬ 1i · , en+1 (x) = ¬ ·

.

. .

e0 (x) = ¬

¯ ·, ei (x) = ¬

¯ ·¯ ·. (3.29)

¬

. .

¬ · ¬ ·

¬.·

xn 0

..

||x||2 1

xi

2

It is easy to verify that this does take values in P . Also, note that the composi-

tion Rn ’ P ’ R gives standard (stereographic) coordinates on R\{∞}. This

partly indicates the notion of “locally equivalent” used in this Proposition. We

now let

u2k e0 (x),

e0 (x, u, p) = ¯

ei (x, u, p) = ei (x) + pi e0 (x),

¯ ¯ (3.30)

||p||2

u’2k (¯n+1 (x) + pj ej (x) +

en+1 (x, u, p) = e ¯ e0 (x)),

¯

2

for some constant k = 0 to be determined shortly. Our use of the dependent

variable u as a scaling factor for e0 re¬‚ects the fact that we expect u to represent

a section of some density line bundle. The formula for en+1 is chosen just so

that our map takes values in P .

Now we can compute directly

2ku’1e0 du + u2k ei dxi

de0 = ¯

2(ku’1du ’ 2 pidxi)e0 + (u2k dxi)ei ,

1

=

so by comparison with the expression in (3.24) we obtain some of the pulled-back

Maurer-Cartan forms:

σ— ωi u2k dxi,

=

σ— ρ ku’1 du ’ 2 pi dxi.

1

=

Similarly, we have

’ en+1 , dei

βi =

||p||2

’ u’2k (¯n+1 + pj ej +

= e ¯ 2 e0 ),

¯

en+1 dxi + pi ek dxk + e0 dpi

¯ ¯ ¯

||p||2

u’2k dpi ’ pi pj dxj + i

= 2 dx .

´

3.2. CONFORMALLY INVARIANT POINCARE-CARTAN FORMS 101

Because we want the projection to M of σ : J 1 (Rn , R+) ’ P to be a contact

mapping, we need σ— ρ to be a multiple of dz ’ pi dxi = (»u)’1 du ’ pidxi, which

holds if we choose

1

k = 2» .

Now, » is still undetermined, but it will shortly be chosen to simplify the ex-

pression for the restriction of Π to a transverse Legendre submanifold. Namely,

we ¬nd that n’2

βi § ω(i) = u » dpi § dx(i) + n’2 ||p||2dx ,

2

and also

n

ω = u » dx.

On transverse Legendre submanifolds, we have

du = »e»z dz = »upi dxi,

so that

1 ‚u

pi = .

»u ‚xi

Di¬erentiating, we obtain

1 ‚2u

1 1 ‚u ‚u

dxj ,

’2 i j

dpi =

u ‚xi ‚xj

» u ‚x ‚x

so that on transverse Legendre submanifolds,

1 ∆u n ’ 2 ’ 2» || u||2 2C

n’2

u2/» dx = 0.

’

Ψ= u» +

2»2 u2 n’2

»u

We can eliminate the ¬rst-order term by choosing

n’2

»= ,

2

and then

n+2

2u

∆u ’ Cu n’2

Ψ= dx,

n’2

which is the desired result.

Note that z = »’1 log u satis¬es a PDE that is slightly more complicated,

but equivalent under a classical transformation. Also, note that (3.28) is usually

given as the Euler-Lagrange equation of the functional

2n

u||2 +

1 n’2

|| Cu n’2 dx,

2 2n

which has the advantage of being ¬rst-order, but the disadvantage of not being

preserved by the full conformal group SO o (n+1, 1). In contrast, our Lagrangian

Λ restricts to transverse Legendre submanifolds (in the coordinates of the pre-

ceding proof) as the variationally equivalent integrand

2n

1 C

2 u∆u ’ n(n’2) u

Λ= dx.

n’2

(n’2)

102 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

3.3 The Conformal Branch of the Equivalence

Problem

Let (M 2n+1, Π) be a manifold with a non-degenerate Poincar´-Cartan form;

e

n+1

that is, Π ∈ „¦ (M ) is closed, and has a linear divisor that is unique modulo

scaling and de¬nes a contact structure. We also assume that n ≥ 3 and that

Π is neo-classical and de¬nite. Then as discussed in §2.4 we may associate to

(M, Π) a G-structure B ’ M , where G is a subgroup of GL(2n + 1, R) whose

Lie algebra consists of matrices of the form

«

(n ’ 2)r 0 0

,

’2rδj + ai

i

0 0 (3.31)

j

j j

nrδi ’ ai

di sij

where ai + aj = 0 and sij = sji, sii = 0. In this section, we show how to

j i

uniquely characterize in terms of the invariants of the G-structure those (M, Π)

which are locally equivalent to the Poincar´-Cartan form for the equation

e

n+2

∆u = Cu n’2 , C = 0. (3.32)

on ¬‚at conformal space. The result may be loosely summarized as follows.

The vanishing of the primary invariants T ijk, U ij , Sj is equivalent

i

to the existence of a foliation B ’ N over a conformal manifold

(N, [ds2 ]), for which [ds2 ] pulls back to the invariant [ (ωi )2 ]. In

this case, under open conditions on further invariants, three suc-

cessive reductions of B ’ M yield a subbundle which is naturally

identi¬ed with the conformal bundle over N . The Poincar´-Cartan

e

form can then be identi¬ed with that associated to a non-linear Pois-

son equation. In case a further invariant is constant, this equation

is equivalent to (3.32).

We ¬nd these conditions by continuing to apply the equivalence method

begun in §2.4, pursuing the case in which all of the non-constant torsion vanishes.

One corollary of the discussion is a characterization of Poincar´-Cartan forms

e

locally equivalent to those for general non-linear Poisson equations of the form

x ∈ N,

∆u = f(x, u), (3.33)

on an n-dimensional conformal manifold (N, [ds2 ]); here and in the following,

∆ is the conformal Laplacian. The condition that (3.33) be non-linear can be

characterized in terms of the geometric invariants associated to (M, Π), as can

the condition that (N, [ds2 ]) be conformally ¬‚at. The characterization of (3.32)

will imply that this equation has maximal symmetry group among non-linear

Euler-Lagrange equations satisfying certain geometric conditions on the torsion.

We will not actually prove the characterization result for general Poisson equa-

tions (3.33), but we will use these equations (in the conformally ¬‚at case, with

‚2

∆= ) as an example at each stage of the following calculations.

‚xi

3.3. CONFORMAL BRANCH OF THE EQUIVALENCE PROBLEM 103

We ¬rst recall the structure equations of the G-structure B ’ M , associated

to a neo-classical, de¬nite Poincar´-Cartan form

e

Π = ’θ § (πi § ω(i) ).

There is a pseudo-connection

«

(n ’ 2)ρ 0 0

±i + ±j = 0,

•= , with

’2ρδj + ±i

i j

0 0 i

j

σij = σji, σii = 0,

j j

nρδi ’ ±i

δi σij

(3.34)

having torsion

« « «

’πi § ωi