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Ψ = βi § ω(i) ’ n’2 ω.



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

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