d» = ’4»ρ + »i ωi (3.21)

for some functions »i . Di¬erentiating again, we ¬nd

d»i = ’2»βi ’ 6»i ρ + »j ±j + »ij ωj , (3.22)

i

for some functions »ij = »ji . Now we can reduce our bundle P ’ N to a

subbundle Pg ‚ P , de¬ned by

Pg = {p ∈ P : »(p) = 1, »1 (p) = · · · = »n (p) = 0}.

Equations (3.21, 3.22) imply that Pg has structure group SO(n, R) ‚ G, and

using bars to denote restrictions to Pg , we have for the pseudo-connection forms

¯ 1¯

βi = 2 »ij ωj , d¯ i = ’¯ i § ωj .

ρ = 0,

¯ ¯ ω ±j ¯

The last of these means that if we identify Pg with the usual orthonormal frame

bundle of (N, g), then ±i gives the Levi-Civita connection. The curvature is by

¯j

de¬nition

d¯ i + ±i § ±k = 1 Ri ωk § ωl ,

±j ¯ k ¯ j 2 jkl ¯ ¯

but we have an expression for the left-hand side coming from the conformal

geometry; namely,

¯¯ ¯

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

±j ¯ k ¯ j ¯ 2 jkl ¯ ¯

¯ ¯¯

Substituting βi = 1 »ij ωj and comparing these two expressions gives

2

¯ ¯ i¯ j¯

Ri = 1 (δli »jk ’ δlj »ik ’ δk »jl + δk »il ) + Ai .

jkl jkl

2

From this we ¬nd the other components of curvature

¯ ¯

Ricjl = Ri = 2 ((2 ’ n)»jl ’ δlj »ii ),

1

jil

¯

R = Ricll = (1 ’ n)»ii .

96 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

n’2

Now we will compute the conformal Laplacian of u ∈ “(D 2n ), but restrict

the computation to Pg . Note that the choice of g amounts to a trivialization

of D and of all of its powers, so in this setting it is correct to think of u as a

function. We have

’(n ’ 2)¯u + uiωi

d¯ =

u ρ¯ ¯ ¯

ui ω i ,

= ¯¯

¯

’ n’2 uβi ’ n¯iρ + uj ±j + uij ωj

d¯i

u = 2¯ u ¯ ¯ ¯i ¯ ¯

¯¯ ω

uj ±j + (¯ij ’ n’2 u»ij )¯ j .

= ¯¯ u

i 4

Denoting by ∆g the Riemannian Laplacian, we now have

¯

uii ’ n’2 u»ii

∆g u =

¯ ¯ 4¯

n’2

= ∆¯ + 4(n’1) R¯.

u u

This is the more familiar expression for the conformal Laplacian, de¬ned in

terms of the Riemannian Laplacian of some representative metric. In the case

of the ¬‚at model of conformal geometry, if one uses standard coordinates on

Rn = R\{∞}, then the Euclidean metric represents the conformal class, and

we can use the ordinary Laplacian ∆ = ( ‚xi )2 . Its transformation properties,

‚

often stated and proved with tedious calculations, can be easily derived from

the present viewpoint.

Of particular interest to us will be non-linear Poisson equations, of the form

∆u = f(xi , u), (3.23)

where we will have an interpretation of ∆ as the conformal Laplacian on a

conformal manifold with coordinates xi. We will therefore want to interpret

n’2

the unknown u as a section of the density bundle D 2n , and we will want to

interpret f(x, u) as a (0th -order) bundle map

n’2 n+2

’D

f :D .

2n 2n

Certain obvious bundle maps f come to mind. One kind is given by multipli-

cation by any section » ∈ “(D 2/n ); this would make (3.23) a linear equation.

Another is the appropriate power map

n+2

u ’ u n’2 .

This yields a non-linear Poisson equation, and we will examine it quite closely

in what follows.

We conclude this discussion with an alternate perspective on the density

bundles Ds/n. First, note that for any conformal manifold (N, [ds2]) with its

associated parallelized bundle P ’ N , the Pfa¬an system

IQ = {ρ, ω1 , . . . , ωn}

´

3.2. CONFORMALLY INVARIANT POINCARE-CARTAN FORMS 97

is integrable, and its associated foliation is simple. The leaf space of this foliation

is just the quotient Q of P by the action of a subgroup of its structure group,

and this Q is also a ¬ber-bundle over N , with ¬ber R— . This generalizes the

space Q of positive null vectors in Ln+2 which appeared in the discussion of the

¬‚at model. Now, the density bundles D s/n are all canonically oriented, and we

claim that Q is canonically identi¬ed with the positive elements of D s/n , for any

s.

To see this, note that any positive u ∈ D s/n, over x ∈ N , is de¬ned as a

positive function on the ¬ber Px ‚ P satisfying (3.17). It is not hard to see

that the locus {p ∈ Px : u(p) = 1} ‚ Px is a leaf of the foliation de¬ned by IQ .

Conversely, let LQ ‚ P be a leaf of the foliation de¬ned by IQ . Then LQ lies

completely in some ¬ber Px of P ’ N , and we can de¬ne a function u on Px by

setting u = 1 on LQ , and extending to Px by the rule (3.17). These are clearly

inverse processes.

s/n

We can extend the identi¬cation as follows. Let J 1 (N, D+ ) be the space of

1-jets of positive sections of D; it is a contact manifold, in the usual manner. Let

M be the leaf space of the simple foliation associated to the integrable Pfa¬an

system on P

IM = {ρ, ω1, . . . , ωn, β1 , . . . , βn}.

This M is also a contact manifold, a with global contact form pulling back to ρ ∈

„¦1 (P ), and it generalizes the contact manifold M mentioned in our discussion of

the ¬‚at model. We claim that there is a canonical contact isomorphism between

s/n

J 1 (N, D+ ) and M .

To see this, note that a 1-jet at x ∈ N of a positive section of D s/n is speci¬ed

by n + 1 functions (u, u1, . . . , un) on the ¬ber Px satisfying (3.17, 3.20). It is

then not hard to see that the locus {p ∈ Px : u(p) = 1, ui (p) = 0} ‚ Px is

a leaf of the foliation de¬ned by IM . Conversely, let LM ‚ P be a leaf of the

foliation de¬ned by IM . Then LM lies completely in some ¬ber Px of P ’ N ,

and we can de¬ne n + 1 functions (u, u1, . . . , un) on Px by setting u = 1 and

ui = 0 on LM , and extending to Px by the rules (3.17, 3.20). These are again

inverse processes, and we leave it to the reader to investigate the correspondence

between contact structures.

3.2 Conformally Invariant Poincar´-Cartan

e

Forms

In this section, we identify the Poincar´-Cartan forms on the contact manifold

e

M over ¬‚at conformal space R that are invariant under the action of the con-

formal group SO o (n + 1, 1). We then specialize to one that is neo-classical, and

determine expressions for the corresponding Euler-Lagrange equation in coordi-

nates; it turns out to be the non-linear Poisson equation with critical exponent

n+2

∆u = Cu n’2 .

98 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

The calculation should clarify some of the more abstract constructions of the

preceding section. It will also be helpful in understanding the branch of the

equivalence problem in which this Poincar´-Cartan form appears, which is the

e

topic of the next section.

We denote by P the set of Lorentz frames for Ln+2 , by M the set of pairs

(e, e ) of positive null vectors with e, e = ’1, by Q the space of positive null

vectors, and by R the ¬‚at conformal space of null lines. There are SO o (n+1, 1)-

equivariant maps

±

πM : P ’ M, (e0 , . . . , en+1) ’ (e0 , en+1 ),

πQ : P ’ Q, (e0 , . . . , en+1) ’ e0 ,

πR : P ’ R, (e0 , . . . , en+1) ’ [e0 ].

For easy reference we recall the structure equations for Lorentz frames

±

de0 = 2e0 ρ + ei ωi ,

±i + ±j = 0,

dej = e0 βj + ei ±i + en+1 ωj , (3.24)

j j i

den+1 = ei βi ’ 2en+1ρ;

± i

1

dρ + 2 βi § ω = 0,

dωi ’ 2ρ § ωi + ±i § ωj = 0,

j

(3.25)

dβi + 2ρ § βi + βj § ±j = 0,

i

d±i + ±i § ±k + βi § ωj ’ βj § ωi = 0.

j j

k

We noted in the previous section that M has a contact 1-form which pulls back

to ρ, and this is the setting for our Poincar´-Cartan forms.

e

Proposition 3.1 The SO o (n + 1, 1)-invariant Poincar´-Cartan forms on M ,

e

pulled back to P , are constant linear combinations of

def

Πk = ρ § βI § ω(I),

|I|=k

where 0 ¤ k ¤ n. Those that are neo-classical with respect to Q are of the form

c0 , c1 ∈ R.

Π = c 1 Π1 + c 0 Π0 , (3.26)

Proof. In this setting, an invariant Poincar´-Cartan form on M , pulled back

e

to P , is an (n + 1)-form that is a multiple of ρ, semibasic over M , invariant

under the left-action of SO o (n + 1, 1), invariant under the right-action of the

isotropy subgroup SO(n, R) of M , and closed. That Π must be semibasic and

SOo (n + 1, 1)-invariant forces it to be a constant linear combination of exterior

products of ρ, βi , ωi. It is then a consequence of the Weyl™s theory of vector

invariants that the further conditions of being a multiple of ρ and SO(n, R)-

invariant force Π to be a linear combination of the given Πk . It follows from

the structure equations of P that dΠk = 0, so each Πk is in fact the pullback of

a Poincar´-Cartan form.

e

´

3.2. CONFORMALLY INVARIANT POINCARE-CARTAN FORMS 99

We note that for the n-form Λk de¬ned by

def

βI § ω(I)

Λk =

|I|=k

we have

dΛk = 2(n ’ 2k)Πk .

This means that for k = n , the Poincar´-Cartan form Πk is associated to an

e

2

o

SO (n + 1, 1)-invariant functional, which in the standard coordinates discussed

below is second-order. For the exceptional case n = 2k, there is no invariant

functional corresponding to Πk , but in the neo-classical case k ¤ 1 with n ≥ 3,

this is not an issue.

We now focus on the neo-classical case (3.26), for which it will be convenient

to rescale and study

2C

Π = ρ § βi § ω(i) ’ ω , (3.27)

n’2

where C is a constant. This is the exterior derivative of the Lagrangian

1 C

§ ω(i) ’

Λ= β ω,

2(n’2) i n(n’2)

and our Monge-Ampere di¬erential system is generated by ρ and the n-form

2C