r j

2

g = 0 ai r’2 ai bk , (3.3)

j k

’2

00 r

where r > 0, ai aj = δ ij . Now, the symmetric di¬erential form on P given by

kk

(ωi )2

q=

is semibasic for πR : P ’ R, and a Lie derivative computation using the struc-

ture equations (3.2) gives, for any vertical vector ¬eld v ∈ Ker (πR )— ,

Lv q = 4(v ρ)q.

This implies that there is a unique conformal structure [ds2] on R whose rep-

—

resentative metrics pull back under πR to multiples of q. By construction, this

conformal structure is invariant under the action of SO o (n + 1, 1), and one can

verify that it gives the same structure as the classical construction described

above.

In §3.1.2, we will follow Cartan in showing that associated to any conformal

structure (N, [ds2]) is a principal bundle P ’ N with 1-forms ±i = ’±j , ρ,

j i

i

ω , and βj , satisfying structure equations like (3.2) but with generally non-zero

curvature terms on the right-hand side.

Before doing this, however, we point out a few more structures in the ¬‚at

model which will have useful generalizations. These correspond to Pfa¬an sys-

tems

IR = {ωi }, IQ = {ωi , ρ}, IM = {ωi , ρ, βj }, IP0 = {ωi , ρ, ±i },

j

3.1. BACKGROUND MATERIAL ON CONFORMAL GEOMETRY 83

each of which is integrable, and in fact has a global quotient; that is, there are

manifolds R, Q, M , and P0, and surjective submersions from P to each of these,

whose leaves are the integral manifolds of IR , IQ , IM , and IP0 , respectively:

P

P0 M

Q

“

R.

We have already seen that the leaves of the system IR are ¬bers of the map

πR : P ’ R. Similarly, the leaves of IQ are ¬bers of the map πQ : P ’ Q given

by

πQ : (e0 , . . . , en+1) ’ e0 .

To understand the leaves of IM , we let M be the set of ordered pairs (e, e )

of positive null vectors satisfying e, e = ’1. We then have a surjective sub-

mersion πM : P ’ M de¬ned by

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

and the ¬bers of this map are the leaves of the Pfa¬an system IM . Note that

the 1-form ρ and its exterior derivative are semibasic for πM : P ’ M , and this

means that there is a 1-form (also called ρ) on M which pulls back to ρ ∈ „¦1(P ).

In fact, the equation for dρ in (3.2) shows that on P ,

ρ § (dρ)n = 0,

so the same is true on M . Therefore, ρ de¬nes an SO o (n+1, 1)-invariant contact

structure on M . The reader can verify that M has the structure of an R— -bundle

over the space G(1,1)(Ln+2 ) parameterizing those oriented 2-planes in Ln+2 on

which the Lorentz metric has signature (1, 1). In this context, 2ρ ∈ „¦1(M ) can

be interpreted as a connection 1-form.

Finally, to understand the leaves of IP0 , we proceed as follows. De¬ne a con-

formal frame for (R, [ds2]) at a point [x] ∈ R to be a positive basis (v1, . . . , vn)

for T[x] R normalizing the conformal inner-product as

ds2 (vi , vj ) = »δij ,

for some » ∈ R— not depending on i, j. The set of conformal frames for (R, [ds2])

is the total space of a principal bundle P0 ’ R, and there is a surjective sub-

mersion P ’ P0. This last is induced by the maps ei : P ’ T R associating to

a Lorentz frame f = (e0 , . . . , en+1 ) an obvious tangent vector ei to R at [e0].

The reader can verify that the ¬bers of the map P ’ P0 are the leaves of the

Pfa¬an system IP0 .

Each of the surjective submersions P ’ R, P ’ Q, P ’ M , P ’ P0 has

the structure of a principal bundle, de¬ned as a quotient of P by a subgroup of

84 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

SOo (n + 1, 1). Additionally, the spaces P , R, Q, M , and P0 are homogeneous

spaces of SOo (n + 1, 1), induced by the standard left-action on Ln+2 .

We conclude with a brief description of the geometry of SO 0 (n + 1, 1) acting

on ¬‚at conformal space R. This will be useful later in understanding the space

of conservation laws of conformally invariant Euler-Lagrange equations. There

are four main types of motions.

• The translations are de¬ned as motions of R induced by left-multiplication

by matrices of the form

«

1 00

wi In 0 . (3.4)

2

||w|| j

w1

2

In the standard coordinates on R\{∞} described in (3.1), this is simply

translation by the vector (w i ).

• The rotations are de¬ned as motions of R induced by matrices of the form

«

100

0 ai 0 ,

j

001

where (ai ) ∈ SO(n, R). In the standard coordinates, this is the usual

j

rotation action of the matrix (ai ).

j

• The dilations are de¬ned as motions of R induced by matrices of the form

«2

r0 0

0 I 0 .

0 0 r’2

In the standard coordinates, this is dilation about the origin by a factor

of r’2.

• The inversions are de¬ned as motions of R induced by matrices of the

form « 2

1 bj ||b||

2

0 I bi .

00 1

Note that these are exactly conjugates of the translation matrices (3.4) by

the matrix «

001

J = 0 I 0 .

100

Now, J itself is not in SO o (n + 1, 1), but it still acts in an obvious way on

R; in standard coordinates, it gives the familiar inversion in the sphere of

√

radius 2. So the inversions can be thought of as conjugates of translation

by the standard sphere-inversion, or alternatively, as “translations with

the origin ¬xed”.

3.1. BACKGROUND MATERIAL ON CONFORMAL GEOMETRY 85

These four subgroups generate SO o (n + 1, 1). Although the conformal isometry

group of R has more that this one component, the others do not appear in

the Lie algebra, so they do not play a role in calculating conservation laws for

conformally invariant Euler-Lagrange equations.

3.1.2 The Conformal Equivalence Problem

We will now apply the method of equivalence to conformal structures of dimen-

sion n ≥ 3. This will involve some of the ideas used in the equivalence problem

for de¬nite Poincar´-Cartan forms discussed in the preceding chapter, but we

e

we will also encounter the new concept of prolongation. This is the step that

one takes when the usual process of absorbing and normalizing the torsion in a

G-structure does not uniquely determine a pseudo-connection.

Let (N, [ds2]) be an oriented conformal manifold of dimension n ≥ 3, and

let P0 ’ N be the bundle of 0th -order oriented conformal coframes ω =

(ω1 , . . . , ωn), which by de¬nition satisfy

[ds2] = (ωi )2 , ω1 § · · · § ωn > 0.

This is a principal bundle with structure group

CO(n, R) = {A ∈ GL+ (n, R) : A tA = »I, for some » ∈ R— },

having Lie algebra

{a ∈ gl(n, R) : a + ta = »I, for some » ∈ R}

co(n, R) =

{(’2rδj + ai ) : ai + aj = 0, ai , r ∈ R}.

i

= j j j

i

We will describe a principal bundle P ’ P0 , called the prolongation of P0 ’ N ,

whose sections correspond to torsion-free pseudo-connections in P0 ’ N , and

construct a canonical parallelism of P which de¬nes a Cartan connection in

P ’ N . In case (N, [ds2 ]) is isomorphic to an open subset of ¬‚at conformal

space, this will correspond to the restriction of the Lorentz frame bundle P ’ R

to that open subset, with parallelism given by the Maurer-Cartan forms of

SOo (n + 1, 1) ∼ P .

=

Recall that a pseudo-connection in P0 ’ N is a co(n, R)-valued 1-form

±i + ±j = 0,

• = (•i ) = (’2ρδj + ±i ),

i

j j j i

whose restriction to each tangent space of a ¬ber of P0 ’ N gives the canonical

identi¬cation with co(n, R) induced by the group action. As discussed previ-

ously (see §2.1), this last requirement means that • satis¬es a structure equation

dωi = ’•i § ωj + 2 Tjk ωj § ωk ,

1i i i

Tjk + Tkj = 0, (3.5)

j

where ωi are the components of the tautological Rn-valued 1-form on P0, and

1i

T ωj §ωk is the semibasic Rn -valued torsion 2-form. We also noted previously

2 jk

86 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

that a psuedo-connection • is a genuine connection if and only if it is Ad-

equivariant for the action of CO(n, R) on P0 , meaning that

R— • = Adg’1 (•),

g

where Rg : P0 ’ P0 is the right-action of g ∈ CO(n, R) and Adg’1 is the

adjoint action on co(n, R), where • takes its values. However, completing this

equivalence problem requires us to consider the more general notion of a pseudo-

connection. Although the parallelism that we eventually construct is sometimes

called the “conformal connection”, there is no canonical way (that is, no way

that is invariant under all conformal automorphisms) to associate to a conformal

structure a linear connection in the usual sense.

What we seek instead is a psuedo-connection •i for which the torsion van-

j

i

ishes, Tjk = 0. We know from the fundamental lemma of Riemannian geome-

try, which guarantees a unique torsion-free connection in the orthonormal frame

bundle of any Riemannian manifold, that whatever structure equation (3.5) we

have with some initial pseudo-connection, we can alter the pseudo-connection-

forms ±i = ’±j to arrange that Tjk = 0. Speci¬cally, we replace

i

j i

j

±i ; ±i + 2 (Tjk ’ Tik ’ Tij )ωk .

i k

1

j j

i

So we can assume that Tjk = 0, and we have simply

dωi = ’•i § ωj = ’(’2ρδj + ±i ) § ωj ,

i

j j

with ±i + ±j = 0. However, in contrast to Riemannian geometry, this condition

j i

on the torsion does not uniquely determine the psuedo-connection forms ρ, ±i .

j

If we write down an undetermined semibasic change in psuedo-connection

ρ ; ρ + tk ω k ,

ti + tj = 0,

±i ; ± i + t i ω k ,

j j jk jk ik

then the condition that the new pseudo-connection be torsion-free is that

(2δj tk ’ ti )ωj § ωk = 0.

i

jk

This boils down eventually to the condition

j

ti = 2(δk ti ’ δk tj ).

i

jk

Therefore, given one torsion-free pseudo-connection •i in P0 ’ N , the most

j

general is obtained by adding

j