i i

(3.6)

where t = (tk ) ∈ Rn is arbitrary. This fact is needed for the next step of the

equivalence method, which consists of prolonging our CO(n, R)-structure. We

3.1. BACKGROUND MATERIAL ON CONFORMAL GEOMETRY 87

now digress to explain this general concept, starting with the abstract machinery

underlying the preceding calculation.

We begin by amplifying the discussion of normalizing torsion in §2.1. Asso-

ciated to any linear Lie algebra g ‚ gl(n, R) is an exact sequence of g-modules

δ 2

0 ’ g(1) ’ g — (Rn)— ’ Rn — (Rn)— ’ H 0,1(g) ’ 0. (3.7)

Here, the map δ is the restriction to the subspace

g — (Rn)— ‚ (Rn — (Rn )— ) — (Rn)—

of the surjective skew-symmetrization map

2

Rn — (Rn)— — (Rn )— ’ Rn — (Rn)— .

The space g(1) is the kernel of this restriction, and is called the prolongation of

g; the cokernel H 0,1(g), a Spencer cohomology group of g, was encountered in

§2.1. Note that g(1) and H 0,1(g) depend on the representation g ’ gl(n, R),

and not just on the abstract Lie algebra g.

Recall from §2.1 that the intrinsic torsion of a G-structure vanishes if and

only if there exist (locally) torsion-free pseudo-connections in that G-structure.

This is a situation in which further canonical reduction of the structure group

is not generally possible. In particular, this will always occur for G-structures

with H 0,1(g) = 0.

In this situation, the torsion-free pseudo-connection is unique if and only if

g = 0. For example, when g = so(n, R), both g(1) = 0 and H 0,1(g) = 0, which

(1)

accounts for the existence and uniqueness of a torsion-free, metric-preserving

connection on any Riemannian manifold. In this favorable situation, we have

essentially completed the method of equivalence, because the tautological form

and the unique torsion-free pseudo-connection constitute a canonical, global

coframing for the total space of our G-structure. Equivalences of G-structures

correspond to isomorphisms of the associated coframings, and there is a sys-

tematic procedure for determining when two parallelized manifolds are locally

isomorphic.

However, one frequently works with a structure group for which g(1) = 0.

The observation that allows us to proceed in this case is that any pseudo-

connection • in a G-structure P ’ N de¬nes a particular type of g•Rn -valued

coframing

• • ω : T P ’ g • Rn (3.8)

of the total space P . Our previous discussion implies that given some torsion-free

pseudo-connection •, any change • lying in g(1) ‚ g — (Rn )— yields a pseudo-

connection • + • which is also torsion-free. This means that the coframings

of P as in (3.8), with • torsion-free, are exactly the sections of a g(1)-structure

P (1) ’ P , where we regard g(1) as an abelian Lie group. This P (1) ’ P

88 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

is by de¬nition the prolongation of the G-structure P ’ N , and di¬erential

invariants of the former are also di¬erential invariants of the latter.1 The next

natural step in studying P ’ N is therefore to start over with P (1) ’ P , by

choosing a pseudo-connection, absorbing and normalizing its torsion, and so

forth.

In practice, completely starting over would be wasteful. The total space

P supports tautological forms • and ω, valued in g and Rn , respectively; and

(1)

the equation dω + • § ω = 0 satis¬ed by any particular torsion-free psuedo-

connection • on P still holds on P (1) with • replaced by a tautological form.

We can therefore di¬erentiate this equation and try to extract results about the

algebraic form of d•. These results can be interpreted as statements about the

intrinsic torsion of P (1) ’ P . Only then do we return to the usual normalization

process. We will now illustrate this, returning to our situation in the conformal

structure equivalence problem.

We have shown the existence of torsion-free pseudo-connections •i in the

j

CO(n, R)-structure P0 ’ N , so the intrinsic torsion of P0 ’ N vanishes.2

We also have that such •i are unique modulo addition of a semibasic co(n, R)-

j

valued 1-form linearly depending on an arbitrary choice of (tk ) ∈ Rn. Therefore

co(n, R)(1) ∼ Rn , and the inclusion co(n, R)(1) ’ co(n, R) — (Rn)— is described

=

def

by (3.6). As explained above, we have an Rn -structure P = (P0 )(1) ’ P0,

whose sections correspond to torsion-free pseudo-connections in P0 ’ N . Any

choice of the latter trivializes P ’ P0, and then (tk ) ∈ Rn is a ¬ber coordinate.

We now search for structure equations on P , with the goal of identifying a

canonical Rn-valued pseudo-connection form for P ’ P0.

The ¬rst structure equation is still

dωi = ’•i § ωj ,

j

where

±i + ±j = 0,

•i = ’2δj ρ + ±i ,

i

j j j i

and ±i , ρ are tautological forms on P . Di¬erentiating this gives

j

(d•i + •i § •k ) § ωj = 0, (3.9)

j k j

so

d•i + •i § •k ≡ 0 (mod {ω1 , . . . , ωn}).

j k j

Taking the trace of this equation of matrix 2-forms shows that dρ ≡ 0, so guided

by the ¬‚at model (3.2), we write

dρ = ’ 1 βi § ωi (3.10)

2

1 Situations with

non-unique torsion-free pseudo-connections are not the only ones that call

for prolongation; sometimes one ¬nds intrinsic torsion lying in the ¬xed set of H 0,1 (g), and

essentially the same process being described here must be used. However, we will not face

such a situation.

2 In fact, what we proved is that δ is surjective for g = co(n, R), so H 0,1 (co(n, R)) = 0.

3.1. BACKGROUND MATERIAL ON CONFORMAL GEOMETRY 89

for some 1-forms βi which are not uniquely determined. We will recognize these

below as pseudo-connection forms in P ’ P0, to be uniquely determined by

conditions on the torsion which we will uncover shortly. Substituting (3.10)

back into (3.9), we have

(d±i + ±i § ±k ’ βj § ωi + βi § ωj ) § ωj = 0,

j k j

and we set

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

j j k j

Note that Ai + Aj = 0. We can write

j i

Ai = ψjk § ωk

i

j

for some 1-forms ψjk = ψkj , in terms of which the condition Ai + Aj = 0 is

i i

j i

j

(ψjk + ψik ) § ωk = 0,

i

which implies

j

i

(mod {ω1 , . . . , ωn}).

ψjk + ψik ≡ 0

Now computing modulo {ω 1 , . . . , ωn } as in Riemannian geometry, we have

j j

i k k i i

ψjk ≡ ’ψik ≡ ’ψki ≡ ψji ≡ ψij ≡ ’ψkj ≡ ’ψjk , (3.11)

i

so ψjk ≡ 0. We can now write

Ai = ψjk § ωk = 2 Ai ωk § ωl ,

i 1

j jkl

and forget about the ψjk , as our real interest is in d±i . We can assume that

i

j

j

i i i

Ajkl + Ajlk = 0, and we necessarily have Ajkl + Aikl = 0. Substituting once

more into Ai § ωj = 0, we ¬nd that

j

Ai + Ai + Ai = 0.

jkl klj ljk

In summary, we have

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

j k j 2 jkl

where Ai has the symmetries of the Riemann curvature tensor.

jkl

In particular, we need only n new 1-forms βi to express the derivatives of dρ,

d±j . The βi are pseudo-connection forms for the prolonged co(n, R)(1)-bundle

i

P ’ P0 , chosen to eliminate torsion in the equation for dρ, while the functions

Ai constitute the torsion in the equations for d±i . Some of this torsion will

j

jkl

now be absorbed in the usual manner, by making a uniquely determined choice

of βi .

Notice that the equation (3.10) for dρ is preserved exactly under substitu-

tions of the form

βi ; βi + sij ωj , sij = sji.

90 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

This substitution will induce a change

Ai ; Ai + (’δli sjk + δlj sik + δk sjl ’ δk sil ).

j

i

jkl jkl

Now, we know from the symmetries of the Riemann curvature tensor that

Al = A l ,

jkl kjl

and on this contraction (the “Ricci” component) our substitution will induce

the change

Al ; Al ’ (n ’ 2)sjk ’ δjk sll .

jkl jkl

As we are assuming n ≥ 3, there is a unique choice of sij which yields

Al = 0.

jkl

It is not di¬cult to compute that the appropriate sij is given by

Al ’ δ Al

1 1

sij = .

2n’2 ij kkl

ijl

n’2

In summary,

On P , there is a unique coframing ω i , ρ, βj , ±i = ’±j , where ωi are

j i

the tautological forms over N , and such that the following structure

equations are satis¬ed:

dωi (2δj ρ ’ ±i ) § ωj ,

i

= j

’ 1 βi § ω i ,

dρ = 2

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

=

j k j 2 jkl

with Al = 0.

jkl

We now seek structure equations for dβj . We start by di¬erentiating the

simplest equation in which βj appears, which is dρ = ’ 2 βj § ωj , and this gives

1

(dβj + 2ρ § βj + βk § ±k ) § ωj = 0.

j

We write

dβj + 2ρ § βj + βk § ±k = Bjk § ωk (3.12)

j

for some 1-forms Bjk = Bkj . Because the equation for dρ did not determine