dβj ; we need to di¬erentiate the equations for d±i , substituting (3.12). This

j

gives

(DAi ’ Bik δlj + Bjk δli + Bil δk ’ Bjl δk ) § ωk § ωl = 0.

j i

jkl

3.1. BACKGROUND MATERIAL ON CONFORMAL GEOMETRY 91

Here we have de¬ned for convenience the “covariant derivative”

DAi = dAi + 4ρAi + ±i Am ’ Ai ±m ’ Ai ±m ’ Ai ±m . (3.13)

jkl jkl jkl m jkl mkl j jml k jkm l

Now we can write

DAi ’ Bik δlj + Bjk δli + Bil δk ’ Bjl δk ≡ 0 (mod {ω1, . . . , ωn }),

j i

jkl

and contracting on il gives

Bjk ≡ 0 (mod {ω1 , . . . , ωn}).

This allows us to write simply

dβi + 2ρ § βi + βj § ±j = 1 Bijk ωj § ωk ,

i 2

for some functions Bijk = ’Bikj . Returning to the equation

0 = d2ρ = ’ 1 d(βj § ωj )

2

now yields the cyclic symmetry

Bijk + Bjki + Bkij = 0.

We now have complete structure equations, which can be summarized in the

matrix form suggested by the ¬‚at model (3.2):

« «

2ρ βj 0 0 Bj 0

def def

φ = ω i ±i βi , ¦ = dφ + φ § φ = 0 Ai Bi , (3.14)

j j

j

’2ρ

0ω 00 0

where

Ai 1i k l

2 Ajkl ω § ω ,

=

j

Ai + A j = Ai + Ai = 0,

jkl jkl jlk

ikl

Ai + A i + Ai = Al = 0,

jkl klj ljk jkl

k l

1

2 Bjkl ω § ω ,

Bj =

Bjkl + Bjlk = Bjkl + Bklj + Bljk = 0.

Furthermore, the action of Rn on P ’ P0 and that of CO(n, R) on P0 ’ N may

be combined, to realize P ’ N as a principal bundle having structure group

G ‚ SOo (n + 1, 1) consisting of matrices of the form (3.3). The matrix 1-form φ

in (3.14) de¬nes an so(n+1, 1)-valued parallelism on P , under which the tangent

spaces of ¬bers of P ’ N are carried to the Lie algebra g ‚ so(n+1, 1) of G, and

φ is equivariant with respect to the adjoint action of G on so(n+1, 1). The data

of (P ’ N, φ) is often called a Cartan connection modelled on g ’ so(n + 1, 1).

92 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

We conclude this discussion by describing some properties of the functions

Ai , Bjkl on P . Di¬erentiating the de¬nition of ¦ (3.14) yields the Bianchi

jkl

identity

d¦ = ¦ § • ’ • § ¦.

The components of this matrix equation yield linear-algebraic consequences

about the derivatives of Ai , Bjkl . First, one ¬nds that

jkl

i

§ ωk § ωl = 2 Bikl ωj § ωk § ωl ’ 1 Bjkl ωi § ωk § ωl .

1 1

2 DAjkl (3.15)

2

Detailed information can be obtained from this equation, but note immediately

the fact that

DAi ≡ 0 (mod {ωi}).

jkl

In particular, referring to the de¬nition (3.13), this shows that the collection

of functions (Ai ) vary along the ¬bers of P ’ N by a linear representation

jkl

of the structure group G. In other words, they correspond to a section of an

associated vector bundle over N . Speci¬cally, we can see that the expression

def 1 i

§ ωj — ωk § ωl ) — (ω1 § · · · § ωn )’2/n

A (ωi

A= 4 jkl

on P is invariant under the group action, so A de¬nes a section of

2

Sym2 ( T — N ) — D’2/n ,

where D is the density line bundle for the conformal structure, to be de¬ned

shortly. This section is called the Weyl tensor of the conformal structure.

Something di¬erent happens with Bjkl . Namely, the Bianchi identity for

dBjkl yields

def

dBjkl + 6ρBjkl ’ Bmkl ±m ’ Bjml ±m ’ Bjkm±m

DBjkl = j k l

’βi Ai (mod {ωi}).

≡ jkl

In particular, the collection (Bjkl ) transforms by a representation of G if and

only if the Weyl tensor A = 0. In case n = 3, the symmetry identities of

Ai imply that A = 0 automatically; there is no Weyl tensor in 3-dimensional

jkl

conformal geometry. In this case, (Bjkl ) de¬nes a section of the vector bun-

2—

dle T — N — T N , which actually lies in a subbundle, consisting of traceless

elements of the kernel of

2 3

T —N — T —N ’ T — N ’ 0.

This section is called the Cotten tensor of the 3-dimensional conformal structure.

If the Cotten tensor vanishes, then the conformal structure is locally equivalent

to the ¬‚at conformal structure on the 3-sphere.

In case n > 3, from (3.15) one can show that the functions Bjkl can be ex-

pressed as linear combinations of the covariant derivatives of Ai . In particular,

jkl

if the Weyl tensor A vanishes, then so do all of the Bjkl , and the conformal struc-

ture of N is locally equivalent to the ¬‚at conformal structure on the n-sphere.

3.1. BACKGROUND MATERIAL ON CONFORMAL GEOMETRY 93

3.1.3 The Conformal Laplacian

To every conformal manifold (N n , [ds2]) is canonically associated a linear di¬er-

ential operator ∆, called the conformal Laplacian. In this section, we de¬ne this

operator and discuss its elementary properties. One subtlety is that ∆ does not

act on functions, but on sections of a certain density line bundle, and our ¬rst

task is to de¬ne this. We will use the parallelized principal bundle π : P ’ N

canonically associated to [ds2 ] as in the preceding discussion.

To begin, note that any n-form σ on N pulls back to P to give a closed

n-form

π— σ = u ω1 § · · · § ωn ∈ „¦n (P ),

where u is a function on P whose values on a ¬ber π ’1 (x) give the coe¬cient of

n —

σx ∈ (Tx N ) with respect to various conformal coframes at x ∈ N . Among

all n-forms on P of the form u ω 1 § · · · § ωn , those that are locally pullbacks

from N are characterized by the property of being closed. Using the structure

equations, we ¬nd that this is equivalent to

(du + 2nuρ) § ω 1 § · · · § ωn = 0,

or

(mod {ω 1, . . . , ωn }).

du ≡ ’2nuρ

This is the in¬nitesimal form of the relation

u(p · g) = r’2nu(p), (3.16)

for p ∈ P and g ∈ G as in (3.3). This is in turn the same as saying that the

function u on P de¬nes a section of the oriented line bundle D ’ N associated

to the 1-dimensional representation g ’ r 2n of the structure group.3 Positive

sections of D correspond to oriented volume forms on N , which in an obvious

way correspond to Riemannian metrics representing the conformal class [ds2].

Because so many of the PDEs studied in the conformal geometry literature

describe conditions on such a metric, we should expect our study of Euler-

Lagrange equations in conformal geometry to involve this density bundle.

s

In analogy with this, we de¬ne for any positive real number s the degree- n

density bundle Ds/n associated to the 1-dimensional representation g ’ r 2s;

the degree-1 density bundle is the preceding D. Sections are represented by

functions u on P satisfying

u(p · g) = r’2su(p), (3.17)

or in¬nitesimally,

(mod {ω 1 , . . . , ωn }).

du ≡ ’2suρ (3.18)

is, D is the quotient of P — R by the equivalence relation (p, u) ∼ (p · g, r ’2n u) for

3 That

p ∈ P , u ∈ R, g ∈ G; a series of elementary exercises shows that this is naturally a line bundle

over N , whose sections correspond to functions u(p) satisfying (3.16).

94 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

Summarizing, we will say that any function u on P satisfying (3.18) de¬nes a

s

section of the degree- n density bundle, and write

u ∈ “(Ds/n ).

We further investigate the local behavior of u ∈ “(D s/n ), writing

du + 2suρ = uiωi

for some “¬rst covariant derivative” functions ui . Di¬erentiating again, and

applying the Cartan lemma, we obtain

dui + suβi + 2(s + 1)uiρ ’ uj ±j = uij ωj , (3.19)

i

for some “second covariant derivatives” uij = uji; this is the in¬nitesimal form

of the transformation rule

ui(p · g) = r’2(s+1)(uj (p)aj ’ sbi u(p)). (3.20)

i

Note that unless s = 0 (so that u is actually a function on N ), the vector-valued

function (ui) on P does not represent a section of any associated vector bundle.

Di¬erentiating again, and factoring out ω k , we obtain modulo {ω 1 , . . . , ωn }

duij ≡ δij uk βk ’ (s + 1)(uj βi + uiβj ) ’ 2(s + 2)uij ρ + ukj ±k + uik ±k ,

i j

so once again, uij is not a section of any associated vector bundle. However, we

can take the trace

(mod {ω1, . . . , ωn}),

duii ≡ (n ’ 2s ’ 2)βk uk ’ 2(s + 2)ρuii

s+2

n’2

and we see that in case s = 2, the function uii on P is a section of D . To

n

summarize,

the map u ’ uii de¬nes a second-order linear di¬erential operator,

called the conformal Laplacian,

n’2 n+2

) ’ “(D

∆ : “(D ).

2n 2n

n’2

Note that for sections u, v ∈ “(D 2n ), the quantity u∆v ∈ “(D 1 ) can be

thought of as an n-form on N , and integrated. Furthermore, the reader can

compute that

(u∆v ’ v∆u)ω = d((uvi ’ vui )ω(i) ).

We interpret this as saying that u∆v ’ v∆u is canonically a divergence, so that

n’2

(·, ∆·) is a symmetric bilinear form on “o (D 2n ), where

n’2 n+2

(·, ·) : “o (D — “o (D ’R

2n ) 2n )

is given by integration on N of the product.

3.1. BACKGROUND MATERIAL ON CONFORMAL GEOMETRY 95

To clarify the meaning of ∆, we can choose a particular Riemannian met-

ric g representing the conformal structure, and compare the second covariant

s

derivatives of an n -density u taken in the conformal sense with those derivatives

taken in the usual sense of Riemannian geometry. By construction of P , the

pulled-back quadratic form π — g ∈ Sym2 (T — P ) may be expressed as

π— g = »((ω1 )2 + · · · + (ωn )2 )

for some function » > 0 on P . Proceeding in a manner similar to the preceding,

we note that

Lv (π— g) = 0

for any vector ¬eld v that is vertical for P ’ N . Knowing the derivatives of ω i

quite explicitly, we can then calculate that