Conformal Differential Geometry

NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY

Michael Eastwoo d

These notes are in no way meant to b e comprehensive neither in treatment nor in

references to the extensive literature They are merely meant as an introduction to a

small selection of topics in the eld They were presented as a series of four lectures at

the Winter School on Geometry and Physics SrnCzech Republic January

Lecture One

Recall that a Riemannian manifold M is really a pair M g consisting of a smo oth

manifold M and a metric g a smo oth and everywhere p ositive denite section of

T M the symmetric tensor pro duct of the cotangent bundle Following standard

practise we shall usually write g instead of g and more generally we shall adorn

tensors with upp er and lower indices in corresp ondence with the tangent or cotangent

bundle We shall also use the Einstein summation convention to denote the natural

pairing of vectors and covectors Thus V denotes a tangent vector or a vector eld

a b

and g V V denotes the square of its length with resp ect to g We shall often raise

a b

and lower indices without commentif V is a vector eld then V g V is the

a ab

corresp onding form More precisely this is Penroses abstract index notation see

for details

A conformal manifold M is a pair M g where g is a Riemannian metric dened

only up to scale In other words g is a section of R T M the bundle of rays

in T M such that a representative g is p ositive denite In yet other words

a conformal manifold is an equivalence class of Riemannian manifolds where two

b b

metrics g and g are said to b e equivalent if g is a multiple of g In this case

ab ab ab ab

it is convenient to write g g for some smo oth function On a conformal

ab ab

manifold one can measure angles b etween vectors but not lengths

This pap er is in nal form and no version of it will b e submitted for publication elsewhere

y ARC Senior Research Fellow University of Adelaide

MICHAEL EASTWOOD

For example in two dimensions an oriented conformal manifold is precisely a Riemann

surface ie a onedimensional complex manifold Certainly if M is a Riemann surface

with lo cal coordinate z x iy then dx dy is a Riemannian metric and if w u iv

is another then the Cauchy Riemann equations

u v v u

x y y x

imply

u v v u

dx dy dx dy du dv

x y x y

u v

dx dy

x x

Conversely there is a theorem of Korn and Lichtenstein which says that any Rieman

nian metric in two dimensions may b e written lo cally in the form

dx dy

for some smo oth function and suitable coordinates x y chosen compatible with

the orientation Taking z x iy denes a lo cal complex coordinate and any

other choice is holomorphically related Indeed the Cauchy Riemann equations are

precisely that the Jacobian matrix

u x u y

v x v y

is prop ortional to an orthogonal one

Notice that a conformal manifold in two dimensions has no lo cal invariantsall

Riemann surfaces are lo cally indistinguishable The only lo cal invariant of a two

dimensional Riemannian manifold is its scalar curvature and this has b een eliminated

by the freedom to scale the metric In higher dimensions it is reasonable to exp ect

some of the rigidity of Riemannian geometry This is indeed the case and evidence in

its favour can b e found as follows

Firstly let is consider rigid motions of R ie the connected comp onent of the group

of isometries of R with its usual metric Of course this is wellknown to b e generated

by translations and rotations One way of seeing this is to consider an innitesimal

a n

motion ie a vector eld V on R with the prop erty that

son ie V V V

ab ab b

where square brackets around indices denote taking the skew part Consider now

son ie V V and V V V

abc abc abc abc bc

b c

x x

NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY

where round brackets denote the symmetric part However this rst prolongation

son vanishes

V V V V V V V

abc bac bca cba cab acb abc

By integration the result concerning rigid motions follows There are some observa

tions to b e made

The calculation is the principal ingredient in constructing the LeviCivita

connection on a Riemannian manifold

The at mo del of Riemannian geometry is R as a homogeneous space

R o SOn rigid motions

SOn stabiliser of a p oint

As an alternative pro of we could show that geo desics ie straight lines with unit

sp eed parameterisation are preserved by isometries and by ring out geo desics

conclude that an isometry xing a p oint to rst order is necessarily the identity

Let us try the same technique for the conformal case in an attempt to identify the

conformal motions of R An innitesimal motion is a vector eld satisfying

con ie V g V

ab ab b

Thus

V con ie V g and V V

bc abc c ab abc abc

This rst prolongation con is no longer zero In fact

V g g g

a abc c ab b ac a bc

identies con with R However

con ie V g and V V V

bcd cd ab abcd

abcd abcd

and this second prolongation con vanishes if the dimension n is greater than

ab ab ab

n g V g V g V V

cd abcd acbd cabd

acbd

ab ab

g g V g V

bd ac cdab cd cdab

cd cd ab cd cd

ng g g g g ng

cd cd ab cd cd

and is tracefree Since n then

n g V

cd cdab

MICHAEL EASTWOOD

implies that the right hand side is symmetric in cd in which case

ab ab

n g V g g

cd ab cd

cdab

which implies that is pure trace Now V and we nd that V is zero by

cd abcd

abcd

applying on the rst three indices

We may conclude that a conformal motion of R for n which xes the origin to

second order is necessarily the identity More precisely we may identify the group P

of conformal motions xing the identity as matrices of the form

a a n

a a

for m SOn r R r m

b b

a a

r r r m

a a b

a n

acting on x R by

a b c a

m x x x r

b c

a b c b

r m x x x r r

a b c b

It is a go o d exercise to check that this really is conformal ie that its derivative

is everywhere prop ortional to an orthogonal matrix Notice that the denominator

b b c a

vanishes when x m r r r On the other hand this formula is forced by the

c a

prolongation argument In order to allow nonzero r we are therefore obliged to

n n n

compactify R with a single p oint to obtain the sphere S R fg Further

reasoning along these lines identies the full group of conformal motions of the round

n n

sphere S conformally containing Euclidean R via stereographic pro jection as the

identity connected comp onent G of SOn The sphere is realised as the space

of future p ointing null rays

space S

of generators

in R The form of P as ab ove is obtained by taking G to preserve the form

x x x x x

n n

and the basep oint of S to b e represented by the null vector

NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY

In summary the at mo del of conformal geometry is

S GP

for G SO n and P a suitable parabolic subgroup The other two observations

in the Riemannian case will turn out to have analogues in the conformal case We

shall nd an invariant connection but not on the tangent bundle In fact we shall

nd a somewhat b etter dierential op erator originally found by TY Thomas We

shall also nd analogues of geo desics in the conformal case They are known as

conformal circles b ecause in the at case they coincide with the round circles on S

equipp ed with their standard pro jective parameterisations Certainly these circles

are preserved by G

Lecture Two

So much for the at case To pro ceed in general the nave approach is to work with

a metric in the conformal class and then see how things change when the metric

is scaled If g is replaced by g g then the LeviCivita connection r is

ab ab ab a

replaced by the connection r acting on forms according to

r r g

a b a b a b b a c ab

where r Indeed this formula surely denes a torsionfree connection

a a

and induces

d d

r r g g

a bc a bc a bc b ac c ba dc ab bd ac

on contravariant tensors clearly annihilating g Corresp ondingly the new

ab ab

connection on vector elds is

b b b b c b

r V r V V V V

a a a a c a

where is the Kronecker delta It is convenient to introduce a line bundle E on

M as follows If a metric g in the conformal class is chosen then E is identied

with the trivial bundle E Equivalently a lo cal section of E may b e regarded as a

function say f If however g is replaced by g g then the function f repres

ab ab ab

enting with resp ect to the metric g is given by f f The w p ower of E will

b e denoted by E w and its sections called conformally weighted functions of weight w

Such a section may b e represented by a function scaling according to f f We

shall write E and E for the tangent and cotangent bundle resp ectively Other tensor

bundles will b e denoted by adorning E with the corresp onding indices The conformal

metric may b e regarded as an invariantly dened section of E Raising and lower

ing indices denes a canonical isomorphism of E w with E w for any weight w

Cho osing a metric induces a connection on E w which transforms according to

r r w

a a a

MICHAEL EASTWOOD

under scaling of the metric

The Riemann curvature is dened by

c c d

r r r r V R V

a b b a ab d

and transforms by

R R g g g g

abcd abcd ac bd bc ad bd ac ad bc

under scaling of the metric where

r g

ab a b a b c ab

In particular the variation is entirely through traces The totally tracefree part C

abcd

of R is therefore invariant and suggests that we write the remaining part in

abcd

terms of a symmetric tensor P according to

R C P g P g P g P g

abcd abcd ac bd bc ad bd ac ad bc

Then the Weyl curvature C has weight and is conformally invariant whilst the

abcd

Rhotensor P has weight and implies that

g P P r

c ab ab ab a b a b

The Rhotensor is a traceadjusted multiple of the Ricci tensor R R

bd ab d

P R g

ab ab ab

n n

is the scalar curvature where R R

Without further ado we can now introduce the conformally invariant connection

alluded to earlier Further details and motivation can b e found in It is a connection

on a vector bundle E which we now dene In the presence of a metric g in the

conformal class it may b e identied as a direct sum

A a

E E E E

but if g is replaced by g g then a lo cal section is identied with

ab ab ab

b b b

its counterpart in the new scale according to

a a a

b b

It is easy to check that this is an equivalence relation and hence that the bundle

E is welldened We shall use the term tractor for tensor p owers of this bundle

and their sections by analogy with the term tensor in Riemannian geomtry The

NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY

structure group of this bundle is the group P encountered in the discussion of the

at case Whereas the tractor bundle corresp onds to the standard representation of

G SO n restricted to P the spin representation if n is o dd or one of the

two spin representations if n is even induces the local twistor bundles introduced

in four dimensions by Penrose see Recall that the at mo del of conformal

geometry is the homogeneous space GP The tractor bundle E in this case is the

homogeneous bundle induced by restricting the dening representation of G on R

n n

to the subgroup P It is therefore simply a pro duct S R

For a given metric the tractor connection on E is dened by

b b

a a a a

r P r

b b b b

r P

b ba

This denition is conformally invariant as can b e veried by direct calculation

b b

r b

b b

a a a

b b

b b b r

r P

b b b

b b

P r

ba b

b b b

a a a c c a a a c a

r P r

b b c c b b b c b

c c a a c

P r g r

c ba b a b a c ba b c

b b b b

a a a c c a

b b b c b

a a a a c c

r P

b b c c b b

c c c a a

r P r g

b b c c ba b a b a c ba

b b

a a a a a

r r P

b b b b b

c c a a c

r r r P P

b c b c b b ba ba c b

b b

a a a

r P

b b b

r P

b ba

This denition is due to TY Thomas It is equivalent to E Cartans conformal

connection on the asso ciated frame bundle Thomass discovery was slightly later

than though indep endent of Cartans The connection induced on the bundle of lo cal

twistors is called local twistor transport see for explicit formulae in dimension

four

In the at case M S GP the tractor connection is the at connection on the

n n

pro duct bundle S R In general the curvature is given by

d c c c

r r r r r P C

a b b a a b ab d

r P

a bd

MICHAEL EASTWOOD

For n the Bianchi identity r R implies that

a bc e

d c

r P r C

ab d

a b

so the tractor curvature is equivalent to the Weyl curvature When n the Weyl

curvature vanishes by symmetry considerations and so we may conclude that r P

a bc

is conformally invariant This is known as the CottonYork tensor In any case it is

straightforward to deduce that these tensors are precisely the obstruction to a given

conformal manifold b eing lo cally equivalent to the at mo del

The tractor bundle carries a nondegenerate symmetric form the tractor metric a

Lorentzian metric characterised by

a a

k k

It is conformally invariant and is preserved by the tractor connection In the at case

it coincides with the Lorentzian metric on R

We are now in a p osition to dene the conformal analogues of geo desics the socalled

conformal circles Though it is p ossible to pro ceed directly as in it is convenient

to use tractors More details can b e found in

Supp ose is a smo oth curve in M parameterised by t a smo oth function on with

nowherevanishing derivative This determines the velocity vector U along by

U requiring that it b e tangent to and that U r t Dene u U a function

a a

b a b

of conformal weight Dene the acceleration vector A U r U A unit sp eed

geo desic is dened by u and A Of course the acceleration vector is not

b b b c b

conformally invariant in fact A A u U U Dene the velocity

B B

tractor U and acceleration tractor A by

B a B a B

U U r and A U r U

a a

These are manifestly conformally invariant denitions Calculation yields

a b b b a b

kAk u U U r A u A A u U A P U U

b a b b ab

automatically invariant Its vanishing may b e regarded as a order ordinary dif

ferential equation along for the function t This gives a preferred family of lo cal

parameterisations of which we call projective If s and t are pro jective parameters

then the third order ODE relating them reduces to the Schwarzian

at b d s d s ds

whence s

dt dt dt ct d

The curve is called a pro jectively parameterised conformal circle if and only if

a B

kAk and U r A a

NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY

Certainly this is a conformally invariant system In fact it reduces to a third order

a b

ODE with leading term U r A See for more details The upshot of this dis

cussion is that for every velocity and acceleration vector at a p oint there is a unique

parameterised conformal circle with these as initial conditions In the at case it

may b e veried that these are the round circles with standard pro jective paramet

erisations As indicated earlier this gives an alternative approach to the conformal

motions of R The details are left to the reader

Lecture Three

Recall from Lecture One that the at mo del of conformal geometry is the sphere S as

a homogeneous space GP where G SO n and P is the subgroup consisting

of matrices of the form It is an exercise in the theory of Verma modules to classify

the Ginvariant dierential op erators on S and this lecture will mostly b e devoted

to indicating this theory and how it applies Of course it is imp ortant to understand

the at mo del else we cannot hop e to understand the curved ie general case More

imp ortantly it will turn out that various elements of the at theory generalise to the

curved case

For the following discussion G and P can b e arbitrary Lie groups A homogeneous

vector bundle on GP is one whose total space is equipp ed with an action of G which

is compatible with the action on GP and which is linear on the bres Such a bundle

may b e reconstructed from its bre over the identity coset In other words supp ose

P AutE

is a nitedimensional representation of P Then

G E

E G E

g e g p p e

is homogeneous and this provides a corresp ondence b etween the nite dimen

sional representations of P and the homogeneous vector bundles on GP For more

details see for example For an arbitrary smo oth vector bundle E on a smo oth

manifold one has the jet bundles

J E J E J E J E E

where J E is dened as the inverse limit of course allowing an innitedimensional

vector bundle in this instance For a homogeneous vector bundle there is the equi

valent diagram of P mo dules

J E J E J E J E E

A linear dierential op erator of order k b etween arbitrary vector bundles E and

F may b e dened see for example as a homomorphism of vector bundles

MICHAEL EASTWOOD

D J E F In the homogeneous case the Ginvariant such op erators corresp ond

to homomorphisms of P mo dules D J E F

For E a representation of P dene the Verma module V E to b e the gmo dule

Ug E

Ugsubmo dule generated by fp f pf g

where is the derivative of the dual representation p is the Lie algebra of P and

g is the Lie algebra of G with universal enveloping algebra Ug Warning this is

somewhat nonstandard terminologysee the parenthetical remark starting at the

b ottom of this page The term induced module is also used see for example

In fact P also acts on V E in a compatible way In the homogeneous case observe

that the action of G on E induces an action of g on J E It is straightforward to

verify that there is a natural isomorphism V E J E as g P mo dules The

universal enveloping algebra Ug is ltered by degree

R U g U g U g U g Ug

k k

with corresp onding gradings

U g J

U g

This induces a ltration of the Verma mo dule

R V E V E V E V E V E

k k

which is preserved by the action of P The corresp onding gradings may b e viewed as

an exact sequences of P mo dules

gp E V E V E

k k

whose dual induces the jet exact sequence see for example

k k k

T M E J E J E

In this way Verma mo dules build the P mo dules J E which arise in constructing

Ginvariant linear dierential op erators However it is usually more convenient to

use the Verma mo dules as a wholea Ginvariant linear dierential op erator b etween

the bundles on GP induced from representations E and F is equivalent to a homo

morphism of g P mo dules

V E V F

This is easily shown from a geometric p oint of p oint in representation theory it is

known as Frobenius reciprocity see for example

To pro ceed further it is necessary to b e more sp ecic concerning G P and E Mat

ters are esp ecially congenial if G is semisimple P a parab olic subgroup and E an

irreducible representation The term generalised Verma mo dule is often used for this

NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY

case see for example with the term Verma mo dule reserved for the case when

P is Borel in which case E b eing irreducible is necessarily onedimensional

It is illuminating to consider the following simple example

G SL R P st r R E R R

with an element of P of the form shown acting on R as multiplication by for

some w R The corresp onding homogeneous bundle is denoted

E w as in TN Baileys lectures

To investigate the Verma mo dule V w introduce the standard generators of sl R

h y x

We may use the standard commutation relations to put elements of Usl R into a

standard order ie apply the PoincareBirkhoWittpro cedure and hence identify

V w Ry where x and h w

In other words the action of y on this p olynomial algebra is by left multiplication

whilst the actions of x and h are obtained by rep eated commutation to bring them to

the right whereup on they act on as sp ecied Thus V w is a highest weight mo dule

generated by The following calculations are typical Consider the case w Then

xy x y y x h

hy h y y h y y y

xy x y y y xy hy y y y

hy h y y y hy y y y

This shows that y is a maximal sometimes called singular weight vector in V

ie an eigenvector for h that is annihilated by y Apart from itself it is easily

veried by similar calculations that up to scale there are no other maximal weight

vectors As the weight of y is it follows that there is a homomorphism of Verma

mo dules

V Ry Ry V

with y and therefore f y y f y Equivalently there is a second

order linear dierential op erator

g E E

More generally these calculations show that up to scale the only nontrivial linear

dierential op erators on the circle which are projectively invariant ie invariant under

the action of SL R are

g E w E w for w Z

MICHAEL EASTWOOD

k th

where g is a k order dierential op erator cf for the complex case The non

linear dierential op erators in this situation are the sub ject of The op erator

g E E is the exterior derivative

Leaving this example for the moment it is useful to note how much of this applies

in general ie when G is semisimple P parab olic and E irreducible Certainly

the search for Ginvariant linear dierential op erators b etween irreducible bundles

on GP is equivalent to the search for maximal weight vectors in V E When

GP S the at mo del of conformal geometry these Verma mo dules are of the

form

V E Ry y y E

where y y y are lowering op erators in g corresp onding to the n p ositions in the

top row opp osite the vector r in The representation E of P b eing irreducible

is generated by a highest weight vector unique up to scale The entire Verma

mo dule is then generated by applying lowering op erators from g to in particular

the op erators y y y which together with lowering op erators from p span the

lowering op erators of g In this generality the search for maximal weight vectors

in V E is not so amenable to direct computation Instead the JantzenZuckerman

translation functor see for example avoids these calculations and also provides

an inductive metho d of constructing the whole family of op erators from their simplest

members such as the exterior derivatives They key to the translation functor is to

consider the action of the centre of Ug

Returning to our example the centre of Usl R contains the element

C h y x h

and in fact consists of p olynomials in C Applying C to V w gives w w

Since V w is generated by and C commutes with the action of g it follows that

C acts on all of V w by multiplication by w w This already restricts the

p ossibilities for nontrivial homomorphisms V w V w for in this case clearly

one must have w w w w The action of C also plays a role as follows

The standard representation of SL R on R gives rise to a Verma mo dule V R

Generally if G AutE is restricted to P then the corresp onding homogeneous

vector bundle E is canonically trivial

E GP E

Thus on the level of Verma mo dules

V R V R

and more generally

V R R V w R w

NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY

The short exact sequence of P mo dules

R R R R

w w w

gives rise to a short exact sequence of gmo dules

V w V w R V w

For w notice that C acts by dierent scalars on V w and V w In

these cases therefore V w R decomp oses as a gmo dule into a direct sum of

the two eigenspaces of C Bearing in mind that a homomorphism of Verma mo dules

corresp onds to a Ginvariant dierential op erator this shows that there are invariant

dierential splittings for example

E E E

I I

A D

where E is the trivial bundle with R as bre In homogeneous coordinates

D x

The exterior derivative g E E also acts on this trivial bundle to give the

diagram

E E E

The comp osition D g D is then g E E Notice that it has b een

constructed in a manifestly invariant manner Iterating this pro cedure generates all

the invariant op erators g inductively

In the at conformal case there are similar invariant dierential splittings of the

tractor bundle Their existence is guaranteed abstractly by analysing the action

of the centre of Ug see for corresp onding splittings of the twistor bundles in

the fourdimensional case This analysis not only gives a complete classication of

the conformally invariant linear dierential op erators b etween irreducible conformally

weighted tensor bundles in the at case but also gives via the translation functor a

way of building families of invariant op erators from their simplest members For the

purp oses of this article the results of this classication are not so imp ortant see for

example or for the fourdimensional case for the general case and for

generalisations

The rest of this article is concerned with the extent to which the translation functor

as a metho d for generating invariant op erators extends to the curved case The

main p oint is that the dierential splittings mentioned ab ove exist in the curved case

to o In fact the basic examples were already written down in by Thomas

As motivation he noted that the tractor connection is an inadequate substitute for

MICHAEL EASTWOOD

the LeviCivita connection In the Riemannian case one can apply the LeviCivita

connection rep eatedly Indeed in some welldened sense which we wont go into

here this captures all the invariant calculus that is present on a Riemannian manifold

A A

In the conformal case having formed r V for a tractor eld V there is no invariant

connection on the cotangent bundle so one is at a loss for a second derivative Thomas

B B

suggested the following replacement Dene D E w E w by

w n w f

n w r f

D f

w Pf

a a

where denotes the Laplacian r r and P P a multiple of the scalar

a a

curvature As usual this denition is written with resp ect to a particular choice

of metric in the conformal class It is however conformally invariant To see this

consider how the Laplacian changes under scaling of the metric g g g

ab ab ab

c c c

r f r r f w f r

a b a a b

r r f w f w r f w f

b a a b a a

r f w f r f w f g

a b b c c ab

Thus

a a a

c c

r f r r f w f n w r f w f r

a a a a a

a a a a

r r f w r f n w r f w n w f

a a a a

r r f n w r f w r n w f

a a a a

On the other hand

a a

P P r

a a

Hence

b b

w P f w Pf n w r f w n w f

b b

whilst

b b b

n w r f n w r f w n w f

These are exactly the transformations of a tractor as required

Notice that the argument is unaected if f is replaced by a tractor eld of conformal

weight w where of course the Laplacian is replaced by the tractor Laplacian an

explicit formula for which is quite complicated In other words the op erator D

A BA

E w E w makes p erfectly go o d sense We can continue in this way to form

B A C B A

D V D D V

for any conformally weighted tractor eld V

The D op erator combines several features of conformal geometry If w then

B A

the rst comp onent of D V vanishes and so the second comp onent is conformally

NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY

invariant This is the tractor connection If w then b oth the rst and second

comp onents vanish The third comp onent is therefore conformally invariant In other

words the dierential op erator

f w Pf Rf

In is conformally invariant when acting on functions or tractor elds of weight

the at mo del D is closely related to dierentiation with resp ect to the coordinates

n A

on R Indeed this asp ect may b e exploited in the curved case with D nding

interpretation in the ambient metric construction of Feerman and Graham

Lecture Four

The investigation of invariant dierential op erators on a conformal manifold is an act

ive area of research see for example The conformally invariant

Laplacian or Yamabe op erator is an example of such an op erator More gener

ally by a conformally invariant dierential op erator we shall mean any p olynomial

expression in the LeviCivita connection and its curvature acting b etween conform

ally weighted tensor bundles which is unchanged when the metric is scaled Some

examples should suce to make clear what is meant here The exterior derivative

r E E

a bd abd

is certainly invariant Consider the op erator

E trace free part of E

g r r f Pf f r r f P f

ab c a b ab

From

c c

r f r r f r f trace terms r

a b

b a b a b a

whilst

b b

Pg r Pg P P

ab a b a b ab ab ab

n n

The op erator is therefore invariant A more exotic invariant op erator is given by

f totally trace free part of r r r f P r f r P f

a c ab c a bc

acting on conformally weighted functions of weight In fact all these op erators are

strongly invariant in the sense that they are also invariant when acting on tractor elds

rather than just scalar functions This is b ecause the transformations under scaling of

the metric simply do not distinguish b etween pure tensors and tensor elds coupled

to the tractor bundle However it is not the case that all invariant op erators are

strongly invariant Consider for example the op erator L E E in dimension

four given by

b a ba ba

Lf r r r P Pg r f

b a

MICHAEL EASTWOOD

Calculation yields

b a b a a b

Lf Lf r r r r r f r r r r r f r r r r r f

a b b a a b b a a b b a

The curvature terms cancel so L is conformally invariant However if f is replaced

C C C

by f a section of E then writing for the tractor curvature cf

ab D

C C b C a D b a C D

Lf Lf r f r f

ab D ab D

so conformal invariance is lost

We shall now describ e a pro cedure the curved translation principle for generating

conformally invariant op erators This pro cedure takes a strongly invariant op erator

and yields new strongly invariant op erators from it We should remark however that

though this is a generally applicable pro cedure there are much more ecient means

for calculating certain series of op erators in particular cases see for example To

pro ceed we need to generalise the D op erator so that it acts on arbitrary conformally

B B

weighted irreducible tensor bundles Thus D F w F w for any irreducible

tensor bundle F The general existence of these op erators follows from their existence

in the at case and this as indicated in Lecture Three follows from the general

theory of Verma mo dules The formulae b ecome quite complicated For example

B B

D E w E w sends a conformally weighted form of weight w to

a a a

n w n w w w

b b b c

n w n w w r n w r w r

a a a c

n w w w P n r r n w P

a a b ab

However these op erators only of second order It is this fact that allows a general

existence argument In fact there is an alternative approach using twistors instead

of tractors where the corresp onding op erators are only of rst order This makes

their existence more straightforward cf A general formula in four dimensions

B B

is given in As with D E w E w the same formulae with the tractor

connection dene conformally invariant op erators when acting on tractorvalued con

formally weighted tensors ie these op erators are strongly invariant Note however

that there is some choice herethe same op erator on tensors may b e dened by two

dierent formulae when derivatives are commuted at the exp ense of curvature in

which case these dierent formulae may give rise to genuinely dierent op erators on

tractors This is another reason why the twistor approach may b e preferred

bC C

There is also a series E F w F w of strongly invariant rst order

op erators for any irreducible tensor bundle F For example

n w

for a lo cal section of E w and

a a

NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY

c c d c

n w w n w

or or

a a d a

b b b

r nr r

ab ab ab

for a lo cal section of E w and either symmetric trace free puretrace or skew

ab ab

resp ectively

B B

Finally there is a series of zeroth order invariant op erators F F w F w

for any irreducible tensor bundle F whose denition is tautological

Dually there are srongly invariant op erators

B C B

D F w F w E F w F w F F w F w

B bC b B

For example

E E w irreducible decomp osition of E w

ab abd

h i

w g

abd da be abd

h i

g r r r

da be a bd d ab

o n

e e

n w g g r

da be da be

n o

w r

abd a bd

The curved translation principle is obtained simply by combining these op erators as

follows Supp ose K F w G w is a strongly invariant linear dierential op erator

for irreducible tensor bundles F and G and conformal weights w and w Strong

invariance implies that the same formula denes

B B

K F w G w

which we may comp ose with the D E and F op erators to obtain new strongly

invariant op erators b etween conformally weighted irreducible tensor bundles The

following example illustrates this pro cedure Let K b e the exterior derivative from

C C

forms to forms Then K E E is given by

a b ab

c c c c

r P

b a b a b a b

r P

a b ca b b

MICHAEL EASTWOOD

aC C

We may comp ose with E tracefree part of E E given by

ab b

and E E irreducible decomp osition of E to obtain after some cal

ab abd

culation

n r g r

a bc ca bd

another strongly invariant op erator this one acting on symmetric tracefree of

weight

This is still rst order However the translation principle can also increase the order

by or decrease by or more For example

D r

a bC

C C

E tracefree part of E E E

a ab

f nr r f nP f g f g Pf

ab ab ab

a b

yields a second order op erator from a rst order op erator whilst

r D

a dC

C C

E highest weight part of E E E

b b

ab abd

n C

a abd c

yields a zeroth order op erator from a rst order op erator

To summarize The existence of the D and E op erators derives from the at case

where this translation principle coincides with the JantzenZuckerman translation

functor The resulting classication in the at case yields various series of op erators

and the curved translation principle shows that most of these series admit curved

analogues ie invariant op erators with the same symbol as an invariant op erator

in the at case Indeed in the o dddimensional case all the at op erators admit

curved analogues and the even dimensional case most can b e obtained by translating

the exterior derivative There are however some exceptions and in dimension four

Graham has shown that the at invariant op erator

E E

has no curved analogue In the at case this op erator may b e obtained by

translating but in the curved case this breaks down since as we observed is

not strongly invariant

NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY

There are many questions yet to b e answered

Precisely which at op erators have curved analogues There is a clear conjec

ture knowing where the curved translation principle breaks down but is the

only explicitly known example

Further investigate the curved analogue of E E n for n even This

op erator has b een shown to exist but Branson has asked for example

whether it can b e chosen to b e selfadjoint

Find more explicit formulae for the curved analogues cf They are

not in general unique

Generalise this theory to other geometries AHS see CR

References

TN Bailey Parabolic invariant theory in geometry Lectures at the Winter

School on Geometry and Physics SrnCzech Republic January this volume

TN Bailey and MG Eastwoo d Conformal circles and parametrizations of curves

in conformal manifolds Pro c Amer Math So c 108

TN Bailey MG Eastwoo d and AR Gover Thomass structure bund le for con

formal projective and related structures Ro cky Mtn Jour Math 24

page numbers not certain

RJ Baston and MG Eastwoo d Invariant Operators in Twistors in Mathematics

and Physics London Mathematical So ciety Lecture Notes vol Cambridge

University Press pp

TP Branson Dierential operators canonically associated to a conformal structure

Math Scand 57

A Cap and J Slovak Invariant operators on manifolds with almost Hermitian sym

metric structures Lectures at the Winter School on Geometry and Physics

SrnCzech Republic January this volume

VK Dobrev q dierence intertwining operators and q conformal invariant equa

tions Lectures at the Winter School on Geometry and Physics Srn Czech

Republic January this volume

MG Eastwoo d and JW Rice Conformal ly invariant dierential operators on

Minkowski space and their curved analogues Commun Math Phys 109

Erratum Commun Math Phys 144

MG Eastwoo d and KP Tod Edtha dierential operator on the sphere Math

Pro c Camb Phil So c 92

C Feerman and CR Graham Conformal invariants in Elie Cartan et les MathematiquesdAujourdui Asterisque pp

MICHAEL EASTWOOD

HD Fegan Conformal ly invariant rst order dierential operators Quart Jour

Math 27

AR Gover Conformal ly invariant operators of standard type Quart Jour Math

CR Graham Conformal ly invariant powers of the Laplacian II Nonexistence

Jour Lond Math So c 46

CR Graham R Jenne LJ Mason and GAJ Sparling Conformal ly invariant

powers of the Laplacian I Existence Jour Lond Math So c 46

HP Jacobsen Conformal invariants Publ RIMS Kyoto Uni 22

I KolarPW Michor and J Slovak Natural Operations in Dierential Geometry

Springer

AW Knapp Introduction to representations in analytic cohomology in The Pen

rose Transform and Analytic Cohomology in Representation Theory Contemporary

Mathematics vol Amer Math So c pp

J Lep owsky A generalization of the BernsteinGelfandGelfand resolution Jour

Algebra 49

R Penrose and W Rindler Spinors and Spacetime vol Cambridge University

Press

R Penrose and W Rindler Spinors and Spacetime vol Cambridge University

Press

J Slovak Invariant operators on conformal manifolds Lecture Notes University of

Vienna

DC Sp encer Overdetermined systems of linear partial dierential equations Bull

Amer Math So c 75

TY Thomas On conformal geometry Pro c Nat Acad Sci 12

TY Thomas Conformal tensors I Pro c Nat Acad Sci 18

DA Vogan Jr Representations of Real Reductive Lie Groups Birkhauser

V W unsch On conformally invariant dierential operators Math Nachr 129

Department of Pure Mathematics

University of Adelaide

South AUSTRALIA

Email meastwoospammathsadelaideeduau